LU decomposition

Example

We factor the following 2 × 2 matrix: $${\displaystyle \left[\begin{array}{cc}4& 3\\ 6& 3\end{array}\right]=\left[\begin{array}{cc}{\ell }_{11}& 0\\ {\ell }_{21}& {\ell }_{22}\end{array}\right]\left[\begin{array}{cc}{u}_{11}& u_{12}\\ 0& u_{22}\end{array}\right].}$$

One way to find the LU decomposition of this simple matrix would be to simply solve the linear equations by inspection. Expanding the matrix multiplication gives $${\displaystyle \{\begin{array}{ccccccccc}{\ell }_{11}\cdot u_{11}& & +& & 0\cdot 0& & =& & 4\\ {\ell }_{11}\cdot u_{12}& & +& & 0\cdot u_{22}& & =& & 3\\ {\ell }_{21}\cdot u_{11}& & +& & {\ell }_{22}\cdot 0& & =& & 6\\ {\ell }_{21}\cdot u_{12}& & +& & {\ell }_{22}\cdot u_{22}& & =& & 3\end{array}}$$

This system of equations is underdetermined. In this case any two nonzero elements of L and U matrices are parameters of the solution and can be set arbitrarily to any nonzero value. Therefore, to find the unique LU decomposition, it is necessary to put some restriction on L and U matrices. For example, we can conveniently require the lower triangular matrix L to be a unit triangular matrix, so that all the entries of its main diagonal are set to one. Then the system of equations has the following solution: $${\displaystyle \begin{array}{cc}{\ell }_{11}& ={\ell }_{22}=1\\ {\ell }_{21}& =1.5\\ u_{11}& =4\\ u_{12}& =3\\ u_{22}& =-1.5\end{array}}$$

Substituting these values into the LU decomposition above yields $${\displaystyle \left[\begin{array}{cc}4& 3\\ 6& 3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 1.5& 1\end{array}\right]\left[\begin{array}{cc}4& 3\\ 0& -1.5\end{array}\right].}$$

Existence and uniqueness

Square matrices

Any square matrix A admits LUP and PLU factorizations. If A is invertible, then it admits an LU (or LDU) factorization if and only if all its leading principal minors are nonzero (for example $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ does not admit an LU or LDU factorization). If A is a singular matrix of rank k, then it admits an LU factorization if the first k leading principal minors are nonzero, although the converse is not true.

If a square, invertible matrix has an LDU factorization (with all diagonal entries of L and U equal to 1), then the factorization is unique. In that case, the LU factorization is also unique if we require that the diagonal of either L or U consists of ones.

In general, any square matrix A_n×n could have one of the following:

1. a unique LU factorization (as mentioned above);
2. infinitely many LU factorizations if any of the first (n − 1) columns are linearly dependent;
3. no LU factorization if the first (n − 1) columns are linearly independent and at least one leading principal minor is zero.

In Case 3, one can approximate an LU factorization by changing a diagonal entry a_ij to a_ij ± ε to avoid a zero leading principal minor.

Symmetric positive-definite matrices

If A is a symmetric (or Hermitian, if A is complex) positive-definite matrix, we can arrange matters so that U is the conjugate transpose of L. That is, we can write A as $${\displaystyle A=L{L}^{\ast }.}$$

This decomposition is called the Cholesky decomposition. If A is positive definite, then the Cholesky decomposition exists and is unique. Furthermore, computing the Cholesky decomposition is more efficient and numerically more stable than computing some other LU decompositions.

General matrices

For a (not necessarily invertible) matrix over any field, the exact necessary and sufficient conditions under which it has an LU factorization are known. The conditions are expressed in terms of the ranks of certain submatrices. The Gaussian elimination algorithm for obtaining LU decomposition has also been extended to this most general case.

Algorithms

Closed formula

When an LDU factorization exists and is unique, there is a closed (explicit) formula for the elements of L, D, and U in terms of ratios of determinants of certain submatrices of the original matrix A. In particular, D₁ = A_1,1, and for i = 2, ... , n, D_i is the ratio of the i-th principal submatrix to the (i − 1)-th principal submatrix. Computation of the determinants is computationally expensive, so this explicit formula is not used in practice.

Using Gaussian elimination

The following algorithm is essentially a modified form of Gaussian elimination. Computing an LU decomposition using this algorithm requires ⁠2/3⁠n³ floating-point operations, ignoring lower-order terms. Partial pivoting adds only a quadratic term; this is not the case for full pivoting.

Generalized explanation

Notation

Given an N × N matrix ${\displaystyle A=(a_{i,j}{)}_{1\le i,j\le N}}$, define ${\displaystyle A^{(0)}}$ as the original, unmodified version of the matrix A. The parenthetical superscript (e.g., (0)) of the matrix A is the version of the matrix. The matrix A⁽ⁿ⁾ is the A matrix in which the elements below the main diagonal have already been eliminated to 0 through Gaussian elimination for the first n columns.

Below is a matrix to observe to help us remember the notation (where each ∗ represents any real number in the matrix):

${\displaystyle A^{(n-1)}=\left(\begin{array}{ccccccc}\ast & & & \cdots & & & \ast \\ 0& \ddots & & & & \\ & \ddots & \ast & & & \\ ⋮& & 0& a_{n,n}^{(n-1)}& & & ⋮\\ & & ⋮& a_{i,n}^{(n-1)}& \ast \\ & & & ⋮& ⋮& \ddots \\ 0& \cdots & 0& a_{i,n}^{(n-1)}& \ast & \cdots & \ast \end{array}\right)}$

Procedure

During this process, we gradually modify the matrix A using row operations until it becomes the matrix U in which all the elements below the main diagonal are equal to zero. During this, we will simultaneously create two separate matrices P and L, such that PA = LU.

We define the final permutation matrix P as the identity matrix which has all the same rows swapped in the same order as the A matrix while it transforms into the matrix U. For our matrix A⁽ⁿ⁻¹⁾, we may start by swapping rows to provide the desired conditions for the n-th column. For example, we might swap rows to perform partial pivoting, or we might do it to set the pivot element a_n,n on the main diagonal to a nonzero number so that we can complete the Gaussian elimination.

For our matrix A⁽ⁿ⁻¹⁾, we want to set every element below ${\displaystyle a_{n,n}^{(n-1)}}$ to zero (where ${\displaystyle a_{n,n}^{(n-1)}}$ is the element in the n-th column of the main diagonal). We will denote each element below ${\displaystyle a_{n,n}^{(n-1)}}$ as ${\displaystyle a_{i,n}^{(n-1)}}$ (where i = n+1, ... , N). To set ${\displaystyle a_{i,n}^{(n-1)}}$ to zero, we set row_i = row_i − (ℓ_i,n)⋅row_n for each row i. For this operation, ${\ell }_{i,n}:=a_{i,n}^{(n-1)}/a_{n,n}^{(n-1)}$. Once we have performed the row operations for the first N − 1 columns, we have obtained an upper triangular matrix A^(N−1) which is denoted by U.

We can also create the lower triangular matrix denoted as L, by directly inputting the previously calculated values of ℓ_i,n via the formula below.

$${\displaystyle L=\left(\begin{array}{cccc}1& 0& \cdots & 0\\ {\ell }_{2,1}& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ {\ell }_{N,1}& \cdots & {\ell }_{N,N-1}& 1\end{array}\right)}$$

Example

If we are given the matrix $${\displaystyle A=\left(\begin{array}{ccc}0& 5& \frac{22}{3}\\ 4& 2& 1\\ 2& 7& 9\end{array}\right),}$$ we will choose to implement partial pivoting and thus swap the first and second row so that our matrix A and the first iteration of our P matrix respectively become $${\displaystyle A^{(0)}=\left(\begin{array}{ccc}4& 2& 1\\ 0& 5& \frac{22}{3}\\ 2& 7& 9\end{array}\right),P^{(0)}=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right).}$$ Once we have swapped the rows, we can eliminate the elements below the main diagonal on the first column by performing $${\displaystyle \begin{array}{}ro{w}_2=ro{w}_2-({\ell }_{2,1})\cdot ro{w}_1\\ ro{w}_3=ro{w}_3-({\ell }_{3,1})\cdot ro{w}_1\end{array}}$$ such that, $${\displaystyle \begin{array}{}{\ell }_{2,1}=\frac{0}{4}=0\\ {\ell }_{3,1}=\frac{2}{4}=0.5\end{array}}$$ Once these rows have been subtracted, we have derived from A⁽¹⁾ the matrix $${\displaystyle A^{(1)}=\left(\begin{array}{ccc}4& 2& 1\\ 0& 5& \frac{22}{3}\\ 0& 6& 8.5\end{array}\right).}$$

Because we are implementing partial pivoting, we swap the second and third rows of our derived matrix and the current version of our P matrix respectively to obtain $${\displaystyle A^{(1)}=\left(\begin{array}{ccc}4& 2& 1\\ 0& 6& 8.5\\ 0& 5& \frac{22}{3}\end{array}\right),P^{(1)}=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right).}$$ Now, we eliminate the elements below the main diagonal on the second column by performing row₃ = row₃ − (ℓ_3,2)⋅row₂ such that ℓ_3,2 = ⁠5/6⁠. Because no nonzero elements exist below the main diagonal in our current iteration of A after this row subtraction, this row subtraction derives our final A matrix (denoted as U) and final P matrix: $${\displaystyle A^{(2)}=A^{(N-1)}=U=\left(\begin{array}{ccc}4& 2& 1\\ 0& 6& 8.5\\ 0& 0& 0.25\end{array}\right),P=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right).}$$ After also switching the corresponding rows, we obtain our final L matrix: $${\displaystyle L=\left(\begin{array}{ccc}1& 0& 0\\ {\ell }_{3,1}& 1& 0\\ {\ell }_{2,1}& {\ell }_{3,2}& 1\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0.5& 1& 0\\ 0& \frac{5}{6}& 1\end{array}\right)}$$

Now these matrices have a relation such that PA = LU.

Relations when no rows are swapped

If we did not swap rows at all during this process, we can perform the row operations simultaneously for each column n by setting ${\displaystyle A^{(n)}:=L_n^{-1}{A}^{(n-1)},}$ where ${\displaystyle L_n^{-1}}$ is the N × N identity matrix with its n-th column replaced by the transposed vector (0 ⋯ 0 1 −ℓ_n+1,n ⋯ −ℓ_N,n)^T.

In other words, the lower triangular matrix $${\displaystyle L_n^{-1}=\left(\begin{array}{cccccc}1& & & & & \\ & \ddots & & & & \\ & & 1& & & \\ & & -{\ell }_{n+1,n}& & & \\ & & ⋮& & \ddots & \\ & & -{\ell }_{N,n}& & & 1\end{array}\right).}$$

Performing all the row operations for the first N − 1 columns using the ${\displaystyle A^{(n)}:=L_n^{-1}{A}^{(n-1)}}$ formula is equivalent to finding the decomposition $${\displaystyle A=L_1{L}_1^{-1}{A}^{(0)}=L_1{A}^{(1)}=L_1{L}_2{L}_2^{-1}{A}^{(1)}=L_1{L}_2{A}^{(2)}=\cdots =L_1\cdots L_{N-1}{A}^{(N-1)}.}$$ Denote L = L₁ ⋯ L_N−1 so that A = LA^(N−1) = LU.

Now let's compute the sequence of L₁ ⋯ L_N−1. We know that L_i has the following formula: $${\displaystyle L_n=\left(\begin{array}{cccccc}1& & & & & \\ & \ddots & & & & \\ & & 1& & & \\ & & {\ell }_{n+1,n}& & & \\ & & ⋮& & \ddots & \\ & & {\ell }_{N,n}& & & 1\end{array}\right)}$$

If there are two lower triangular matrices with 1s in the main diagonal, and neither have a nonzero item below the main diagonal in the same column as the other, then we can include all nonzero items at their same location in the product of the two matrices. For example:

${\displaystyle \left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 77& 1& 0& 0& 0\\ 12& 0& 1& 0& 0\\ 63& 0& 0& 1& 0\\ 7& 0& 0& 0& 1\end{array}\right)\left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 22& 1& 0& 0\\ 0& 33& 0& 1& 0\\ 0& 44& 0& 0& 1\end{array}\right)=\left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 77& 1& 0& 0& 0\\ 12& 22& 1& 0& 0\\ 63& 33& 0& 1& 0\\ 7& 44& 0& 0& 1\end{array}\right)}$

Finally, multiply L₁ together and generate the fused matrix denoted as L (as previously mentioned). Using the matrix L, we obtain A = LU.

It is clear that in order for this algorithm to work, one needs to have ${\displaystyle a_{n,n}^{(n-1)}\ne 0}$ at each step (see the definition of ℓ_i,n. If this assumption fails at some point, one needs to interchange n-th row with another row below it before continuing. This is why an LU decomposition in general looks like P⁻¹A = LU.

LU Banachiewicz decomposition

Although Banachiewicz (1938) LU decomposition algorithm preceded the advent of programmed electronic computers, it was ready made for direct implementation into code as index swapping, transpose and column by column multiplication remain native built capabilities of the most programming languages and they are handled by compilers alone with little delay of the actual execution. The peculiar matrix notation used by Banachiewicz enabled him to multiply matrices column by column, a convenient feature for mechanical calculations as he could reveal consecutive factors by sliding a ruler to next rows of matrices. For human readers however, his equations are best transformed into standard matrix notation. To obtain from a full matrix A triangle matrices U and L calculations, start by copying top row and leftmost column of A respectively into corresponding positions of matrices U and L. The known unit diagonal elements of L are not stored nor used throughout the whole process. Next calculations continue for the subsequent rows and columns till the bottom right corner of A.

The figure illustrates calculations for 3rd row and column, assuming previous stages were already completed. Involved matrices are named above squares marking their content. Matrix products and subtractions are applied only to elements in the thick frame boxes. Green filled thin frame boxes indicate values already known, from previous stages. Blue boxes indicate places in U and L matrices for storing of results. Note that at each stage the result elements of L need to be divided by the corresponding pivot element on the main diagonal of U. This applies to the leftmost column of L too.

Note that after completion of 3rd stage the involved elements of matrix A are no longer used and neither those from the previous stages. This enables replacement of these elements with the result values of U and L, i.e. execution of LU decomposition in place, so that the whole A is replaced with U and L except for the unit diagonal of L. Banachiewicz LU algorithm is well suited for partial pivoting by choosing the absolute maximum pivot from the newly calculated row of U and subsequently swapping its columns so that it lands on the main diagonal. More details can be figured out from inspection of the enclosed Fortran90 code.

All partial pivoting LU algorithms cost roughly the same amount, of order $O\left(\frac{2}{3}{n}^3\right)$ operations, where n is the number of rows or columns of A.

LU Crout decomposition

Note that the decomposition obtained through this procedure is a Doolittle decomposition: the main diagonal of L is composed solely of ones. If one would proceed by removing elements above the main diagonal by adding multiples of the columns (instead of removing elements below the diagonal by adding multiples of the rows), we would obtain a Crout decomposition, where the main diagonal of U is of ones.

Another (equivalent) way of producing a Crout decomposition of a given matrix A is to obtain a Doolittle decomposition of the transpose of A. Indeed, if A^T = L₀U₀ is the LU-decomposition obtained through the algorithm presented in this section, then by taking L = U^T
₀ and U = L^T
₀, we have that A = LU is a Crout decomposition.

Randomized algorithm

It is possible to find a low rank approximation to an LU decomposition using a randomized algorithm. Given an input matrix A and a desired low rank k, the randomized LU returns permutation matrices P, Q and lower/upper trapezoidal matrices L, U of size m × k and k × n respectively, such that with high probability ‖PAQ − LU‖₂ ≤ Cσ_k+1, where C is a constant that depends on the parameters of the algorithm and σ_k+1 is the (k+1)-th singular value of the input matrix A.

Theoretical complexity

If two matrices of order n can be multiplied in time M(n), where M(n) ≥ n^a for some a > 2, then an LU-decomposition can be computed in time O(M(n)). This means, for example, that an O(n^2.376) algorithm exists based on the Coppersmith–Winograd algorithm. See also for fast matrix multiplication algorithms article for more details.

Sparse-matrix decomposition

Special algorithms have been developed for factorizing large sparse matrices. These algorithms attempt to find sparse factors L and U. Ideally, the cost of computation is determined by the number of nonzero entries, rather than by the size of the matrix.

These algorithms use the freedom to exchange rows and columns to minimize fill-in (entries that change from an initial zero to a nonzero value during the execution of an algorithm).

General treatment of orderings that minimize fill-in can be addressed using graph theory.

Applications

Solving linear equations

Given a system of linear equations in matrix form $${\displaystyle A\mathbf{x}=\mathbf{b},}$$

we want to solve the equation for x, given A and b. Suppose we have already obtained the LUP decomposition of A such that PA = LU, so LUx = Pb.

In this case the solution is done in two logical steps:

1. First, we solve the equation Ly = Pb for y.
2. Second, we solve the equation Ux = y for x.

In both cases we are dealing with triangular matrices (L and U), which can be solved directly by forward and backward substitution without using the Gaussian elimination process (however we do need this process or equivalent to compute the LU decomposition itself).

The above procedure can be repeatedly applied to solve the equation multiple times for different b. In this case it is faster (and more convenient) to do an LU decomposition of the matrix A once and then solve the triangular matrices for the different b, rather than using Gaussian elimination each time. The matrices L and U could be thought to have "encoded" the Gaussian elimination process.

The cost of solving a system of linear equations is approximately ⁠2/3⁠n³ floating-point operations if the matrix A has size n. This makes it twice as fast as algorithms based on QR decomposition, which costs about ⁠4/3⁠n³ floating-point operations when Householder reflections are used. For this reason, LU decomposition is usually preferred.

Inverting a matrix

When solving systems of equations, b is usually treated as a vector with a length equal to the height of matrix A. In matrix inversion however, instead of vector b, we have matrix B, where B is an n × p matrix, so that we are trying to find a matrix X (also a n × p matrix): $${\displaystyle AX=LUX=B.}$$

We can use the same algorithm presented earlier to solve for each column of matrix X. Now suppose that B is the identity matrix of size n, I_n. It follows that the result X must be the inverse of A.

Computing the determinant

Given the LUP decomposition A = P⁻¹LU of a square matrix A, the determinant of A can be computed straightforwardly as $${\displaystyle \begin{array}{cc}det(A)& =det\left(P^{-1}\right)det(L)det(U)\\ & =(-1{)}^S\left(\prod _{i=1}^n{l}_{ii}\right)\left(\prod _{i=1}^n{u}_{ii}\right).\end{array}}$$

The second equation follows from the fact that the determinant of a triangular matrix is simply the product of its diagonal entries, and that the determinant of a permutation matrix is equal to (−1)^S where S is the number of row exchanges in the decomposition.

In the case of LU decomposition with full pivoting, det(A) also equals the right-hand side of the above equation, if we let S be the total number of row and column exchanges.

The same method readily applies to LU decomposition by setting P equal to the identity matrix.

History

The LU decomposition is related to elimination of linear systems of equations, as e.g. described by Ralston. The solution of N linear equations in N unknowns by elimination was already known to ancient Chinese. Before Gauss many mathematicians in Eurasia were performing and perfecting it, yet as the method became relegated to school grade, few of them left any detailed descriptions. Thus the name Gaussian elimination is only a convenient abbreviation of a complex history.

Polish astronomer Tadeusz Banachiewicz introduced the LU decomposition in 1938. About Banachiewicz, Paul Dwyer has stated:

It appears that Gauss and Doolittle applied the method [of elimination] only to symmetric equations. More recent authors, for example, Aitken, Banachiewicz, Dwyer, and Crout ... have emphasized the use of the method, or variations of it, in connection with non-symmetric problems ... Banachiewicz ... saw the point ... that the basic problem is really one of matrix factorization, or "decomposition" as he called it.

— Paul Dwyer, Linear Computations (1951)

Banachiewicz was the first to consider elimination in terms of matrices and in this way formulated LU decomposition, as demonstrated by his graphic illustration. His calculations follow ordinary matrix ones, yet notation deviates in that he preferred to write one factor transposed, to be able to multiply them mechanically column by column, by sliding ruler down consecutive rows of both (using arithmometer). Combined with swapped order of indices his formulae in modern notation read $${\displaystyle \begin{array}{cc}\mathbf{x}\cdot I{A}^{\prime }=\mathbf{0}& \to A^{\prime }\mathbf{x}=0\equiv (A|\mathbf{l})\mathbf{x},\\ A=G\cdot H& \to A^T=G^{T}H,\end{array}}$$

where IA → A^T; x ≡ [x₁, ... , x_n, −1]; A′ refers to A extended with the last column; and the last component of x is −1. Matrix formulae to calculate rows and columns of LU factors by recursion are given in the remaining part of Banachiewicz's paper as Eq. (2.3) and (2.4) . This paper by Banachiewicz contains both derivation of LU and R^TR factors of respectively non-symmetric and symmetric matrices. They are sometimes confused as later publications tend to tie his name solely with the rediscovery of Cholesky decomposition. Banachiewicz himself can be excused of inaction as already next year he suffered from persecution by the Nazi German occupiers of Poland, spending three months in Sachsenhausen concentration camp, on release from which he carried himself from a train his collaborator and co-prisoner Antoni Wilk, who died of exhaustion a week later.

Code examples

Fortran90 code example

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By Banachiewicz (1938, hereafter B38) LU decomposition method calculates such</span> <span class="c" id="mwAtg">! triangles L=G^T, and U=H that square B=A^T=G^TH=LU. Partial pivoting</span> <span class="c" id="mwAtk">! by column permutation IP(:) is modern addition.</span> <span class="c" id="mwAto">! Within the code a, g correspond to B38 A^T and G^T, so that a=gh holds.</span> <span class="c" id="mwAts">!</span> <span class="c" id="mwAtw">! Normal use is for square A, however for RHS l already known</span> <span class="c" id="mwAt0">! input of (A|l)^T yields (L|y^T)^T where x in L^Tx=y is solution of Ax=l.</span> <span class="w" id="mwAt4"> </span><span class="kt" id="mwAt8">Real</span><span class="w" id="mwAuA"> </span><span class="p" id="mwAuE">(</span><span class="n" id="mwAuI">SP</span><span class="p" id="mwAuM">),</span><span class="w" id="mwAuQ"> </span><span class="k" id="mwAuU">Intent</span><span class="w" id="mwAuY"> </span><span class="p" id="mwAuc">(</span><span class="n" id="mwAug">In</span><span class="p" id="mwAuk">)</span><span class="w" id="mwAuo"> </span><span class="kd" id="mwAus">::</span><span class="w" id="mwAuw"> </span><span class="n" id="mwAu0">a</span><span class="w" id="mwAu4"> </span><span class="p" id="mwAu8">(:,</span><span class="w" id="mwAvA"> </span><span class="p" id="mwAvE">:)</span><span class="c" id="mwAvI">! input matrix A(m,n), n<=m</span> <span class="w" id="mwAvM"> </span><span class="kt" id="mwAvQ">Real</span><span class="w" id="mwAvU"> </span><span class="p" id="mwAvY">(</span><span class="n" id="mwAvc">SP</span><span class="p" id="mwAvg">),</span><span class="w" id="mwAvk"> </span><span class="k" id="mwAvo">Intent</span><span class="w" id="mwAvs"> </span><span class="p" id="mwAvw">(</span><span class="n" id="mwAv0">In</span><span class="p" id="mwAv4">)</span><span class="w" id="mwAv8"> </span><span class="kd" id="mwAwA">::</span><span class="w" id="mwAwE"> </span><span class="n" id="mwAwI">tol</span><span class="w" id="mwAwM"> </span><span class="c" id="mwAwQ">! tolerance for near zero pivot</span> <span class="w" id="mwAwU"> </span><span class="kt" id="mwAwY">Real</span><span class="w" id="mwAwc"> </span><span class="p" id="mwAwg">(</span><span class="n" id="mwAwk">SP</span><span class="p" id="mwAwo">),</span><span class="w" id="mwAws"> </span><span class="k" id="mwAww">Intent</span><span class="w" id="mwAw0"> </span><span class="p" id="mwAw4">(</span><span class="n" id="mwAw8">Out</span><span class="p" id="mwAxA">)</span><span class="w" id="mwAxE"> </span><span class="kd" id="mwAxI">::</span><span class="w" id="mwAxM"> </span><span class="n" id="mwAxQ">g</span><span class="w" id="mwAxU"> </span><span class="p" id="mwAxY">(</span><span class="n" id="mwAxc">size</span><span class="p" id="mwAxg">(</span><span class="n" id="mwAxk">a</span><span class="p" id="mwAxo">,</span><span class="nb" id="mwAxs">dim</span><span class="o" id="mwAxw">=</span><span class="mi" id="mwAx0">1</span><span class="p" id="mwAx4">),</span><span class="w" id="mwAx8"> </span><span class="n" id="mwAyA">size</span><span class="p" id="mwAyE">(</span><span class="n" id="mwAyI">a</span><span class="p" id="mwAyM">,</span><span class="nb" id="mwAyQ">dim</span><span class="o" id="mwAyU">=</span><span class="mi" id="mwAyY">2</span><span class="p" id="mwAyc">))</span><span class="w" id="mwAyg"> </span><span class="c" id="mwAyk">! L(m,n)</span> <span class="w" id="mwAyo"> </span><span class="kt" id="mwAys">Real</span><span class="w" id="mwAyw"> </span><span class="p" id="mwAy0">(</span><span class="n" id="mwAy4">SP</span><span class="p" id="mwAy8">),</span><span class="w" id="mwAzA"> </span><span class="k" id="mwAzE">Intent</span><span class="w" id="mwAzI"> </span><span class="p" id="mwAzM">(</span><span class="n" id="mwAzQ">Out</span><span class="p" id="mwAzU">)</span><span class="w" id="mwAzY"> </span><span class="kd" id="mwAzc">::</span><span class="w" id="mwAzg"> </span><span class="n" id="mwAzk">h</span><span class="w" id="mwAzo"> </span><span class="p" id="mwAzs">(</span><span class="n" id="mwAzw">size</span><span class="p" id="mwAz0">(</span><span class="n" id="mwAz4">a</span><span class="p" id="mwAz8">,</span><span class="nb" id="mwA0A">dim</span><span class="o" id="mwA0E">=</span><span class="mi" id="mwA0I">2</span><span class="p" id="mwA0M">),</span><span class="w" id="mwA0Q"> </span><span class="n" id="mwA0U">size</span><span class="p" id="mwA0Y">(</span><span class="n" id="mwA0c">a</span><span class="p" id="mwA0g">,</span><span class="nb" id="mwA0k">dim</span><span class="o" id="mwA0o">=</span><span class="mi" id="mwA0s">2</span><span class="p" id="mwA0w">))</span><span class="w" id="mwA00"> </span><span class="c" id="mwA04">! U(n,n)</span> <span class="w" id="mwA08"> </span><span class="c" id="mwA1A">! note U columns are permuted</span> <span class="w" id="mwA1E"> </span><span class="kt" id="mwA1I">Real</span><span class="w" id="mwA1M"> </span><span class="p" id="mwA1Q">(</span><span class="n" id="mwA1U">SP</span><span class="p" id="mwA1Y">),</span><span class="w" id="mwA1c"> </span><span class="k" id="mwA1g">Intent</span><span class="w" id="mwA1k"> </span><span class="p" id="mwA1o">(</span><span class="n" id="mwA1s">Out</span><span class="p" id="mwA1w">)</span><span class="w" id="mwA10"> </span><span class="kd" id="mwA14">::</span><span class="w" id="mwA18"> </span><span class="n" id="mwA2A">condinv</span><span class="w" id="mwA2E"> </span><span class="c" id="mwA2I">! 1/cond(A), 0 for singular A</span> <span class="w" id="mwA2M"> </span><span class="kt" id="mwA2Q">Real</span><span class="w" id="mwA2U"> </span><span class="p" id="mwA2Y">(</span><span class="n" id="mwA2c">SP</span><span class="p" id="mwA2g">),</span><span class="w" id="mwA2k"> </span><span class="k" id="mwA2o">Intent</span><span class="w" id="mwA2s"> </span><span class="p" id="mwA2w">(</span><span class="n" id="mwA20">Out</span><span class="p" id="mwA24">)</span><span class="w" id="mwA28"> </span><span class="kd" id="mwA3A">::</span><span class="w" id="mwA3E"> </span><span class="n" id="mwA3I">detnth</span><span class="w" id="mwA3M"> </span><span class="c" id="mwA3Q">! sign*Abs(det(A))**(1/n)</span> <span class="w" id="mwA3U"> </span><span class="kt" id="mwA3Y">Integer</span><span class="p" id="mwA3c">,</span><span class="w" id="mwA3g"> </span><span class="k" id="mwA3k">Intent</span><span class="w" id="mwA3o"> </span><span class="p" id="mwA3s">(</span><span class="n" id="mwA3w">Out</span><span class="p" id="mwA30">)</span><span class="w" id="mwA34"> </span><span class="kd" id="mwA38">::</span><span class="w" id="mwA4A"> </span><span class="n" id="mwA4E">ip</span><span class="w" id="mwA4I"> </span><span class="p" id="mwA4M">(</span><span class="n" id="mwA4Q">size</span><span class="p" id="mwA4U">(</span><span class="n" id="mwA4Y">a</span><span class="p" id="mwA4c">,</span><span class="w" id="mwA4g"> </span><span class="nb" id="mwA4k">dim</span><span class="o" id="mwA4o">=</span><span class="mi" id="mwA4s">2</span><span class="p" id="mwA4w">))</span><span class="w" id="mwA40"> </span><span class="c" id="mwA44">! columns permutation</span> <span class="c" id="mwA48">!</span> <span class="w" id="mwA5A"> </span><span class="kt" id="mwA5E">Integer</span><span class="w" id="mwA5I"> </span><span class="kd" id="mwA5M">::</span><span class="w" id="mwA5Q"> </span><span class="n" id="mwA5U">k</span><span class="p" id="mwA5Y">,</span><span class="w" id="mwA5c"> </span><span class="n" id="mwA5g">n</span><span class="p" id="mwA5k">,</span><span class="w" id="mwA5o"> </span><span class="n" id="mwA5s">j</span><span class="p" id="mwA5w">,</span><span class="w" id="mwA50"> </span><span class="n" id="mwA54">l</span><span class="p" id="mwA58">,</span><span class="w" id="mwA6A"> </span><span class="n" id="mwA6E">isig</span> <span class="w" id="mwA6I"> </span><span class="kt" id="mwA6M">Real</span><span class="w" id="mwA6Q"> </span><span class="p" id="mwA6U">(</span><span class="n" id="mwA6Y">SP</span><span class="p" id="mwA6c">)</span><span class="w" id="mwA6g"> </span><span class="kd" id="mwA6k">::</span><span class="w" id="mwA6o"> </span><span class="n" id="mwA6s">tol0</span><span class="p" id="mwA6w">,</span><span class="w" id="mwA60"> </span><span class="n" id="mwA64">pivmax</span><span class="p" id="mwA68">,</span><span class="w" id="mwA7A"> </span><span class="n" id="mwA7E">pivmin</span><span class="p" id="mwA7I">,</span><span class="w" id="mwA7M"> </span><span class="n" id="mwA7Q">piv</span> <span class="c" id="mwA7U">!</span> <span class="w" id="mwA7Y"> </span><span class="n" id="mwA7c">n</span><span class="w" id="mwA7g"> </span><span class="o" id="mwA7k">=</span><span class="w" id="mwA7o"> </span><span class="n" id="mwA7s">size</span><span class="w" id="mwA7w"> </span><span class="p" id="mwA70">(</span><span class="n" id="mwA74">a</span><span class="p" id="mwA78">,</span><span class="w" id="mwA8A"> </span><span class="nb" id="mwA8E">dim</span><span class="o" id="mwA8I">=</span><span class="mi" id="mwA8M">2</span><span class="p" id="mwA8Q">)</span> <span class="w" id="mwA8U"> </span><span class="n" id="mwA8Y">tol0</span><span class="w" id="mwA8c"> </span><span class="o" id="mwA8g">=</span><span class="w" id="mwA8k"> </span><span class="nb" id="mwA8o">Max</span><span class="w" id="mwA8s"> </span><span class="p" id="mwA8w">(</span><span class="n" id="mwA80">tol</span><span class="p" id="mwA84">,</span><span class="w" id="mwA88"> </span><span class="mf" id="mwA9A">3._SP</span><span class="o" id="mwA9E">*</span><span class="nb" id="mwA9I">epsilon</span><span class="p" id="mwA9M">(</span><span class="n" id="mwA9Q">tol0</span><span class="p" id="mwA9U">))</span><span class="c" id="mwA9Y">! use default for tol=0</span> <span class="c" id="mwA9c">!</span> <span class="c" id="mwA9g">! Rectangular A and G are permitted under condition:</span> <span class="w" id="mwA9k"> </span><span class="k" id="mwA9o">If</span><span class="w" id="mwA9s"> </span><span class="p" id="mwA9w">(</span><span class="n" id="mwA90">n</span><span class="w" id="mwA94"> </span><span class="o" id="mwA98">></span><span class="w" id="mwA-A"> </span><span class="n" id="mwA-E">size</span><span class="p" id="mwA-I">(</span><span class="n" id="mwA-M">a</span><span class="p" id="mwA-Q">,</span><span class="w" id="mwA-U"> </span><span class="nb" id="mwA-Y">dim</span><span class="o" id="mwA-c">=</span><span class="mi" id="mwA-g">1</span><span class="p" id="mwA-k">)</span><span class="w" id="mwA-o"> </span><span class="p" id="mwA-s">.</span><span class="nb" id="mwA-w">Or</span><span class="p" id="mwA-0">.</span><span class="w" id="mwA-4"> </span><span class="n" id="mwA-8">n</span><span class="w" id="mwA_A"> </span><span class="o" id="mwA_E"><</span><span class="w" id="mwA_I"> </span><span class="mi" id="mwA_M">1</span><span class="p" id="mwA_Q">)</span><span class="w" id="mwA_U"> </span><span class="k" id="mwA_Y">Stop </span><span class="mi" id="mwA_c">91</span> <span class="w" id="mwA_g"> </span><span class="k" id="mwA_k">Forall</span><span class="w" id="mwA_o"> </span><span class="p" id="mwA_s">(</span><span class="n" id="mwA_w">k</span><span class="o" id="mwA_0">=</span><span class="mi" id="mwA_4">1</span><span class="p" id="mwA_8">:</span><span class="n" id="mwBAA">n</span><span class="p" id="mwBAE">)</span><span class="w" id="mwBAI"> </span><span class="n" id="mwBAM">ip</span><span class="w" id="mwBAQ"> </span><span class="p" id="mwBAU">(</span><span class="n" id="mwBAY">k</span><span class="p" id="mwBAc">)</span><span class="w" id="mwBAg"> </span><span class="o" id="mwBAk">=</span><span class="w" id="mwBAo"> </span><span class="n" id="mwBAs">k</span> <span class="w" id="mwBAw"> </span><span class="n" id="mwBA0">h</span><span class="w" id="mwBA4"> </span><span class="o" id="mwBA8">=</span><span class="w" id="mwBBA"> </span><span class="mf" id="mwBBE">0._SP</span> <span class="w" id="mwBBI"> </span><span class="n" id="mwBBM">g</span><span class="w" id="mwBBQ"> </span><span class="o" id="mwBBU">=</span><span class="w" id="mwBBY"> </span><span class="mf" id="mwBBc">0._SP</span> <span class="w" id="mwBBg"> </span><span class="n" id="mwBBk">isig</span><span class="w" id="mwBBo"> </span><span class="o" id="mwBBs">=</span><span class="w" id="mwBBw"> </span><span class="mi" id="mwBB0">1</span> <span class="w" id="mwBB4"> </span><span class="n" id="mwBB8">detnth</span><span class="w" id="mwBCA"> </span><span class="o" id="mwBCE">=</span><span class="w" id="mwBCI"> </span><span class="mf" id="mwBCM">0._SP</span> <span class="w" id="mwBCQ"> </span><span class="n" id="mwBCU">pivmax</span><span class="w" id="mwBCY"> </span><span class="o" id="mwBCc">=</span><span class="w" id="mwBCg"> </span><span class="nb" id="mwBCk">Maxval</span><span class="w" id="mwBCo"> </span><span class="p" id="mwBCs">(</span><span class="nb" id="mwBCw">Abs</span><span class="w" id="mwBC0"> </span><span class="p" id="mwBC4">(</span><span class="n" id="mwBC8">a</span><span class="p" id="mwBDA">(</span><span class="mi" id="mwBDE">1</span><span class="p" id="mwBDI">,</span><span class="w" id="mwBDM"> </span><span class="p" id="mwBDQ">:)))</span> <span class="w" id="mwBDU"> </span><span class="n" id="mwBDY">pivmin</span><span class="w" id="mwBDc"> </span><span class="o" id="mwBDg">=</span><span class="w" id="mwBDk"> </span><span class="n" id="mwBDo">pivmax</span> <span class="c" id="mwBDs">!</span> <span class="w" id="mwBDw"> </span><span class="k" id="mwBD0">Do </span><span class="n" id="mwBD4">k</span><span class="w" id="mwBD8"> </span><span class="o" id="mwBEA">=</span><span class="w" id="mwBEE"> </span><span class="mi" id="mwBEI">1</span><span class="p" id="mwBEM">,</span><span class="w" id="mwBEQ"> </span><span class="n" id="mwBEU">n</span> <span class="c" id="mwBEY">! Banachiewicz (1938) Eq. (2.3)</span> <span class="w" id="mwBEc"> </span><span class="n" id="mwBEg">h</span><span class="p" id="mwBEk">(</span><span class="n" id="mwBEo">k</span><span class="p" id="mwBEs">,</span><span class="w" id="mwBEw"> </span><span class="n" id="mwBE0">ip</span><span class="p" id="mwBE4">(</span><span class="n" id="mwBE8">k</span><span class="p" id="mwBFA">:))</span><span class="w" id="mwBFE"> </span><span class="o" id="mwBFI">=</span><span class="w" id="mwBFM"> </span><span class="n" id="mwBFQ">a</span><span class="p" id="mwBFU">(</span><span class="n" id="mwBFY">k</span><span class="p" id="mwBFc">,</span><span class="w" id="mwBFg"> </span><span class="n" id="mwBFk">ip</span><span class="p" id="mwBFo">(</span><span class="n" id="mwBFs">k</span><span class="p" id="mwBFw">:))</span><span class="w" id="mwBF0"> </span><span class="o" id="mwBF4">-</span><span class="w" id="mwBF8"> </span><span class="nb" id="mwBGA">Matmul</span><span class="p" id="mwBGE">(</span><span class="n" id="mwBGI">g</span><span class="p" id="mwBGM">(</span><span class="n" id="mwBGQ">k</span><span class="p" id="mwBGU">,</span><span class="w" id="mwBGY"> </span><span class="p" id="mwBGc">:</span><span class="n" id="mwBGg">k</span><span class="o" id="mwBGk">-</span><span class="mi" id="mwBGo">1</span><span class="p" id="mwBGs">),</span><span class="w" id="mwBGw"> </span><span class="n" id="mwBG0">h</span><span class="p" id="mwBG4">(:</span><span class="n" id="mwBG8">k</span><span class="o" id="mwBHA">-</span><span class="mi" id="mwBHE">1</span><span class="p" id="mwBHI">,</span><span class="w" id="mwBHM"> </span><span class="n" id="mwBHQ">ip</span><span class="p" id="mwBHU">(</span><span class="n" id="mwBHY">k</span><span class="p" id="mwBHc">:)))</span> <span class="c" id="mwBHg">!</span> <span class="c" id="mwBHk">! Find row pivot</span> <span class="w" id="mwBHo"> </span><span class="n" id="mwBHs">j</span><span class="w" id="mwBHw"> </span><span class="o" id="mwBH0">=</span><span class="w" id="mwBH4"> </span><span class="p" id="mwBH8">(</span><span class="nb" id="mwBIA">Maxloc</span><span class="p" id="mwBIE">(</span><span class="nb" id="mwBII">Abs</span><span class="p" id="mwBIM">(</span><span class="n" id="mwBIQ">h</span><span class="p" id="mwBIU">(</span><span class="n" id="mwBIY">k</span><span class="p" id="mwBIc">,</span><span class="w" id="mwBIg"> </span><span class="n" id="mwBIk">ip</span><span class="p" id="mwBIo">(</span><span class="n" id="mwBIs">k</span><span class="p" id="mwBIw">:))),</span><span class="w" id="mwBI0"> </span><span class="nb" id="mwBI4">dim</span><span class="o" id="mwBI8">=</span><span class="mi" id="mwBJA">1</span><span class="p" id="mwBJE">)</span><span class="w" id="mwBJI"> </span><span class="o" id="mwBJM">+</span><span class="w" id="mwBJQ"> </span><span class="n" id="mwBJU">k</span><span class="o" id="mwBJY">-</span><span class="mi" id="mwBJc">1</span><span class="p" id="mwBJg">)</span> <span class="w" id="mwBJk"> </span><span class="k" id="mwBJo">If</span><span class="w" id="mwBJs"> </span><span class="p" id="mwBJw">(</span><span class="n" id="mwBJ0">j</span><span class="w" id="mwBJ4"> </span><span class="o" id="mwBJ8">/=</span><span class="w" id="mwBKA"> </span><span class="n" id="mwBKE">k</span><span class="p" id="mwBKI">)</span><span class="w" id="mwBKM"> </span><span class="k" id="mwBKQ">Then</span><span class="w" id="mwBKU"> </span><span class="c" id="mwBKY">! Swap columns j and k</span> <span class="w" id="mwBKc"> </span><span class="n" id="mwBKg">isig</span><span class="w" id="mwBKk"> </span><span class="o" id="mwBKo">=</span><span class="w" id="mwBKs"> </span><span class="o" id="mwBKw">-</span><span class="w" id="mwBK0"> </span><span class="n" id="mwBK4">isig</span><span class="w" id="mwBK8"> </span><span class="c" id="mwBLA">! Change Det(A) sign because of permutation</span> <span class="w" id="mwBLE"> </span><span class="n" id="mwBLI">l</span><span class="w" id="mwBLM"> </span><span class="o" id="mwBLQ">=</span><span class="w" id="mwBLU"> </span><span class="n" id="mwBLY">ip</span><span class="w" id="mwBLc"> </span><span class="p" id="mwBLg">(</span><span class="n" id="mwBLk">k</span><span class="p" id="mwBLo">)</span> <span class="w" id="mwBLs"> </span><span class="n" id="mwBLw">ip</span><span class="w" id="mwBL0"> </span><span class="p" id="mwBL4">(</span><span class="n" id="mwBL8">k</span><span class="p" id="mwBMA">)</span><span class="w" id="mwBME"> </span><span class="o" id="mwBMI">=</span><span class="w" id="mwBMM"> </span><span class="n" id="mwBMQ">ip</span><span class="w" id="mwBMU"> </span><span class="p" id="mwBMY">(</span><span class="n" id="mwBMc">j</span><span class="p" id="mwBMg">)</span> <span class="w" id="mwBMk"> </span><span class="n" id="mwBMo">ip</span><span class="w" id="mwBMs"> </span><span class="p" id="mwBMw">(</span><span class="n" id="mwBM0">j</span><span class="p" id="mwBM4">)</span><span class="w" id="mwBM8"> </span><span class="o" id="mwBNA">=</span><span class="w" id="mwBNE"> </span><span class="n" id="mwBNI">l</span> <span class="w" id="mwBNM"> </span><span class="k" id="mwBNQ">End If</span> <span class="k" id="mwBNU"> </span><span class="n" id="mwBNY">piv</span><span class="w" id="mwBNc"> </span><span class="o" id="mwBNg">=</span><span class="w" id="mwBNk"> </span><span class="nb" id="mwBNo">Abs</span><span class="w" id="mwBNs"> </span><span class="p" id="mwBNw">(</span><span class="n" id="mwBN0">h</span><span class="p" id="mwBN4">(</span><span class="n" id="mwBN8">k</span><span class="p" id="mwBOA">,</span><span class="w" id="mwBOE"> </span><span class="n" id="mwBOI">ip</span><span class="p" id="mwBOM">(</span><span class="n" id="mwBOQ">k</span><span class="p" id="mwBOU">)))</span> <span class="w" id="mwBOY"> </span><span class="n" id="mwBOc">pivmax</span><span class="w" id="mwBOg"> </span><span class="o" id="mwBOk">=</span><span class="w" id="mwBOo"> </span><span class="nb" id="mwBOs">Max</span><span class="w" id="mwBOw"> </span><span class="p" id="mwBO0">(</span><span class="n" id="mwBO4">piv</span><span class="p" id="mwBO8">,</span><span class="w" id="mwBPA"> </span><span class="n" id="mwBPE">pivmax</span><span class="p" id="mwBPI">)</span><span class="w" id="mwBPM"> </span><span class="c" id="mwBPQ">! Adjust condinv</span> <span class="w" id="mwBPU"> </span><span class="n" id="mwBPY">pivmin</span><span class="w" id="mwBPc"> </span><span class="o" id="mwBPg">=</span><span class="w" id="mwBPk"> </span><span class="nb" id="mwBPo">Min</span><span class="w" id="mwBPs"> </span><span class="p" id="mwBPw">(</span><span class="n" id="mwBP0">piv</span><span class="p" id="mwBP4">,</span><span class="w" id="mwBP8"> </span><span class="n" id="mwBQA">pivmin</span><span class="p" id="mwBQE">)</span> <span class="w" id="mwBQI"> </span><span class="k" id="mwBQM">If</span><span class="w" id="mwBQQ"> </span><span class="p" id="mwBQU">(</span><span class="n" id="mwBQY">piv</span><span class="w" id="mwBQc"> </span><span class="o" id="mwBQg"><</span><span class="w" id="mwBQk"> </span><span class="n" id="mwBQo">tol0</span><span class="p" id="mwBQs">)</span><span class="w" id="mwBQw"> </span><span class="k" id="mwBQ0">Then</span><span class="w" id="mwBQ4"> </span><span class="c" id="mwBQ8">! singular matrix</span> <span class="w" id="mwBRA"> </span><span class="n" id="mwBRE">isig</span><span class="w" id="mwBRI"> </span><span class="o" id="mwBRM">=</span><span class="w" id="mwBRQ"> </span><span class="mi" id="mwBRU">0</span> <span class="w" id="mwBRY"> </span><span class="n" id="mwBRc">pivmax</span><span class="w" id="mwBRg"> </span><span class="o" id="mwBRk">=</span><span class="w" id="mwBRo"> </span><span class="mf" id="mwBRs">1._SP</span> <span class="w" id="mwBRw"> </span><span class="k" id="mwBR0">Exit</span> <span class="k" id="mwBR4"> Else</span><span class="w" id="mwBR8"> </span><span class="c" id="mwBSA">! Account for pivot contribution to Det(A) sign and value</span> <span class="w" id="mwBSE"> </span><span class="k" id="mwBSI">If</span><span class="w" id="mwBSM"> </span><span class="p" id="mwBSQ">(</span><span class="n" id="mwBSU">h</span><span class="p" id="mwBSY">(</span><span class="n" id="mwBSc">k</span><span class="p" id="mwBSg">,</span><span class="w" id="mwBSk"> </span><span class="n" id="mwBSo">ip</span><span class="p" id="mwBSs">(</span><span class="n" id="mwBSw">k</span><span class="p" id="mwBS0">))</span><span class="w" id="mwBS4"> </span><span class="o" id="mwBS8"><</span><span class="w" id="mwBTA"> </span><span class="mf" id="mwBTE">0._SP</span><span class="p" id="mwBTI">)</span><span class="w" id="mwBTM"> </span><span class="n" id="mwBTQ">isig</span><span class="w" id="mwBTU"> </span><span class="o" id="mwBTY">=</span><span class="w" id="mwBTc"> </span><span class="o" id="mwBTg">-</span><span class="w" id="mwBTk"> </span><span class="n" id="mwBTo">isig</span> <span class="w" id="mwBTs"> </span><span class="n" id="mwBTw">detnth</span><span class="w" id="mwBT0"> </span><span class="o" id="mwBT4">=</span><span class="w" id="mwBT8"> </span><span class="n" id="mwBUA">detnth</span><span class="w" id="mwBUE"> </span><span class="o" id="mwBUI">+</span><span class="w" id="mwBUM"> </span><span class="nb" id="mwBUQ">Log</span><span class="w" id="mwBUU"> </span><span class="p" id="mwBUY">(</span><span class="n" id="mwBUc">piv</span><span class="p" id="mwBUg">)</span> <span class="w" id="mwBUk"> </span><span class="k" id="mwBUo">End If</span> <span class="c" id="mwBUs">!</span> <span class="c" id="mwBUw">! Transposed Banachiewicz (1938) Eq. (2.4)</span> <span class="w" id="mwBU0"> </span><span class="n" id="mwBU4">g</span><span class="w" id="mwBU8"> </span><span class="p" id="mwBVA">(</span><span class="n" id="mwBVE">k</span><span class="o" id="mwBVI">+</span><span class="mi" id="mwBVM">1</span><span class="p" id="mwBVQ">:,</span><span class="w" id="mwBVU"> </span><span class="n" id="mwBVY">k</span><span class="p" id="mwBVc">)</span><span class="w" id="mwBVg"> </span><span class="o" id="mwBVk">=</span><span class="w" id="mwBVo"> </span><span class="p" id="mwBVs">(</span><span class="n" id="mwBVw">a</span><span class="p" id="mwBV0">(</span><span class="n" id="mwBV4">k</span><span class="o" id="mwBV8">+</span><span class="mi" id="mwBWA">1</span><span class="p" id="mwBWE">:,</span><span class="w" id="mwBWI"> </span><span class="n" id="mwBWM">ip</span><span class="p" id="mwBWQ">(</span><span class="n" id="mwBWU">k</span><span class="p" id="mwBWY">))</span><span class="w" id="mwBWc"> </span><span class="o" id="mwBWg">-</span><span class="w" id="mwBWk"> </span><span class="p" id="mwBWo">&</span> <span class="w" id="mwBWs"> </span><span class="nb" id="mwBWw">Matmul</span><span class="p" id="mwBW0">(</span><span class="n" id="mwBW4">g</span><span class="p" id="mwBW8">(</span><span class="n" id="mwBXA">k</span><span class="o" id="mwBXE">+</span><span class="mi" id="mwBXI">1</span><span class="p" id="mwBXM">:,</span><span class="w" id="mwBXQ"> </span><span class="p" id="mwBXU">:</span><span class="n" id="mwBXY">k</span><span class="o" id="mwBXc">-</span><span class="mi" id="mwBXg">1</span><span class="p" id="mwBXk">),</span><span class="w" id="mwBXo"> </span><span class="n" id="mwBXs">h</span><span class="p" id="mwBXw">(:</span><span class="n" id="mwBX0">k</span><span class="o" id="mwBX4">-</span><span class="mi" id="mwBX8">1</span><span class="p" id="mwBYA">,</span><span class="w" id="mwBYE"> </span><span class="n" id="mwBYI">ip</span><span class="p" id="mwBYM">(</span><span class="n" id="mwBYQ">k</span><span class="p" id="mwBYU">))))</span><span class="w" id="mwBYY"> </span><span class="o" id="mwBYc">/</span><span class="w" id="mwBYg"> </span><span class="n" id="mwBYk">h</span><span class="w" id="mwBYo"> </span><span class="p" id="mwBYs">(</span><span class="n" id="mwBYw">k</span><span class="p" id="mwBY0">,</span><span class="w" id="mwBY4"> </span><span class="n" id="mwBY8">ip</span><span class="p" id="mwBZA">(</span><span class="n" id="mwBZE">k</span><span class="p" id="mwBZI">))</span> <span class="w" id="mwBZM"> </span><span class="n" id="mwBZQ">g</span><span class="w" id="mwBZU"> </span><span class="p" id="mwBZY">(</span><span class="n" id="mwBZc">k</span><span class="p" id="mwBZg">,</span><span class="w" id="mwBZk"> </span><span class="n" id="mwBZo">k</span><span class="p" id="mwBZs">)</span><span class="w" id="mwBZw"> </span><span class="o" id="mwBZ0">=</span><span class="w" id="mwBZ4"> </span><span class="mf" id="mwBZ8">1._SP</span> <span class="w" id="mwBaA"> </span><span class="k" id="mwBaE">End Do</span> <span class="c" id="mwBaI">!</span> <span class="w" id="mwBaM"> </span><span class="n" id="mwBaQ">detnth</span><span class="w" id="mwBaU"> </span><span class="o" id="mwBaY">=</span><span class="w" id="mwBac"> </span><span class="n" id="mwBag">isig</span><span class="w" id="mwBak"> </span><span class="o" id="mwBao">*</span><span class="w" id="mwBas"> </span><span class="nb" id="mwBaw">Exp</span><span class="w" id="mwBa0"> </span><span class="p" id="mwBa4">(</span><span class="n" id="mwBa8">detnth</span><span class="o" id="mwBbA">/</span><span class="n" id="mwBbE">n</span><span class="p" id="mwBbI">)</span> <span class="w" id="mwBbM"> </span><span class="n" id="mwBbQ">condinv</span><span class="w" id="mwBbU"> </span><span class="o" id="mwBbY">=</span><span class="w" id="mwBbc"> </span><span class="nb" id="mwBbg">Abs</span><span class="w" id="mwBbk"> </span><span class="p" id="mwBbo">(</span><span class="n" id="mwBbs">isig</span><span class="p" id="mwBbw">)</span><span class="w" id="mwBb0"> </span><span class="o" id="mwBb4">*</span><span class="w" id="mwBb8"> </span><span class="n" id="mwBcA">pivmin</span><span class="w" id="mwBcE"> </span><span class="o" id="mwBcI">/</span><span class="w" id="mwBcM"> </span><span class="n" id="mwBcQ">pivmax</span> <span class="c" id="mwBcU">! Test for square A(n,n) by uncommenting below</span> <span class="c" id="mwBcY">! Print *, '|AQ-LU| ',Maxval (Abs(a(:,ip(:))-Matmul(g, h(:,ip(:)))))</span> <span class="w" id="mwBcc"> </span><span class="k" id="mwBcg">End Subroutine </span><span class="n" id="mwBck">luban</span> <span class="w" id="mwBco"> </span> <span class="w" id="mwBcs"> </span><span class="k" id="mwBcw">Subroutine </span><span class="n" id="mwBc0">lusolve</span><span class="p" id="mwBc4">(</span><span class="n" id="mwBc8">l</span><span class="p" id="mwBdA">,</span><span class="n" id="mwBdE">u</span><span class="p" id="mwBdI">,</span><span class="n" id="mwBdM">ip</span><span class="p" id="mwBdQ">,</span><span class="n" id="mwBdU">x</span><span class="p" id="mwBdY">)</span> <span class="c" id="mwBdc">! Solves Ax=b system using triangle factors LU=A </span> <span class="w" id="mwBdg"> </span><span class="kt" id="mwBdk">Real</span><span class="w" id="mwBdo"> </span><span class="p" id="mwBds">(</span><span class="n" id="mwBdw">SP</span><span class="p" id="mwBd0">),</span><span class="w" id="mwBd4"> </span><span class="k" id="mwBd8">Intent</span><span class="w" id="mwBeA"> </span><span class="p" id="mwBeE">(</span><span class="n" id="mwBeI">In</span><span class="p" id="mwBeM">)</span><span class="w" id="mwBeQ"> </span><span class="kd" id="mwBeU">::</span><span class="w" id="mwBeY"> </span><span class="n" id="mwBec">l</span><span class="w" id="mwBeg"> </span><span class="p" id="mwBek">(:,</span><span class="w" id="mwBeo"> </span><span class="p" id="mwBes">:)</span><span class="w" id="mwBew"> </span><span class="c" id="mwBe0">! lower triangle matrix L(n,n)</span> <span class="w" id="mwBe4"> </span><span class="kt" id="mwBe8">Real</span><span class="w" id="mwBfA"> </span><span class="p" id="mwBfE">(</span><span class="n" id="mwBfI">SP</span><span class="p" id="mwBfM">),</span><span class="w" id="mwBfQ"> </span><span class="k" id="mwBfU">Intent</span><span class="w" id="mwBfY"> </span><span class="p" id="mwBfc">(</span><span class="n" id="mwBfg">In</span><span class="p" id="mwBfk">)</span><span class="w" id="mwBfo"> </span><span class="kd" id="mwBfs">::</span><span class="w" id="mwBfw"> </span><span class="n" id="mwBf0">u</span><span class="w" id="mwBf4"> </span><span class="p" id="mwBf8">(:,</span><span class="w" id="mwBgA"> </span><span class="p" id="mwBgE">:)</span><span class="w" id="mwBgI"> </span><span class="c" id="mwBgM">! upper triangle matrix U(n,n)</span> <span class="w" id="mwBgQ"> </span><span class="kt" id="mwBgU">Integer</span><span class="p" id="mwBgY">,</span><span class="w" id="mwBgc"> </span><span class="k" id="mwBgg">Intent</span><span class="w" id="mwBgk"> </span><span class="p" id="mwBgo">(</span><span class="n" id="mwBgs">In</span><span class="p" id="mwBgw">)</span><span class="w" id="mwBg0"> </span><span class="kd" id="mwBg4">::</span><span class="w" id="mwBg8"> </span><span class="n" id="mwBhA">ip</span><span class="w" id="mwBhE"> </span><span class="p" id="mwBhI">(:)</span><span class="w" id="mwBhM"> </span><span class="c" id="mwBhQ">! columns permutation IP(n)</span> <span class="w" id="mwBhU"> </span><span class="kt" id="mwBhY">Real</span><span class="w" id="mwBhc"> </span><span class="p" id="mwBhg">(</span><span class="n" id="mwBhk">SP</span><span class="p" id="mwBho">),</span><span class="w" id="mwBhs"> </span><span class="k" id="mwBhw">Intent</span><span class="w" id="mwBh0"> </span><span class="p" id="mwBh4">(</span><span class="n" id="mwBh8">InOut</span><span class="p" id="mwBiA">)</span><span class="w" id="mwBiE"> </span><span class="kd" id="mwBiI">::</span><span class="w" id="mwBiM"> </span><span class="n" id="mwBiQ">x</span><span class="w" id="mwBiU"> </span><span class="p" id="mwBiY">(:,</span><span class="w" id="mwBic"> </span><span class="p" id="mwBig">:)</span><span class="w" id="mwBik"> </span><span class="c" id="mwBio">! Input: m sets of RHSs B(n,m), </span> <span class="w" id="mwBis"> </span><span class="c" id="mwBiw">! Output: the corresponding sets of unknowns X(n,m)</span> <span class="w" id="mwBi0"> </span> <span class="w" id="mwBi4"> </span><span class="kt" id="mwBi8">Integer</span><span class="w" id="mwBjA"> </span><span class="kd" id="mwBjE">::</span><span class="w" id="mwBjI"> </span><span class="n" id="mwBjM">n</span><span class="p" id="mwBjQ">,</span><span class="w" id="mwBjU"> </span><span class="n" id="mwBjY">m</span><span class="p" id="mwBjc">,</span><span class="w" id="mwBjg"> </span><span class="n" id="mwBjk">i</span><span class="p" id="mwBjo">,</span><span class="w" id="mwBjs"> </span><span class="n" id="mwBjw">j</span> <span class="w" id="mwBj0"> </span> <span class="w" id="mwBj4"> </span><span class="n" id="mwBj8">n</span><span class="w" id="mwBkA"> </span><span class="o" id="mwBkE">=</span><span class="w" id="mwBkI"> </span><span class="n" id="mwBkM">size</span><span class="p" id="mwBkQ">(</span><span class="n" id="mwBkU">ip</span><span class="p" id="mwBkY">)</span> <span class="w" id="mwBkc"> </span><span class="n" id="mwBkg">m</span><span class="w" id="mwBkk"> </span><span class="o" id="mwBko">=</span><span class="w" id="mwBks"> </span><span class="n" id="mwBkw">size</span><span class="p" id="mwBk0">(</span><span class="n" id="mwBk4">x</span><span class="p" id="mwBk8">,</span><span class="w" id="mwBlA"> </span><span class="nb" id="mwBlE">dim</span><span class="o" id="mwBlI">=</span><span class="mi" id="mwBlM">2</span><span class="p" id="mwBlQ">)</span> <span class="w" id="mwBlU"> </span><span class="k" id="mwBlY">If</span><span class="w" id="mwBlc"> </span><span class="p" id="mwBlg">(</span><span class="n" id="mwBlk">n</span><span class="o" id="mwBlo"><</span><span class="mf" id="mwBls">1.</span><span class="nb" id="mwBlw">Or</span><span class="p" id="mwBl0">.</span><span class="n" id="mwBl4">m</span><span class="o" id="mwBl8"><</span><span class="mf" id="mwBmA">1.</span><span class="nb" id="mwBmE">Or</span><span class="p" id="mwBmI">.</span><span class="nb" id="mwBmM">Any</span><span class="p" id="mwBmQ">([</span><span class="n" id="mwBmU">n</span><span class="p" id="mwBmY">,</span><span class="n" id="mwBmc">n</span><span class="p" id="mwBmg">]</span><span class="o" id="mwBmk">/=</span><span class="nb" id="mwBmo">shape</span><span class="p" id="mwBms">(</span><span class="n" id="mwBmw">l</span><span class="p" id="mwBm0">)).</span><span class="nb" id="mwBm4">Or</span><span class="p" id="mwBm8">.</span><span class="nb" id="mwBnA">Any</span><span class="p" id="mwBnE">(</span><span class="nb" id="mwBnI">shape</span><span class="p" id="mwBnM">(</span><span class="n" id="mwBnQ">l</span><span class="p" id="mwBnU">)</span><span class="o" id="mwBnY">/=</span><span class="nb" id="mwBnc">shape</span><span class="p" id="mwBng">(</span><span class="n" id="mwBnk">u</span><span class="p" id="mwBno">)).</span><span class="nb" id="mwBns">Or</span><span class="p" id="mwBnw">.</span><span class="w" id="mwBn0"> </span><span class="p" id="mwBn4">&</span> <span class="w" id="mwBn8"> </span><span class="n" id="mwBoA">n</span><span class="o" id="mwBoE">/=</span><span class="n" id="mwBoI">size</span><span class="p" id="mwBoM">(</span><span class="n" id="mwBoQ">x</span><span class="p" id="mwBoU">,</span><span class="nb" id="mwBoY">dim</span><span class="o" id="mwBoc">=</span><span class="mi" id="mwBog">1</span><span class="p" id="mwBok">))</span><span class="w" id="mwBoo"> </span><span class="k" id="mwBos">Stop </span><span class="mi" id="mwBow">91</span> <span class="w" id="mwBo0"> </span><span class="k" id="mwBo4">Do </span><span class="n" id="mwBo8">i</span><span class="w" id="mwBpA"> </span><span class="o" id="mwBpE">=</span><span class="w" id="mwBpI"> </span><span class="mi" id="mwBpM">1</span><span class="p" id="mwBpQ">,</span><span class="w" id="mwBpU"> </span><span class="n" id="mwBpY">m</span> <span class="w" id="mwBpc"> </span><span class="k" id="mwBpg">Do </span><span class="n" id="mwBpk">j</span><span class="w" id="mwBpo"> </span><span class="o" id="mwBps">=</span><span class="w" id="mwBpw"> </span><span class="mi" id="mwBp0">1</span><span class="p" id="mwBp4">,</span><span class="w" id="mwBp8"> </span><span class="n" id="mwBqA">n</span> <span class="w" id="mwBqE"> </span><span class="n" id="mwBqI">x</span><span class="p" id="mwBqM">(</span><span class="n" id="mwBqQ">j</span><span class="p" id="mwBqU">,</span><span class="n" id="mwBqY">i</span><span class="p" id="mwBqc">)</span><span class="w" id="mwBqg"> </span><span class="o" id="mwBqk">=</span><span class="w" id="mwBqo"> </span><span class="n" id="mwBqs">x</span><span class="p" id="mwBqw">(</span><span class="n" id="mwBq0">j</span><span class="p" id="mwBq4">,</span><span class="n" id="mwBq8">i</span><span class="p" id="mwBrA">)</span><span class="o" id="mwBrE">-</span><span class="nb" id="mwBrI">dot_product</span><span class="p" id="mwBrM">(</span><span class="n" id="mwBrQ">x</span><span class="p" id="mwBrU">(:</span><span class="n" id="mwBrY">j</span><span class="o" id="mwBrc">-</span><span class="mi" id="mwBrg">1</span><span class="p" id="mwBrk">,</span><span class="n" id="mwBro">i</span><span class="p" id="mwBrs">),</span><span class="n" id="mwBrw">l</span><span class="p" id="mwBr0">(</span><span class="n" id="mwBr4">j</span><span class="p" id="mwBr8">,:</span><span class="n" id="mwBsA">j</span><span class="o" id="mwBsE">-</span><span class="mi" id="mwBsI">1</span><span class="p" id="mwBsM">))</span> <span class="w" id="mwBsQ"> </span><span class="k" id="mwBsU">End Do</span> <span class="k" id="mwBsY"> Do </span><span class="n" id="mwBsc">j</span><span class="w" id="mwBsg"> </span><span class="o" id="mwBsk">=</span><span class="w" id="mwBso"> </span><span class="n" id="mwBss">n</span><span class="p" id="mwBsw">,</span><span class="w" id="mwBs0"> </span><span class="mi" id="mwBs4">1</span><span class="p" id="mwBs8">,</span><span class="w" id="mwBtA"> </span><span class="o" id="mwBtE">-</span><span class="mi" id="mwBtI">1</span> <span class="w" id="mwBtM"> </span><span class="n" id="mwBtQ">x</span><span class="p" id="mwBtU">(</span><span class="n" id="mwBtY">j</span><span class="p" id="mwBtc">,</span><span class="n" id="mwBtg">i</span><span class="p" id="mwBtk">)</span><span class="w" id="mwBto"> </span><span class="o" id="mwBts">=</span><span class="w" id="mwBtw"> </span><span class="p" id="mwBt0">(</span><span class="n" id="mwBt4">x</span><span class="p" id="mwBt8">(</span><span class="n" id="mwBuA">j</span><span class="p" id="mwBuE">,</span><span class="n" id="mwBuI">i</span><span class="p" id="mwBuM">)</span><span class="o" id="mwBuQ">-</span><span class="nb" id="mwBuU">dot_product</span><span class="p" id="mwBuY">(</span><span class="n" id="mwBuc">x</span><span class="p" id="mwBug">(</span><span class="n" id="mwBuk">j</span><span class="o" id="mwBuo">+</span><span class="mi" id="mwBus">1</span><span class="p" id="mwBuw">:,</span><span class="n" id="mwBu0">i</span><span class="p" id="mwBu4">),</span><span class="n" id="mwBu8">u</span><span class="p" id="mwBvA">(</span><span class="n" id="mwBvE">j</span><span class="p" id="mwBvI">,</span><span class="n" id="mwBvM">ip</span><span class="p" id="mwBvQ">(</span><span class="n" id="mwBvU">j</span><span class="o" id="mwBvY">+</span><span class="mi" id="mwBvc">1</span><span class="p" id="mwBvg">:))))</span><span class="w" id="mwBvk"> </span><span class="o" id="mwBvo">/</span><span class="w" id="mwBvs"> </span><span class="p" id="mwBvw">&</span> <span class="w" id="mwBv0"> </span><span class="n" id="mwBv4">u</span><span class="p" id="mwBv8">(</span><span class="n" id="mwBwA">j</span><span class="p" id="mwBwE">,</span><span class="n" id="mwBwI">ip</span><span class="p" id="mwBwM">(</span><span class="n" id="mwBwQ">j</span><span class="p" id="mwBwU">))</span> <span class="w" id="mwBwY"> </span><span class="k" id="mwBwc">End Do</span> <span class="k" id="mwBwg"> End Do</span> <span class="k" id="mwBwk"> End Subroutine </span><span class="n" id="mwBwo">lusolve</span><span class="w" id="mwBws"> </span> <span class="k" id="mwBww">End Module </span><span class="n" id="mwBw0">mlu</span>

C code example

<span id="mwBxE"></span><span class="cm" id="mwBxI">/* INPUT: A - array of pointers to rows of a square matrix having dimension N</span> <span class="cm" id="mwBxM"> * Tol - small tolerance number to detect failure when the matrix is near degenerate</span> <span class="cm" id="mwBxQ"> * OUTPUT: Matrix A is changed, it contains a copy of both matrices L-E and U as A=(L-E)+U such that P*A=L*U.</span> <span class="cm" id="mwBxU"> * The permutation matrix is not stored as a matrix, but in an integer vector P of size N+1 </span> <span class="cm" id="mwBxY"> * containing column indexes where the permutation matrix has "1". The last element P[N]=S+N, </span> <span class="cm" id="mwBxc"> * where S is the number of row exchanges needed for determinant computation, det(P)=(-1)^S </span> <span class="cm" id="mwBxg"> */</span> <span class="kt" id="mwBxk">int</span><span class="w" id="mwBxo"> </span><span class="nf" id="mwBxs">LUPDecompose</span><span class="p" id="mwBxw">(</span><span class="kt" id="mwBx0">double</span><span class="w" id="mwBx4"> </span><span class="o" id="mwBx8">**</span><span class="n" id="mwByA">A</span><span class="p" id="mwByE">,</span><span class="w" id="mwByI"> </span><span class="kt" id="mwByM">int</span><span class="w" id="mwByQ"> </span><span class="n" id="mwByU">N</span><span class="p" id="mwByY">,</span><span class="w" id="mwByc"> </span><span class="kt" id="mwByg">double</span><span class="w" id="mwByk"> </span><span class="n" id="mwByo">Tol</span><span class="p" id="mwBys">,</span><span class="w" id="mwByw"> </span><span class="kt" id="mwBy0">int</span><span class="w" id="mwBy4"> </span><span class="o" id="mwBy8">*</span><span class="n" id="mwBzA">P</span><span class="p" id="mwBzE">)</span><span class="w" id="mwBzI"> </span><span class="p" id="mwBzM">{</span> <span class="w" id="mwBzQ"> </span><span class="kt" id="mwBzU">int</span><span class="w" id="mwBzY"> </span><span class="n" id="mwBzc">i</span><span class="p" id="mwBzg">,</span><span class="w" id="mwBzk"> </span><span class="n" id="mwBzo">j</span><span class="p" id="mwBzs">,</span><span class="w" id="mwBzw"> </span><span class="n" id="mwBz0">k</span><span class="p" id="mwBz4">,</span><span class="w" id="mwBz8"> </span><span class="n" id="mwB0A">imax</span><span class="p" id="mwB0E">;</span><span class="w" id="mwB0I"> </span> <span class="w" id="mwB0M"> </span><span class="kt" id="mwB0Q">double</span><span class="w" id="mwB0U"> </span><span class="n" id="mwB0Y">maxA</span><span class="p" id="mwB0c">,</span><span class="w" id="mwB0g"> </span><span class="o" id="mwB0k">*</span><span class="n" id="mwB0o">ptr</span><span class="p" id="mwB0s">,</span><span class="w" id="mwB0w"> </span><span class="n" id="mwB00">absA</span><span class="p" id="mwB04">;</span> <span class="w" id="mwB08"> </span><span class="k" id="mwB1A">for</span><span class="w" id="mwB1E"> </span><span class="p" id="mwB1I">(</span><span class="n" id="mwB1M">i</span><span class="w" id="mwB1Q"> </span><span class="o" id="mwB1U">=</span><span class="w" id="mwB1Y"> </span><span class="mi" id="mwB1c">0</span><span class="p" id="mwB1g">;</span><span class="w" id="mwB1k"> </span><span class="n" id="mwB1o">i</span><span class="w" id="mwB1s"> </span><span class="o" id="mwB1w"><=</span><span class="w" id="mwB10"> </span><span class="n" id="mwB14">N</span><span class="p" id="mwB18">;</span><span class="w" id="mwB2A"> </span><span class="n" id="mwB2E">i</span><span class="o" id="mwB2I">++</span><span class="p" id="mwB2M">)</span> <span class="w" id="mwB2Q"> </span><span class="n" id="mwB2U">P</span><span class="p" id="mwB2Y">[</span><span class="n" id="mwB2c">i</span><span class="p" id="mwB2g">]</span><span class="w" id="mwB2k"> </span><span class="o" id="mwB2o">=</span><span class="w" id="mwB2s"> </span><span class="n" id="mwB2w">i</span><span class="p" id="mwB20">;</span><span class="w" id="mwB24"> </span><span class="c1" id="mwB28">//Unit permutation matrix, P[N] initialized with N</span> <span class="w" id="mwB3A"> </span><span class="k" id="mwB3E">for</span><span class="w" id="mwB3I"> </span><span class="p" id="mwB3M">(</span><span class="n" id="mwB3Q">i</span><span class="w" id="mwB3U"> </span><span class="o" id="mwB3Y">=</span><span class="w" id="mwB3c"> </span><span class="mi" id="mwB3g">0</span><span class="p" id="mwB3k">;</span><span class="w" id="mwB3o"> </span><span class="n" id="mwB3s">i</span><span class="w" id="mwB3w"> </span><span class="o" id="mwB30"><</span><span class="w" id="mwB34"> </span><span class="n" id="mwB38">N</span><span class="p" id="mwB4A">;</span><span class="w" id="mwB4E"> </span><span class="n" id="mwB4I">i</span><span class="o" id="mwB4M">++</span><span class="p" id="mwB4Q">)</span><span class="w" id="mwB4U"> </span><span class="p" id="mwB4Y">{</span> <span class="w" id="mwB4c"> </span><span class="n" id="mwB4g">maxA</span><span class="w" id="mwB4k"> </span><span class="o" id="mwB4o">=</span><span class="w" id="mwB4s"> </span><span class="mf" id="mwB4w">0.0</span><span class="p" id="mwB40">;</span> <span class="w" id="mwB44"> </span><span class="n" id="mwB48">imax</span><span class="w" id="mwB5A"> </span><span class="o" id="mwB5E">=</span><span class="w" id="mwB5I"> </span><span class="n" id="mwB5M">i</span><span class="p" id="mwB5Q">;</span> <span class="w" id="mwB5U"> </span><span class="k" id="mwB5Y">for</span><span class="w" id="mwB5c"> </span><span class="p" id="mwB5g">(</span><span class="n" id="mwB5k">k</span><span class="w" id="mwB5o"> </span><span class="o" id="mwB5s">=</span><span class="w" id="mwB5w"> </span><span class="n" id="mwB50">i</span><span class="p" id="mwB54">;</span><span class="w" id="mwB58"> </span><span class="n" id="mwB6A">k</span><span class="w" id="mwB6E"> </span><span class="o" id="mwB6I"><</span><span class="w" id="mwB6M"> </span><span class="n" id="mwB6Q">N</span><span class="p" id="mwB6U">;</span><span class="w" id="mwB6Y"> </span><span class="n" id="mwB6c">k</span><span class="o" id="mwB6g">++</span><span class="p" id="mwB6k">)</span> <span class="w" id="mwB6o"> </span><span class="k" id="mwB6s">if</span><span class="w" id="mwB6w"> </span><span class="p" id="mwB60">((</span><span class="n" id="mwB64">absA</span><span class="w" id="mwB68"> </span><span class="o" id="mwB7A">=</span><span class="w" id="mwB7E"> </span><span class="n" id="mwB7I">fabs</span><span class="p" id="mwB7M">(</span><span class="n" id="mwB7Q">A</span><span class="p" id="mwB7U">[</span><span class="n" id="mwB7Y">k</span><span class="p" id="mwB7c">][</span><span class="n" id="mwB7g">i</span><span class="p" id="mwB7k">]))</span><span class="w" id="mwB7o"> </span><span class="o" id="mwB7s">></span><span class="w" id="mwB7w"> </span><span class="n" id="mwB70">maxA</span><span class="p" id="mwB74">)</span><span class="w" id="mwB78"> </span><span class="p" id="mwB8A">{</span><span class="w" id="mwB8E"> </span> <span class="w" id="mwB8I"> </span><span class="n" id="mwB8M">maxA</span><span class="w" id="mwB8Q"> </span><span class="o" id="mwB8U">=</span><span class="w" id="mwB8Y"> </span><span class="n" id="mwB8c">absA</span><span class="p" id="mwB8g">;</span> <span class="w" id="mwB8k"> </span><span class="n" id="mwB8o">imax</span><span class="w" id="mwB8s"> </span><span class="o" id="mwB8w">=</span><span class="w" id="mwB80"> </span><span class="n" id="mwB84">k</span><span class="p" id="mwB88">;</span> <span class="w" id="mwB9A"> </span><span class="p" id="mwB9E">}</span> <span class="w" id="mwB9I"> </span><span class="k" id="mwB9M">if</span><span class="w" id="mwB9Q"> </span><span class="p" id="mwB9U">(</span><span class="n" id="mwB9Y">maxA</span><span class="w" id="mwB9c"> </span><span class="o" id="mwB9g"><</span><span class="w" id="mwB9k"> </span><span class="n" id="mwB9o">Tol</span><span class="p" id="mwB9s">)</span><span class="w" id="mwB9w"> </span><span class="k" id="mwB90">return</span><span class="w" id="mwB94"> </span><span class="mi" id="mwB98">0</span><span class="p" id="mwB-A">;</span><span class="w" id="mwB-E"> </span><span class="c1" id="mwB-I">//failure, matrix is degenerate</span> <span class="w" id="mwB-M"> </span><span class="k" id="mwB-Q">if</span><span class="w" id="mwB-U"> </span><span class="p" id="mwB-Y">(</span><span class="n" id="mwB-c">imax</span><span class="w" id="mwB-g"> </span><span class="o" id="mwB-k">!=</span><span class="w" id="mwB-o"> </span><span class="n" id="mwB-s">i</span><span class="p" id="mwB-w">)</span><span class="w" id="mwB-0"> </span><span class="p" id="mwB-4">{</span> <span class="w" id="mwB-8"> </span><span class="c1" id="mwB_A">//pivoting P</span> <span class="w" id="mwB_E"> </span><span class="n" id="mwB_I">j</span><span class="w" id="mwB_M"> </span><span class="o" id="mwB_Q">=</span><span class="w" id="mwB_U"> </span><span class="n" id="mwB_Y">P</span><span class="p" id="mwB_c">[</span><span class="n" id="mwB_g">i</span><span class="p" id="mwB_k">];</span> <span class="w" id="mwB_o"> </span><span class="n" id="mwB_s">P</span><span class="p" id="mwB_w">[</span><span class="n" id="mwB_0">i</span><span class="p" id="mwB_4">]</span><span class="w" id="mwB_8"> </span><span class="o" id="mwCAA">=</span><span class="w" id="mwCAE"> </span><span class="n" id="mwCAI">P</span><span class="p" id="mwCAM">[</span><span class="n" id="mwCAQ">imax</span><span class="p" id="mwCAU">];</span> <span class="w" id="mwCAY"> </span><span class="n" id="mwCAc">P</span><span class="p" id="mwCAg">[</span><span class="n" id="mwCAk">imax</span><span class="p" id="mwCAo">]</span><span class="w" id="mwCAs"> </span><span class="o" id="mwCAw">=</span><span class="w" id="mwCA0"> </span><span class="n" id="mwCA4">j</span><span class="p" id="mwCA8">;</span> <span class="w" id="mwCBA"> </span><span class="c1" id="mwCBE">//pivoting rows of A</span> <span class="w" id="mwCBI"> </span><span class="n" id="mwCBM">ptr</span><span class="w" id="mwCBQ"> </span><span class="o" id="mwCBU">=</span><span class="w" id="mwCBY"> </span><span class="n" id="mwCBc">A</span><span class="p" id="mwCBg">[</span><span class="n" id="mwCBk">i</span><span class="p" id="mwCBo">];</span> <span class="w" id="mwCBs"> </span><span class="n" id="mwCBw">A</span><span class="p" id="mwCB0">[</span><span class="n" id="mwCB4">i</span><span class="p" id="mwCB8">]</span><span class="w" id="mwCCA"> </span><span class="o" id="mwCCE">=</span><span class="w" id="mwCCI"> </span><span class="n" id="mwCCM">A</span><span class="p" id="mwCCQ">[</span><span class="n" id="mwCCU">imax</span><span class="p" id="mwCCY">];</span> <span class="w" id="mwCCc"> </span><span class="n" id="mwCCg">A</span><span class="p" id="mwCCk">[</span><span class="n" id="mwCCo">imax</span><span class="p" id="mwCCs">]</span><span class="w" id="mwCCw"> </span><span class="o" id="mwCC0">=</span><span class="w" id="mwCC4"> </span><span class="n" id="mwCC8">ptr</span><span class="p" id="mwCDA">;</span> <span class="w" id="mwCDE"> </span><span class="c1" id="mwCDI">//counting pivots starting from N (for determinant)</span> <span class="w" id="mwCDM"> </span><span class="n" id="mwCDQ">P</span><span class="p" id="mwCDU">[</span><span class="n" id="mwCDY">N</span><span class="p" id="mwCDc">]</span><span class="o" id="mwCDg">++</span><span class="p" id="mwCDk">;</span> <span class="w" id="mwCDo"> </span><span class="p" id="mwCDs">}</span> <span class="w" id="mwCDw"> </span><span class="k" id="mwCD0">for</span><span class="w" id="mwCD4"> </span><span class="p" id="mwCD8">(</span><span class="n" id="mwCEA">j</span><span class="w" id="mwCEE"> </span><span class="o" id="mwCEI">=</span><span class="w" id="mwCEM"> </span><span class="n" id="mwCEQ">i</span><span class="w" id="mwCEU"> </span><span class="o" id="mwCEY">+</span><span class="w" id="mwCEc"> </span><span class="mi" id="mwCEg">1</span><span class="p" id="mwCEk">;</span><span class="w" id="mwCEo"> </span><span class="n" id="mwCEs">j</span><span class="w" id="mwCEw"> </span><span class="o" id="mwCE0"><</span><span class="w" id="mwCE4"> </span><span class="n" id="mwCE8">N</span><span class="p" id="mwCFA">;</span><span class="w" id="mwCFE"> </span><span class="n" id="mwCFI">j</span><span class="o" id="mwCFM">++</span><span class="p" id="mwCFQ">)</span><span class="w" id="mwCFU"> </span><span class="p" id="mwCFY">{</span> <span class="w" id="mwCFc"> </span><span class="n" id="mwCFg">A</span><span class="p" id="mwCFk">[</span><span class="n" id="mwCFo">j</span><span class="p" id="mwCFs">][</span><span class="n" id="mwCFw">i</span><span class="p" id="mwCF0">]</span><span class="w" id="mwCF4"> </span><span class="o" id="mwCF8">/=</span><span class="w" id="mwCGA"> </span><span class="n" id="mwCGE">A</span><span class="p" id="mwCGI">[</span><span class="n" id="mwCGM">i</span><span class="p" id="mwCGQ">][</span><span class="n" id="mwCGU">i</span><span class="p" id="mwCGY">];</span> <span class="w" id="mwCGc"> </span><span class="k" id="mwCGg">for</span><span class="w" id="mwCGk"> </span><span class="p" id="mwCGo">(</span><span class="n" id="mwCGs">k</span><span class="w" id="mwCGw"> </span><span class="o" id="mwCG0">=</span><span class="w" id="mwCG4"> </span><span class="n" id="mwCG8">i</span><span class="w" id="mwCHA"> </span><span class="o" id="mwCHE">+</span><span class="w" id="mwCHI"> </span><span class="mi" id="mwCHM">1</span><span class="p" id="mwCHQ">;</span><span class="w" id="mwCHU"> </span><span class="n" id="mwCHY">k</span><span class="w" id="mwCHc"> </span><span class="o" id="mwCHg"><</span><span class="w" id="mwCHk"> </span><span class="n" id="mwCHo">N</span><span class="p" id="mwCHs">;</span><span class="w" id="mwCHw"> </span><span class="n" id="mwCH0">k</span><span class="o" id="mwCH4">++</span><span class="p" id="mwCH8">)</span> <span class="w" id="mwCIA"> </span><span class="n" id="mwCIE">A</span><span class="p" id="mwCII">[</span><span class="n" id="mwCIM">j</span><span class="p" id="mwCIQ">][</span><span class="n" id="mwCIU">k</span><span class="p" id="mwCIY">]</span><span class="w" id="mwCIc"> </span><span class="o" id="mwCIg">-=</span><span class="w" id="mwCIk"> </span><span class="n" id="mwCIo">A</span><span class="p" id="mwCIs">[</span><span class="n" id="mwCIw">j</span><span class="p" id="mwCI0">][</span><span class="n" id="mwCI4">i</span><span class="p" id="mwCI8">]</span><span class="w" id="mwCJA"> </span><span class="o" id="mwCJE">*</span><span class="w" id="mwCJI"> </span><span class="n" id="mwCJM">A</span><span class="p" id="mwCJQ">[</span><span class="n" id="mwCJU">i</span><span class="p" id="mwCJY">][</span><span class="n" id="mwCJc">k</span><span class="p" id="mwCJg">];</span> <span class="w" id="mwCJk"> </span><span class="p" id="mwCJo">}</span> <span class="w" id="mwCJs"> </span><span class="p" id="mwCJw">}</span> <span class="w" id="mwCJ0"> </span><span class="k" id="mwCJ4">return</span><span class="w" id="mwCJ8"> </span><span class="mi" id="mwCKA">1</span><span class="p" id="mwCKE">;</span><span class="w" id="mwCKI"> </span><span class="c1" id="mwCKM">//decomposition done </span> <span class="p" id="mwCKQ">}</span> <span class="cm" id="mwCKU">/* INPUT: A,P filled in LUPDecompose; b - rhs vector; N - dimension</span> <span class="cm" id="mwCKY"> * OUTPUT: x - solution vector of A*x=b</span> <span class="cm" id="mwCKc"> */</span> <span class="kt" id="mwCKg">void</span><span class="w" id="mwCKk"> </span><span class="nf" id="mwCKo">LUPSolve</span><span class="p" id="mwCKs">(</span><span class="kt" id="mwCKw">double</span><span class="w" id="mwCK0"> </span><span class="o" id="mwCK4">**</span><span class="n" id="mwCK8">A</span><span class="p" id="mwCLA">,</span><span class="w" id="mwCLE"> </span><span class="kt" id="mwCLI">int</span><span class="w" id="mwCLM"> </span><span class="o" id="mwCLQ">*</span><span class="n" id="mwCLU">P</span><span class="p" id="mwCLY">,</span><span class="w" id="mwCLc"> </span><span class="kt" id="mwCLg">double</span><span class="w" id="mwCLk"> </span><span class="o" id="mwCLo">*</span><span class="n" id="mwCLs">b</span><span class="p" id="mwCLw">,</span><span class="w" id="mwCL0"> </span><span class="kt" id="mwCL4">int</span><span class="w" id="mwCL8"> </span><span class="n" id="mwCMA">N</span><span class="p" id="mwCME">,</span><span class="w" id="mwCMI"> </span><span class="kt" id="mwCMM">double</span><span class="w" id="mwCMQ"> </span><span class="o" id="mwCMU">*</span><span class="n" id="mwCMY">x</span><span class="p" id="mwCMc">)</span><span class="w" id="mwCMg"> </span><span class="p" id="mwCMk">{</span> <span class="w" id="mwCMo"> </span><span class="k" id="mwCMs">for</span><span class="w" id="mwCMw"> </span><span class="p" id="mwCM0">(</span><span class="kt" id="mwCM4">int</span><span class="w" id="mwCM8"> </span><span class="n" id="mwCNA">i</span><span class="w" id="mwCNE"> </span><span class="o" id="mwCNI">=</span><span class="w" id="mwCNM"> </span><span class="mi" id="mwCNQ">0</span><span class="p" id="mwCNU">;</span><span class="w" id="mwCNY"> </span><span class="n" id="mwCNc">i</span><span class="w" id="mwCNg"> </span><span class="o" id="mwCNk"><</span><span class="w" id="mwCNo"> </span><span class="n" id="mwCNs">N</span><span class="p" id="mwCNw">;</span><span class="w" id="mwCN0"> </span><span class="n" id="mwCN4">i</span><span class="o" id="mwCN8">++</span><span class="p" id="mwCOA">)</span><span class="w" id="mwCOE"> </span><span class="p" id="mwCOI">{</span> <span class="w" id="mwCOM"> </span><span class="n" id="mwCOQ">x</span><span class="p" id="mwCOU">[</span><span class="n" id="mwCOY">i</span><span class="p" id="mwCOc">]</span><span class="w" id="mwCOg"> </span><span class="o" id="mwCOk">=</span><span class="w" id="mwCOo"> </span><span class="n" id="mwCOs">b</span><span class="p" id="mwCOw">[</span><span class="n" id="mwCO0">P</span><span class="p" id="mwCO4">[</span><span class="n" id="mwCO8">i</span><span class="p" id="mwCPA">]];</span> <span class="w" id="mwCPE"> </span><span class="k" id="mwCPI">for</span><span class="w" id="mwCPM"> </span><span class="p" id="mwCPQ">(</span><span class="kt" id="mwCPU">int</span><span class="w" id="mwCPY"> </span><span class="n" id="mwCPc">k</span><span class="w" id="mwCPg"> </span><span class="o" id="mwCPk">=</span><span class="w" id="mwCPo"> </span><span class="mi" id="mwCPs">0</span><span class="p" id="mwCPw">;</span><span class="w" id="mwCP0"> </span><span class="n" id="mwCP4">k</span><span class="w" id="mwCP8"> </span><span class="o" id="mwCQA"><</span><span class="w" id="mwCQE"> </span><span class="n" id="mwCQI">i</span><span class="p" id="mwCQM">;</span><span class="w" id="mwCQQ"> </span><span class="n" id="mwCQU">k</span><span class="o" id="mwCQY">++</span><span class="p" id="mwCQc">)</span> <span class="w" id="mwCQg"> </span><span class="n" id="mwCQk">x</span><span class="p" id="mwCQo">[</span><span class="n" id="mwCQs">i</span><span class="p" id="mwCQw">]</span><span class="w" id="mwCQ0"> </span><span class="o" id="mwCQ4">-=</span><span class="w" id="mwCQ8"> </span><span class="n" id="mwCRA">A</span><span class="p" id="mwCRE">[</span><span class="n" id="mwCRI">i</span><span class="p" id="mwCRM">][</span><span class="n" id="mwCRQ">k</span><span class="p" id="mwCRU">]</span><span class="w" id="mwCRY"> </span><span class="o" id="mwCRc">*</span><span class="w" id="mwCRg"> </span><span class="n" id="mwCRk">x</span><span class="p" id="mwCRo">[</span><span class="n" id="mwCRs">k</span><span class="p" id="mwCRw">];</span> <span class="w" id="mwCR0"> </span><span class="p" id="mwCR4">}</span> <span class="w" id="mwCR8"> </span><span class="k" id="mwCSA">for</span><span class="w" id="mwCSE"> </span><span class="p" id="mwCSI">(</span><span class="kt" id="mwCSM">int</span><span class="w" id="mwCSQ"> </span><span class="n" id="mwCSU">i</span><span class="w" id="mwCSY"> </span><span class="o" id="mwCSc">=</span><span class="w" id="mwCSg"> </span><span class="n" id="mwCSk">N</span><span class="w" id="mwCSo"> </span><span class="o" id="mwCSs">-</span><span class="w" id="mwCSw"> </span><span class="mi" id="mwCS0">1</span><span class="p" id="mwCS4">;</span><span class="w" id="mwCS8"> </span><span class="n" id="mwCTA">i</span><span class="w" id="mwCTE"> </span><span class="o" id="mwCTI">>=</span><span class="w" id="mwCTM"> </span><span class="mi" id="mwCTQ">0</span><span class="p" id="mwCTU">;</span><span class="w" id="mwCTY"> </span><span class="n" id="mwCTc">i</span><span class="o" id="mwCTg">--</span><span class="p" id="mwCTk">)</span><span class="w" id="mwCTo"> </span><span class="p" id="mwCTs">{</span> <span class="w" id="mwCTw"> </span><span class="k" id="mwCT0">for</span><span class="w" id="mwCT4"> </span><span class="p" id="mwCT8">(</span><span class="kt" id="mwCUA">int</span><span class="w" id="mwCUE"> </span><span class="n" id="mwCUI">k</span><span class="w" id="mwCUM"> </span><span class="o" id="mwCUQ">=</span><span class="w" id="mwCUU"> </span><span class="n" id="mwCUY">i</span><span class="w" id="mwCUc"> </span><span class="o" id="mwCUg">+</span><span class="w" id="mwCUk"> </span><span class="mi" id="mwCUo">1</span><span class="p" id="mwCUs">;</span><span class="w" id="mwCUw"> </span><span class="n" id="mwCU0">k</span><span class="w" id="mwCU4"> </span><span class="o" id="mwCU8"><</span><span class="w" id="mwCVA"> </span><span class="n" id="mwCVE">N</span><span class="p" id="mwCVI">;</span><span class="w" id="mwCVM"> </span><span class="n" id="mwCVQ">k</span><span class="o" id="mwCVU">++</span><span class="p" id="mwCVY">)</span> <span class="w" id="mwCVc"> </span><span class="n" id="mwCVg">x</span><span class="p" id="mwCVk">[</span><span class="n" id="mwCVo">i</span><span class="p" id="mwCVs">]</span><span class="w" id="mwCVw"> </span><span class="o" id="mwCV0">-=</span><span class="w" id="mwCV4"> </span><span class="n" id="mwCV8">A</span><span class="p" id="mwCWA">[</span><span class="n" id="mwCWE">i</span><span class="p" id="mwCWI">][</span><span class="n" id="mwCWM">k</span><span class="p" id="mwCWQ">]</span><span class="w" id="mwCWU"> </span><span class="o" id="mwCWY">*</span><span class="w" id="mwCWc"> </span><span class="n" id="mwCWg">x</span><span class="p" id="mwCWk">[</span><span class="n" id="mwCWo">k</span><span class="p" id="mwCWs">];</span> <span class="w" id="mwCWw"> </span><span class="n" id="mwCW0">x</span><span class="p" id="mwCW4">[</span><span class="n" id="mwCW8">i</span><span class="p" id="mwCXA">]</span><span class="w" id="mwCXE"> </span><span class="o" id="mwCXI">/=</span><span class="w" id="mwCXM"> </span><span class="n" id="mwCXQ">A</span><span class="p" id="mwCXU">[</span><span class="n" id="mwCXY">i</span><span class="p" id="mwCXc">][</span><span class="n" id="mwCXg">i</span><span class="p" id="mwCXk">];</span> <span class="w" id="mwCXo"> </span><span class="p" id="mwCXs">}</span> <span class="p" id="mwCXw">}</span> <span class="cm" id="mwCX0">/* INPUT: A,P filled in LUPDecompose; N - dimension</span> <span class="cm" id="mwCX4"> * OUTPUT: IA is the inverse of the initial matrix</span> <span class="cm" id="mwCX8"> */</span> <span class="kt" id="mwCYA">void</span><span class="w" id="mwCYE"> </span><span class="nf" id="mwCYI">LUPInvert</span><span class="p" id="mwCYM">(</span><span class="kt" id="mwCYQ">double</span><span class="w" id="mwCYU"> </span><span class="o" id="mwCYY">**</span><span class="n" id="mwCYc">A</span><span class="p" id="mwCYg">,</span><span class="w" id="mwCYk"> </span><span class="kt" id="mwCYo">int</span><span class="w" id="mwCYs"> </span><span class="o" id="mwCYw">*</span><span class="n" id="mwCY0">P</span><span class="p" id="mwCY4">,</span><span class="w" id="mwCY8"> </span><span class="kt" id="mwCZA">int</span><span class="w" id="mwCZE"> </span><span class="n" id="mwCZI">N</span><span class="p" id="mwCZM">,</span><span class="w" id="mwCZQ"> </span><span class="kt" id="mwCZU">double</span><span class="w" id="mwCZY"> </span><span class="o" id="mwCZc">**</span><span class="n" id="mwCZg">IA</span><span class="p" id="mwCZk">)</span><span class="w" id="mwCZo"> </span><span class="p" id="mwCZs">{</span> <span class="w" id="mwCZw"> </span> <span class="w" id="mwCZ0"> </span><span class="k" id="mwCZ4">for</span><span class="w" id="mwCZ8"> </span><span class="p" id="mwCaA">(</span><span class="kt" id="mwCaE">int</span><span class="w" id="mwCaI"> </span><span class="n" id="mwCaM">j</span><span class="w" id="mwCaQ"> </span><span class="o" id="mwCaU">=</span><span class="w" id="mwCaY"> </span><span class="mi" id="mwCac">0</span><span class="p" id="mwCag">;</span><span class="w" id="mwCak"> </span><span class="n" id="mwCao">j</span><span class="w" id="mwCas"> </span><span class="o" id="mwCaw"><</span><span class="w" id="mwCa0"> </span><span class="n" id="mwCa4">N</span><span class="p" id="mwCa8">;</span><span class="w" id="mwCbA"> </span><span class="n" id="mwCbE">j</span><span class="o" id="mwCbI">++</span><span class="p" id="mwCbM">)</span><span class="w" id="mwCbQ"> </span><span class="p" id="mwCbU">{</span> <span class="w" id="mwCbY"> </span><span class="k" id="mwCbc">for</span><span class="w" id="mwCbg"> </span><span class="p" id="mwCbk">(</span><span class="kt" id="mwCbo">int</span><span class="w" id="mwCbs"> </span><span class="n" id="mwCbw">i</span><span class="w" id="mwCb0"> </span><span class="o" id="mwCb4">=</span><span class="w" id="mwCb8"> </span><span class="mi" id="mwCcA">0</span><span class="p" id="mwCcE">;</span><span class="w" id="mwCcI"> </span><span class="n" id="mwCcM">i</span><span class="w" id="mwCcQ"> </span><span class="o" id="mwCcU"><</span><span class="w" id="mwCcY"> </span><span class="n" id="mwCcc">N</span><span class="p" id="mwCcg">;</span><span class="w" id="mwCck"> </span><span class="n" id="mwCco">i</span><span class="o" id="mwCcs">++</span><span class="p" id="mwCcw">)</span><span class="w" id="mwCc0"> </span><span class="p" id="mwCc4">{</span> <span class="w" id="mwCc8"> </span><span class="n" id="mwCdA">IA</span><span class="p" id="mwCdE">[</span><span class="n" id="mwCdI">i</span><span class="p" id="mwCdM">][</span><span class="n" id="mwCdQ">j</span><span class="p" id="mwCdU">]</span><span class="w" id="mwCdY"> </span><span class="o" id="mwCdc">=</span><span class="w" id="mwCdg"> </span><span class="n" id="mwCdk">P</span><span class="p" id="mwCdo">[</span><span class="n" id="mwCds">i</span><span class="p" id="mwCdw">]</span><span class="w" id="mwCd0"> </span><span class="o" id="mwCd4">==</span><span class="w" id="mwCd8"> </span><span class="n" id="mwCeA">j</span><span class="w" id="mwCeE"> </span><span class="o" id="mwCeI">?</span><span class="w" id="mwCeM"> </span><span class="mf" id="mwCeQ">1.0</span><span class="w" id="mwCeU"> </span><span class="o" id="mwCeY">:</span><span class="w" id="mwCec"> </span><span class="mf" id="mwCeg">0.0</span><span class="p" id="mwCek">;</span> <span class="w" id="mwCeo"> </span><span class="k" id="mwCes">for</span><span class="w" id="mwCew"> </span><span class="p" id="mwCe0">(</span><span class="kt" id="mwCe4">int</span><span class="w" id="mwCe8"> </span><span class="n" id="mwCfA">k</span><span class="w" id="mwCfE"> </span><span class="o" id="mwCfI">=</span><span class="w" id="mwCfM"> </span><span class="mi" id="mwCfQ">0</span><span class="p" id="mwCfU">;</span><span class="w" id="mwCfY"> </span><span class="n" id="mwCfc">k</span><span class="w" id="mwCfg"> </span><span class="o" id="mwCfk"><</span><span class="w" id="mwCfo"> </span><span class="n" id="mwCfs">i</span><span class="p" id="mwCfw">;</span><span class="w" id="mwCf0"> </span><span class="n" id="mwCf4">k</span><span class="o" id="mwCf8">++</span><span class="p" id="mwCgA">)</span> <span class="w" id="mwCgE"> </span><span class="n" id="mwCgI">IA</span><span class="p" id="mwCgM">[</span><span class="n" id="mwCgQ">i</span><span class="p" id="mwCgU">][</span><span class="n" id="mwCgY">j</span><span class="p" id="mwCgc">]</span><span class="w" id="mwCgg"> </span><span class="o" id="mwCgk">-=</span><span class="w" id="mwCgo"> </span><span class="n" id="mwCgs">A</span><span class="p" id="mwCgw">[</span><span class="n" id="mwCg0">i</span><span class="p" id="mwCg4">][</span><span class="n" id="mwCg8">k</span><span class="p" id="mwChA">]</span><span class="w" id="mwChE"> </span><span class="o" id="mwChI">*</span><span class="w" id="mwChM"> </span><span class="n" id="mwChQ">IA</span><span class="p" id="mwChU">[</span><span class="n" id="mwChY">k</span><span class="p" id="mwChc">][</span><span class="n" id="mwChg">j</span><span class="p" id="mwChk">];</span> <span class="w" id="mwCho"> </span><span class="p" id="mwChs">}</span> <span class="w" id="mwChw"> </span><span class="k" id="mwCh0">for</span><span class="w" id="mwCh4"> </span><span class="p" id="mwCh8">(</span><span class="kt" id="mwCiA">int</span><span class="w" id="mwCiE"> </span><span class="n" id="mwCiI">i</span><span class="w" id="mwCiM"> </span><span class="o" id="mwCiQ">=</span><span class="w" id="mwCiU"> </span><span class="n" id="mwCiY">N</span><span class="w" id="mwCic"> </span><span class="o" id="mwCig">-</span><span class="w" id="mwCik"> </span><span class="mi" id="mwCio">1</span><span class="p" id="mwCis">;</span><span class="w" id="mwCiw"> </span><span class="n" id="mwCi0">i</span><span class="w" id="mwCi4"> </span><span class="o" id="mwCi8">>=</span><span class="w" id="mwCjA"> </span><span class="mi" id="mwCjE">0</span><span class="p" id="mwCjI">;</span><span class="w" id="mwCjM"> </span><span class="n" id="mwCjQ">i</span><span class="o" id="mwCjU">--</span><span class="p" id="mwCjY">)</span><span class="w" id="mwCjc"> </span><span class="p" id="mwCjg">{</span> <span class="w" id="mwCjk"> </span><span class="k" id="mwCjo">for</span><span class="w" id="mwCjs"> </span><span class="p" id="mwCjw">(</span><span class="kt" id="mwCj0">int</span><span class="w" id="mwCj4"> </span><span class="n" id="mwCj8">k</span><span class="w" id="mwCkA"> </span><span class="o" id="mwCkE">=</span><span class="w" id="mwCkI"> </span><span class="n" id="mwCkM">i</span><span class="w" id="mwCkQ"> </span><span class="o" id="mwCkU">+</span><span class="w" id="mwCkY"> </span><span class="mi" id="mwCkc">1</span><span class="p" id="mwCkg">;</span><span class="w" id="mwCkk"> </span><span class="n" id="mwCko">k</span><span class="w" id="mwCks"> </span><span class="o" id="mwCkw"><</span><span class="w" id="mwCk0"> </span><span class="n" id="mwCk4">N</span><span class="p" id="mwCk8">;</span><span class="w" id="mwClA"> </span><span class="n" id="mwClE">k</span><span class="o" id="mwClI">++</span><span class="p" id="mwClM">)</span> <span class="w" id="mwClQ"> </span><span class="n" id="mwClU">IA</span><span class="p" id="mwClY">[</span><span class="n" id="mwClc">i</span><span class="p" id="mwClg">][</span><span class="n" id="mwClk">j</span><span class="p" id="mwClo">]</span><span class="w" id="mwCls"> </span><span class="o" id="mwClw">-=</span><span class="w" id="mwCl0"> </span><span class="n" id="mwCl4">A</span><span class="p" id="mwCl8">[</span><span class="n" id="mwCmA">i</span><span class="p" id="mwCmE">][</span><span class="n" id="mwCmI">k</span><span class="p" id="mwCmM">]</span><span class="w" id="mwCmQ"> </span><span class="o" id="mwCmU">*</span><span class="w" id="mwCmY"> </span><span class="n" id="mwCmc">IA</span><span class="p" id="mwCmg">[</span><span class="n" id="mwCmk">k</span><span class="p" id="mwCmo">][</span><span class="n" id="mwCms">j</span><span class="p" id="mwCmw">];</span> <span class="w" id="mwCm0"> </span><span class="n" id="mwCm4">IA</span><span class="p" id="mwCm8">[</span><span class="n" id="mwCnA">i</span><span class="p" id="mwCnE">][</span><span class="n" id="mwCnI">j</span><span class="p" id="mwCnM">]</span><span class="w" id="mwCnQ"> </span><span class="o" id="mwCnU">/=</span><span class="w" id="mwCnY"> </span><span class="n" id="mwCnc">A</span><span class="p" id="mwCng">[</span><span class="n" id="mwCnk">i</span><span class="p" id="mwCno">][</span><span class="n" id="mwCns">i</span><span class="p" id="mwCnw">];</span> <span class="w" id="mwCn0"> </span><span class="p" id="mwCn4">}</span> <span class="w" id="mwCn8"> </span><span class="p" id="mwCoA">}</span> <span class="p" id="mwCoE">}</span> <span class="cm" id="mwCoI">/* INPUT: A,P filled in LUPDecompose; N - dimension. </span> <span class="cm" id="mwCoM"> * OUTPUT: Function returns the determinant of the initial matrix</span> <span class="cm" id="mwCoQ"> */</span> <span class="kt" id="mwCoU">double</span><span class="w" id="mwCoY"> </span><span class="nf" id="mwCoc">LUPDeterminant</span><span class="p" id="mwCog">(</span><span class="kt" id="mwCok">double</span><span class="w" id="mwCoo"> </span><span class="o" id="mwCos">**</span><span class="n" id="mwCow">A</span><span class="p" id="mwCo0">,</span><span class="w" id="mwCo4"> </span><span class="kt" id="mwCo8">int</span><span class="w" id="mwCpA"> </span><span class="o" id="mwCpE">*</span><span class="n" id="mwCpI">P</span><span class="p" id="mwCpM">,</span><span class="w" id="mwCpQ"> </span><span class="kt" id="mwCpU">int</span><span class="w" id="mwCpY"> </span><span class="n" id="mwCpc">N</span><span class="p" id="mwCpg">)</span><span class="w" id="mwCpk"> </span><span class="p" id="mwCpo">{</span> <span class="w" id="mwCps"> </span><span class="kt" id="mwCpw">double</span><span class="w" id="mwCp0"> </span><span class="n" id="mwCp4">det</span><span class="w" id="mwCp8"> </span><span class="o" id="mwCqA">=</span><span class="w" id="mwCqE"> </span><span class="n" id="mwCqI">A</span><span class="p" id="mwCqM">[</span><span class="mi" id="mwCqQ">0</span><span class="p" id="mwCqU">][</span><span class="mi" id="mwCqY">0</span><span class="p" id="mwCqc">];</span> <span class="w" id="mwCqg"> </span><span class="k" id="mwCqk">for</span><span class="w" id="mwCqo"> </span><span class="p" id="mwCqs">(</span><span class="kt" id="mwCqw">int</span><span class="w" id="mwCq0"> </span><span class="n" id="mwCq4">i</span><span class="w" id="mwCq8"> </span><span class="o" id="mwCrA">=</span><span class="w" id="mwCrE"> </span><span class="mi" id="mwCrI">1</span><span class="p" id="mwCrM">;</span><span class="w" id="mwCrQ"> </span><span class="n" id="mwCrU">i</span><span class="w" id="mwCrY"> </span><span class="o" id="mwCrc"><</span><span class="w" id="mwCrg"> </span><span class="n" id="mwCrk">N</span><span class="p" id="mwCro">;</span><span class="w" id="mwCrs"> </span><span class="n" id="mwCrw">i</span><span class="o" id="mwCr0">++</span><span class="p" id="mwCr4">)</span> <span class="w" id="mwCr8"> </span><span class="n" id="mwCsA">det</span><span class="w" id="mwCsE"> </span><span class="o" id="mwCsI">*=</span><span class="w" id="mwCsM"> </span><span class="n" id="mwCsQ">A</span><span class="p" id="mwCsU">[</span><span class="n" id="mwCsY">i</span><span class="p" id="mwCsc">][</span><span class="n" id="mwCsg">i</span><span class="p" id="mwCsk">];</span> <span class="w" id="mwCso"> </span><span class="k" id="mwCss">return</span><span class="w" id="mwCsw"> </span><span class="p" id="mwCs0">(</span><span class="n" id="mwCs4">P</span><span class="p" id="mwCs8">[</span><span class="n" id="mwCtA">N</span><span class="p" id="mwCtE">]</span><span class="w" id="mwCtI"> </span><span class="o" id="mwCtM">-</span><span class="w" id="mwCtQ"> </span><span class="n" id="mwCtU">N</span><span class="p" id="mwCtY">)</span><span class="w" id="mwCtc"> </span><span class="o" id="mwCtg">%</span><span class="w" id="mwCtk"> </span><span class="mi" id="mwCto">2</span><span class="w" id="mwCts"> </span><span class="o" id="mwCtw">==</span><span class="w" id="mwCt0"> </span><span class="mi" id="mwCt4">0</span><span class="w" id="mwCt8"> </span><span class="o" id="mwCuA">?</span><span class="w" id="mwCuE"> </span><span class="n" id="mwCuI">det</span><span class="w" id="mwCuM"> </span><span class="o" id="mwCuQ">:</span><span class="w" id="mwCuU"> </span><span class="o" id="mwCuY">-</span><span class="n" id="mwCuc">det</span><span class="p" id="mwCug">;</span> <span class="p" id="mwCuk">}</span>

C# code example

<span id="mwCu0"></span><span class="k" id="mwCu4">public</span><span class="w" id="mwCu8"> </span><span class="k" id="mwCvA">class</span><span class="w" id="mwCvE"> </span><span class="nc" id="mwCvI">SystemOfLinearEquations</span> <span class="p" id="mwCvM">{</span> <span class="w" id="mwCvQ"> </span><span class="k" id="mwCvU">public</span><span class="w" id="mwCvY"> </span><span class="kt" id="mwCvc">double</span><span class="p" id="mwCvg">[]</span><span class="w" id="mwCvk"> </span><span class="nf" id="mwCvo">SolveUsingLU</span><span class="p" id="mwCvs">(</span><span class="kt" id="mwCvw">double</span><span class="p" id="mwCv0">[,]</span><span class="w" id="mwCv4"> </span><span class="n" id="mwCv8">matrix</span><span class="p" id="mwCwA">,</span><span class="w" id="mwCwE"> </span><span class="kt" id="mwCwI">double</span><span class="p" id="mwCwM">[]</span><span class="w" id="mwCwQ"> </span><span class="n" id="mwCwU">rightPart</span><span class="p" id="mwCwY">,</span><span class="w" id="mwCwc"> </span><span class="kt" id="mwCwg">int</span><span class="w" id="mwCwk"> </span><span class="n" id="mwCwo">n</span><span class="p" id="mwCws">)</span> <span class="w" id="mwCww"> </span><span class="p" id="mwCw0">{</span> <span class="w" id="mwCw4"> </span><span class="c1" id="mwCw8">// decomposition of matrix</span> <span class="w" id="mwCxA"> </span><span class="kt" id="mwCxE">double</span><span class="p" id="mwCxI">[,]</span><span class="w" id="mwCxM"> </span><span class="n" id="mwCxQ">lu</span><span class="w" id="mwCxU"> </span><span class="o" id="mwCxY">=</span><span class="w" id="mwCxc"> </span><span class="k" id="mwCxg">new</span><span class="w" id="mwCxk"> </span><span class="kt" id="mwCxo">double</span><span class="p" id="mwCxs">[</span><span class="n" id="mwCxw">n</span><span class="p" id="mwCx0">,</span><span class="w" id="mwCx4"> </span><span class="n" id="mwCx8">n</span><span class="p" id="mwCyA">];</span> <span class="w" id="mwCyE"> </span><span class="kt" id="mwCyI">double</span><span class="w" id="mwCyM"> </span><span class="n" id="mwCyQ">sum</span><span class="w" id="mwCyU"> </span><span class="o" id="mwCyY">=</span><span class="w" id="mwCyc"> </span><span class="mi" id="mwCyg">0</span><span class="p" id="mwCyk">;</span> <span class="w" id="mwCyo"> </span><span class="k" id="mwCys">for</span><span class="w" id="mwCyw"> </span><span class="p" id="mwCy0">(</span><span class="kt" id="mwCy4">int</span><span class="w" id="mwCy8"> </span><span class="n" id="mwCzA">i</span><span class="w" id="mwCzE"> </span><span class="o" id="mwCzI">=</span><span class="w" id="mwCzM"> </span><span class="mi" id="mwCzQ">0</span><span class="p" id="mwCzU">;</span><span class="w" id="mwCzY"> </span><span class="n" id="mwCzc">i</span><span class="w" id="mwCzg"> </span><span class="o" id="mwCzk"><</span><span class="w" id="mwCzo"> </span><span class="n" id="mwCzs">n</span><span class="p" id="mwCzw">;</span><span class="w" id="mwCz0"> </span><span class="n" id="mwCz4">i</span><span class="o" id="mwCz8">++</span><span class="p" id="mwC0A">)</span> <span class="w" id="mwC0E"> </span><span class="p" id="mwC0I">{</span> <span class="w" id="mwC0M"> </span><span class="k" id="mwC0Q">for</span><span class="w" id="mwC0U"> </span><span class="p" id="mwC0Y">(</span><span class="kt" id="mwC0c">int</span><span class="w" id="mwC0g"> </span><span class="n" id="mwC0k">j</span><span class="w" id="mwC0o"> </span><span class="o" id="mwC0s">=</span><span class="w" id="mwC0w"> </span><span class="n" id="mwC00">i</span><span class="p" id="mwC04">;</span><span class="w" id="mwC08"> </span><span class="n" id="mwC1A">j</span><span class="w" id="mwC1E"> </span><span class="o" id="mwC1I"><</span><span class="w" id="mwC1M"> </span><span class="n" id="mwC1Q">n</span><span class="p" id="mwC1U">;</span><span class="w" id="mwC1Y"> </span><span class="n" id="mwC1c">j</span><span class="o" id="mwC1g">++</span><span class="p" id="mwC1k">)</span> <span class="w" id="mwC1o"> </span><span class="p" id="mwC1s">{</span> <span class="w" id="mwC1w"> </span><span class="n" id="mwC10">sum</span><span class="w" id="mwC14"> </span><span class="o" id="mwC18">=</span><span class="w" id="mwC2A"> </span><span class="mi" id="mwC2E">0</span><span class="p" id="mwC2I">;</span> <span class="w" id="mwC2M"> </span><span class="k" id="mwC2Q">for</span><span class="w" id="mwC2U"> </span><span class="p" id="mwC2Y">(</span><span class="kt" id="mwC2c">int</span><span class="w" id="mwC2g"> </span><span class="n" id="mwC2k">k</span><span class="w" id="mwC2o"> </span><span class="o" id="mwC2s">=</span><span class="w" id="mwC2w"> </span><span class="mi" id="mwC20">0</span><span class="p" id="mwC24">;</span><span class="w" id="mwC28"> </span><span class="n" id="mwC3A">k</span><span class="w" id="mwC3E"> </span><span class="o" id="mwC3I"><</span><span class="w" id="mwC3M"> </span><span class="n" id="mwC3Q">i</span><span class="p" id="mwC3U">;</span><span class="w" id="mwC3Y"> </span><span class="n" id="mwC3c">k</span><span class="o" id="mwC3g">++</span><span class="p" id="mwC3k">)</span> <span class="w" id="mwC3o"> </span><span class="n" id="mwC3s">sum</span><span class="w" id="mwC3w"> </span><span class="o" id="mwC30">+=</span><span class="w" id="mwC34"> </span><span class="n" id="mwC38">lu</span><span class="p" id="mwC4A">[</span><span class="n" id="mwC4E">i</span><span class="p" id="mwC4I">,</span><span class="w" id="mwC4M"> </span><span class="n" id="mwC4Q">k</span><span class="p" id="mwC4U">]</span><span class="w" id="mwC4Y"> </span><span class="o" id="mwC4c">*</span><span class="w" id="mwC4g"> </span><span class="n" id="mwC4k">lu</span><span class="p" id="mwC4o">[</span><span class="n" id="mwC4s">k</span><span class="p" id="mwC4w">,</span><span class="w" id="mwC40"> </span><span class="n" id="mwC44">j</span><span class="p" id="mwC48">];</span> <span class="w" id="mwC5A"> </span><span class="n" id="mwC5E">lu</span><span class="p" id="mwC5I">[</span><span class="n" id="mwC5M">i</span><span class="p" id="mwC5Q">,</span><span class="w" id="mwC5U"> </span><span class="n" id="mwC5Y">j</span><span class="p" id="mwC5c">]</span><span class="w" id="mwC5g"> </span><span class="o" id="mwC5k">=</span><span class="w" id="mwC5o"> </span><span class="n" id="mwC5s">matrix</span><span class="p" id="mwC5w">[</span><span class="n" id="mwC50">i</span><span class="p" id="mwC54">,</span><span class="w" id="mwC58"> </span><span class="n" id="mwC6A">j</span><span class="p" id="mwC6E">]</span><span class="w" id="mwC6I"> </span><span class="o" id="mwC6M">-</span><span class="w" id="mwC6Q"> </span><span class="n" id="mwC6U">sum</span><span class="p" id="mwC6Y">;</span> <span class="w" id="mwC6c"> </span><span class="p" id="mwC6g">}</span> <span class="w" id="mwC6k"> </span><span class="k" id="mwC6o">for</span><span class="w" id="mwC6s"> </span><span class="p" id="mwC6w">(</span><span class="kt" id="mwC60">int</span><span class="w" id="mwC64"> </span><span class="n" id="mwC68">j</span><span class="w" id="mwC7A"> </span><span class="o" id="mwC7E">=</span><span class="w" id="mwC7I"> </span><span class="n" id="mwC7M">i</span><span class="w" id="mwC7Q"> </span><span class="o" id="mwC7U">+</span><span class="w" id="mwC7Y"> </span><span class="mi" id="mwC7c">1</span><span class="p" id="mwC7g">;</span><span class="w" id="mwC7k"> </span><span class="n" id="mwC7o">j</span><span class="w" id="mwC7s"> </span><span class="o" id="mwC7w"><</span><span class="w" id="mwC70"> </span><span class="n" id="mwC74">n</span><span class="p" id="mwC78">;</span><span class="w" id="mwC8A"> </span><span class="n" id="mwC8E">j</span><span class="o" id="mwC8I">++</span><span class="p" id="mwC8M">)</span> <span class="w" id="mwC8Q"> </span><span class="p" id="mwC8U">{</span> <span class="w" id="mwC8Y"> </span><span class="n" id="mwC8c">sum</span><span class="w" id="mwC8g"> </span><span class="o" id="mwC8k">=</span><span class="w" id="mwC8o"> </span><span class="mi" id="mwC8s">0</span><span class="p" id="mwC8w">;</span> <span class="w" id="mwC80"> </span><span class="k" id="mwC84">for</span><span class="w" id="mwC88"> </span><span class="p" id="mwC9A">(</span><span class="kt" id="mwC9E">int</span><span class="w" id="mwC9I"> </span><span class="n" id="mwC9M">k</span><span class="w" id="mwC9Q"> </span><span class="o" id="mwC9U">=</span><span class="w" id="mwC9Y"> </span><span class="mi" id="mwC9c">0</span><span class="p" id="mwC9g">;</span><span class="w" id="mwC9k"> </span><span class="n" id="mwC9o">k</span><span class="w" id="mwC9s"> </span><span class="o" id="mwC9w"><</span><span class="w" id="mwC90"> </span><span class="n" id="mwC94">i</span><span class="p" id="mwC98">;</span><span class="w" id="mwC-A"> </span><span class="n" id="mwC-E">k</span><span class="o" id="mwC-I">++</span><span class="p" id="mwC-M">)</span> <span class="w" id="mwC-Q"> </span><span class="n" id="mwC-U">sum</span><span class="w" id="mwC-Y"> </span><span class="o" id="mwC-c">+=</span><span class="w" id="mwC-g"> </span><span class="n" id="mwC-k">lu</span><span class="p" id="mwC-o">[</span><span class="n" id="mwC-s">j</span><span class="p" id="mwC-w">,</span><span class="w" id="mwC-0"> </span><span class="n" id="mwC-4">k</span><span class="p" id="mwC-8">]</span><span class="w" id="mwC_A"> </span><span class="o" id="mwC_E">*</span><span class="w" id="mwC_I"> </span><span class="n" id="mwC_M">lu</span><span class="p" id="mwC_Q">[</span><span class="n" id="mwC_U">k</span><span class="p" id="mwC_Y">,</span><span class="w" id="mwC_c"> </span><span class="n" id="mwC_g">i</span><span class="p" id="mwC_k">];</span> <span class="w" id="mwC_o"> </span><span class="n" id="mwC_s">lu</span><span class="p" id="mwC_w">[</span><span class="n" id="mwC_0">j</span><span class="p" id="mwC_4">,</span><span class="w" id="mwC_8"> </span><span class="n" id="mwDAA">i</span><span class="p" id="mwDAE">]</span><span class="w" id="mwDAI"> </span><span class="o" id="mwDAM">=</span><span class="w" id="mwDAQ"> </span><span class="p" id="mwDAU">(</span><span class="mi" id="mwDAY">1</span><span class="w" id="mwDAc"> </span><span class="o" id="mwDAg">/</span><span class="w" id="mwDAk"> </span><span class="n" id="mwDAo">lu</span><span class="p" id="mwDAs">[</span><span class="n" id="mwDAw">i</span><span class="p" id="mwDA0">,</span><span class="w" id="mwDA4"> </span><span class="n" id="mwDA8">i</span><span class="p" id="mwDBA">])</span><span class="w" id="mwDBE"> </span><span class="o" id="mwDBI">*</span><span class="w" id="mwDBM"> </span><span class="p" id="mwDBQ">(</span><span class="n" id="mwDBU">matrix</span><span class="p" id="mwDBY">[</span><span class="n" id="mwDBc">j</span><span class="p" id="mwDBg">,</span><span class="w" id="mwDBk"> </span><span class="n" id="mwDBo">i</span><span class="p" id="mwDBs">]</span><span class="w" id="mwDBw"> </span><span class="o" id="mwDB0">-</span><span class="w" id="mwDB4"> </span><span class="n" id="mwDB8">sum</span><span class="p" id="mwDCA">);</span> <span class="w" id="mwDCE"> </span><span class="p" id="mwDCI">}</span> <span class="w" id="mwDCM"> </span><span class="p" id="mwDCQ">}</span> <span class="w" id="mwDCU"> </span><span class="c1" id="mwDCY">// lu = L+U-I</span> <span class="w" id="mwDCc"> </span><span class="c1" id="mwDCg">// find solution of Ly = b</span> <span class="w" id="mwDCk"> </span><span class="kt" id="mwDCo">double</span><span class="p" id="mwDCs">[]</span><span class="w" id="mwDCw"> </span><span class="n" id="mwDC0">y</span><span class="w" id="mwDC4"> </span><span class="o" id="mwDC8">=</span><span class="w" id="mwDDA"> </span><span class="k" id="mwDDE">new</span><span class="w" id="mwDDI"> </span><span class="kt" id="mwDDM">double</span><span class="p" id="mwDDQ">[</span><span class="n" id="mwDDU">n</span><span class="p" id="mwDDY">];</span> <span class="w" id="mwDDc"> </span><span class="k" id="mwDDg">for</span><span class="w" id="mwDDk"> </span><span class="p" id="mwDDo">(</span><span class="kt" id="mwDDs">int</span><span class="w" id="mwDDw"> </span><span class="n" id="mwDD0">i</span><span class="w" id="mwDD4"> </span><span class="o" id="mwDD8">=</span><span class="w" id="mwDEA"> </span><span class="mi" id="mwDEE">0</span><span class="p" id="mwDEI">;</span><span class="w" id="mwDEM"> </span><span class="n" id="mwDEQ">i</span><span class="w" id="mwDEU"> </span><span class="o" id="mwDEY"><</span><span class="w" id="mwDEc"> </span><span class="n" id="mwDEg">n</span><span class="p" id="mwDEk">;</span><span class="w" id="mwDEo"> </span><span class="n" id="mwDEs">i</span><span class="o" id="mwDEw">++</span><span class="p" id="mwDE0">)</span> <span class="w" id="mwDE4"> </span><span class="p" id="mwDE8">{</span> <span class="w" id="mwDFA"> </span><span class="n" id="mwDFE">sum</span><span class="w" id="mwDFI"> </span><span class="o" id="mwDFM">=</span><span class="w" id="mwDFQ"> </span><span class="mi" id="mwDFU">0</span><span class="p" id="mwDFY">;</span> <span class="w" id="mwDFc"> </span><span class="k" id="mwDFg">for</span><span class="w" id="mwDFk"> </span><span class="p" id="mwDFo">(</span><span class="kt" id="mwDFs">int</span><span class="w" id="mwDFw"> </span><span class="n" id="mwDF0">k</span><span class="w" id="mwDF4"> </span><span class="o" id="mwDF8">=</span><span class="w" id="mwDGA"> </span><span class="mi" id="mwDGE">0</span><span class="p" id="mwDGI">;</span><span class="w" id="mwDGM"> </span><span class="n" id="mwDGQ">k</span><span class="w" id="mwDGU"> </span><span class="o" id="mwDGY"><</span><span class="w" id="mwDGc"> </span><span class="n" id="mwDGg">i</span><span class="p" id="mwDGk">;</span><span class="w" id="mwDGo"> </span><span class="n" id="mwDGs">k</span><span class="o" id="mwDGw">++</span><span class="p" id="mwDG0">)</span> <span class="w" id="mwDG4"> </span><span class="n" id="mwDG8">sum</span><span class="w" id="mwDHA"> </span><span class="o" id="mwDHE">+=</span><span class="w" id="mwDHI"> </span><span class="n" id="mwDHM">lu</span><span class="p" id="mwDHQ">[</span><span class="n" id="mwDHU">i</span><span class="p" id="mwDHY">,</span><span class="w" id="mwDHc"> </span><span class="n" id="mwDHg">k</span><span class="p" id="mwDHk">]</span><span class="w" id="mwDHo"> </span><span class="o" id="mwDHs">*</span><span class="w" id="mwDHw"> </span><span class="n" id="mwDH0">y</span><span class="p" id="mwDH4">[</span><span class="n" id="mwDH8">k</span><span class="p" id="mwDIA">];</span> <span class="w" id="mwDIE"> </span><span class="n" id="mwDII">y</span><span class="p" id="mwDIM">[</span><span class="n" id="mwDIQ">i</span><span class="p" id="mwDIU">]</span><span class="w" id="mwDIY"> </span><span class="o" id="mwDIc">=</span><span class="w" id="mwDIg"> </span><span class="n" id="mwDIk">rightPart</span><span class="p" id="mwDIo">[</span><span class="n" id="mwDIs">i</span><span class="p" id="mwDIw">]</span><span class="w" id="mwDI0"> </span><span class="o" id="mwDI4">-</span><span class="w" id="mwDI8"> </span><span class="n" id="mwDJA">sum</span><span class="p" id="mwDJE">;</span> <span class="w" id="mwDJI"> </span><span class="p" id="mwDJM">}</span> <span class="w" id="mwDJQ"> </span><span class="c1" id="mwDJU">// find solution of Ux = y</span> <span class="w" id="mwDJY"> </span><span class="kt" id="mwDJc">double</span><span class="p" id="mwDJg">[]</span><span class="w" id="mwDJk"> </span><span class="n" id="mwDJo">x</span><span class="w" id="mwDJs"> </span><span class="o" id="mwDJw">=</span><span class="w" id="mwDJ0"> </span><span class="k" id="mwDJ4">new</span><span class="w" id="mwDJ8"> </span><span class="kt" id="mwDKA">double</span><span class="p" id="mwDKE">[</span><span class="n" id="mwDKI">n</span><span class="p" id="mwDKM">];</span> <span class="w" id="mwDKQ"> </span><span class="k" id="mwDKU">for</span><span class="w" id="mwDKY"> </span><span class="p" id="mwDKc">(</span><span class="kt" id="mwDKg">int</span><span class="w" id="mwDKk"> </span><span class="n" id="mwDKo">i</span><span class="w" id="mwDKs"> </span><span class="o" id="mwDKw">=</span><span class="w" id="mwDK0"> </span><span class="n" id="mwDK4">n</span><span class="w" id="mwDK8"> </span><span class="o" id="mwDLA">-</span><span class="w" id="mwDLE"> </span><span class="mi" id="mwDLI">1</span><span class="p" id="mwDLM">;</span><span class="w" id="mwDLQ"> </span><span class="n" id="mwDLU">i</span><span class="w" id="mwDLY"> </span><span class="o" id="mwDLc">>=</span><span class="w" id="mwDLg"> </span><span class="mi" id="mwDLk">0</span><span class="p" id="mwDLo">;</span><span class="w" id="mwDLs"> </span><span class="n" id="mwDLw">i</span><span class="o" id="mwDL0">--</span><span class="p" id="mwDL4">)</span> <span class="w" id="mwDL8"> </span><span class="p" id="mwDMA">{</span> <span class="w" id="mwDME"> </span><span class="n" id="mwDMI">sum</span><span class="w" id="mwDMM"> </span><span class="o" id="mwDMQ">=</span><span class="w" id="mwDMU"> </span><span class="mi" id="mwDMY">0</span><span class="p" id="mwDMc">;</span> <span class="w" id="mwDMg"> </span><span class="k" id="mwDMk">for</span><span class="w" id="mwDMo"> </span><span class="p" id="mwDMs">(</span><span class="kt" id="mwDMw">int</span><span class="w" id="mwDM0"> </span><span class="n" id="mwDM4">k</span><span class="w" id="mwDM8"> </span><span class="o" id="mwDNA">=</span><span class="w" id="mwDNE"> </span><span class="n" id="mwDNI">i</span><span class="w" id="mwDNM"> </span><span class="o" id="mwDNQ">+</span><span class="w" id="mwDNU"> </span><span class="mi" id="mwDNY">1</span><span class="p" id="mwDNc">;</span><span class="w" id="mwDNg"> </span><span class="n" id="mwDNk">k</span><span class="w" id="mwDNo"> </span><span class="o" id="mwDNs"><</span><span class="w" id="mwDNw"> </span><span class="n" id="mwDN0">n</span><span class="p" id="mwDN4">;</span><span class="w" id="mwDN8"> </span><span class="n" id="mwDOA">k</span><span class="o" id="mwDOE">++</span><span class="p" id="mwDOI">)</span> <span class="w" id="mwDOM"> </span><span class="n" id="mwDOQ">sum</span><span class="w" id="mwDOU"> </span><span class="o" id="mwDOY">+=</span><span class="w" id="mwDOc"> </span><span class="n" id="mwDOg">lu</span><span class="p" id="mwDOk">[</span><span class="n" id="mwDOo">i</span><span class="p" id="mwDOs">,</span><span class="w" id="mwDOw"> </span><span class="n" id="mwDO0">k</span><span class="p" id="mwDO4">]</span><span class="w" id="mwDO8"> </span><span class="o" id="mwDPA">*</span><span class="w" id="mwDPE"> </span><span class="n" id="mwDPI">x</span><span class="p" id="mwDPM">[</span><span class="n" id="mwDPQ">k</span><span class="p" id="mwDPU">];</span> <span class="w" id="mwDPY"> </span><span class="n" id="mwDPc">x</span><span class="p" id="mwDPg">[</span><span class="n" id="mwDPk">i</span><span class="p" id="mwDPo">]</span><span class="w" id="mwDPs"> </span><span class="o" id="mwDPw">=</span><span class="w" id="mwDP0"> </span><span class="p" id="mwDP4">(</span><span class="mi" id="mwDP8">1</span><span class="w" id="mwDQA"> </span><span class="o" id="mwDQE">/</span><span class="w" id="mwDQI"> </span><span class="n" id="mwDQM">lu</span><span class="p" id="mwDQQ">[</span><span class="n" id="mwDQU">i</span><span class="p" id="mwDQY">,</span><span class="w" id="mwDQc"> </span><span class="n" id="mwDQg">i</span><span class="p" id="mwDQk">])</span><span class="w" id="mwDQo"> </span><span class="o" id="mwDQs">*</span><span class="w" id="mwDQw"> </span><span class="p" id="mwDQ0">(</span><span class="n" id="mwDQ4">y</span><span class="p" id="mwDQ8">[</span><span class="n" id="mwDRA">i</span><span class="p" id="mwDRE">]</span><span class="w" id="mwDRI"> </span><span class="o" id="mwDRM">-</span><span class="w" id="mwDRQ"> </span><span class="n" id="mwDRU">sum</span><span class="p" id="mwDRY">);</span> <span class="w" id="mwDRc"> </span><span class="p" id="mwDRg">}</span> <span class="w" id="mwDRk"> </span><span class="k" id="mwDRo">return</span><span class="w" id="mwDRs"> </span><span class="n" id="mwDRw">x</span><span class="p" id="mwDR0">;</span> <span class="w" id="mwDR4"> </span><span class="p" id="mwDR8">}</span> <span class="p" id="mwDSA">}</span>

MATLAB code example

<span id="mwDSQ"></span><span class="k" id="mwDSU">function</span><span class="w" id="mwDSY"> </span>LU<span class="w" id="mwDSc"> </span><span class="p" id="mwDSg">=</span><span class="w" id="mwDSk"> </span><span class="nf" id="mwDSo">LUDecompDoolittle</span><span class="p" id="mwDSs">(</span>A<span class="p" id="mwDSw">)</span> <span class="w" id="mwDS0"> </span><span class="n" id="mwDS4">n</span><span class="w" id="mwDS8"> </span><span class="p" id="mwDTA">=</span><span class="w" id="mwDTE"> </span><span class="nb" id="mwDTI">length</span><span class="p" id="mwDTM">(</span><span class="n" id="mwDTQ">A</span><span class="p" id="mwDTU">);</span> <span class="w" id="mwDTY"> </span><span class="n" id="mwDTc">LU</span><span class="w" id="mwDTg"> </span><span class="p" id="mwDTk">=</span><span class="w" id="mwDTo"> </span><span class="n" id="mwDTs">A</span><span class="p" id="mwDTw">;</span> <span class="w" id="mwDT0"> </span><span class="k" id="mwDT4">for</span><span class="w" id="mwDT8"> </span><span class="n" id="mwDUA">k</span><span class="w" id="mwDUE"> </span><span class="p" id="mwDUI">=</span><span class="w" id="mwDUM"> </span><span class="mi" id="mwDUQ">2</span><span class="p" id="mwDUU">:</span><span class="n" id="mwDUY">n</span> <span class="w" id="mwDUc"> </span><span class="k" id="mwDUg">for</span><span class="w" id="mwDUk"> </span><span class="nb" id="mwDUo">i</span><span class="w" id="mwDUs"> </span><span class="p" id="mwDUw">=</span><span class="w" id="mwDU0"> </span><span class="mi" id="mwDU4">1</span><span class="w" id="mwDU8"> </span><span class="p" id="mwDVA">:</span><span class="w" id="mwDVE"> </span><span class="n" id="mwDVI">k</span><span class="o" id="mwDVM">-</span><span class="mi" id="mwDVQ">1</span> <span class="w" id="mwDVU"> </span><span class="n" id="mwDVY">lamda</span><span class="w" id="mwDVc"> </span><span class="p" id="mwDVg">=</span><span class="w" id="mwDVk"> </span><span class="n" id="mwDVo">LU</span><span class="p" id="mwDVs">(</span><span class="n" id="mwDVw">k</span><span class="p" id="mwDV0">,</span><span class="nb" id="mwDV4">i</span><span class="p" id="mwDV8">)</span><span class="w" id="mwDWA"> </span><span class="o" id="mwDWE">/</span><span class="w" id="mwDWI"> </span><span class="n" id="mwDWM">LU</span><span class="w" id="mwDWQ"> </span><span class="p" id="mwDWU">(</span><span class="nb" id="mwDWY">i</span><span class="p" id="mwDWc">,</span><span class="w" id="mwDWg"> </span><span class="nb" id="mwDWk">i</span><span class="p" id="mwDWo">);</span> <span class="w" id="mwDWs"> </span><span class="n" id="mwDWw">LU</span><span class="p" id="mwDW0">(</span><span class="n" id="mwDW4">k</span><span class="p" id="mwDW8">,</span><span class="nb" id="mwDXA">i</span><span class="p" id="mwDXE">)</span><span class="w" id="mwDXI"> </span><span class="p" id="mwDXM">=</span><span class="w" id="mwDXQ"> </span><span class="n" id="mwDXU">lamda</span><span class="p" id="mwDXY">;</span> <span class="w" id="mwDXc"> </span><span class="n" id="mwDXg">LU</span><span class="p" id="mwDXk">(</span><span class="n" id="mwDXo">k</span><span class="p" id="mwDXs">,</span><span class="nb" id="mwDXw">i</span><span class="o" id="mwDX0">+</span><span class="mi" id="mwDX4">1</span><span class="p" id="mwDX8">:</span><span class="n" id="mwDYA">n</span><span class="p" id="mwDYE">)</span><span class="w" id="mwDYI"> </span><span class="p" id="mwDYM">=</span><span class="w" id="mwDYQ"> </span><span class="n" id="mwDYU">LU</span><span class="p" id="mwDYY">(</span><span class="n" id="mwDYc">k</span><span class="p" id="mwDYg">,</span><span class="nb" id="mwDYk">i</span><span class="o" id="mwDYo">+</span><span class="mi" id="mwDYs">1</span><span class="p" id="mwDYw">:</span><span class="n" id="mwDY0">n</span><span class="p" id="mwDY4">)</span><span class="w" id="mwDY8"> </span><span class="o" id="mwDZA">-</span><span class="w" id="mwDZE"> </span><span class="n" id="mwDZI">LU</span><span class="p" id="mwDZM">(</span><span class="nb" id="mwDZQ">i</span><span class="p" id="mwDZU">,</span><span class="nb" id="mwDZY">i</span><span class="o" id="mwDZc">+</span><span class="mi" id="mwDZg">1</span><span class="p" id="mwDZk">:</span><span class="n" id="mwDZo">n</span><span class="p" id="mwDZs">)</span><span class="w" id="mwDZw"> </span><span class="o" id="mwDZ0">*</span><span class="w" id="mwDZ4"> </span><span class="n" id="mwDZ8">lamda</span><span class="p" id="mwDaA">;</span> <span class="w" id="mwDaE"> </span><span class="k" id="mwDaI">end</span> <span class="w" id="mwDaM"> </span><span class="k" id="mwDaQ">end</span> <span class="k" id="mwDaU">end</span> <span class="k" id="mwDaY">function</span><span class="w" id="mwDac"> </span>x<span class="w" id="mwDag"> </span><span class="p" id="mwDak">=</span><span class="w" id="mwDao"> </span><span class="nf" id="mwDas">SolveLinearSystem</span><span class="p" id="mwDaw">(</span>LU, B<span class="p" id="mwDa0">)</span> <span class="w" id="mwDa4"> </span><span class="n" id="mwDa8">n</span><span class="w" id="mwDbA"> </span><span class="p" id="mwDbE">=</span><span class="w" id="mwDbI"> </span><span class="nb" id="mwDbM">length</span><span class="p" id="mwDbQ">(</span><span class="n" id="mwDbU">LU</span><span class="p" id="mwDbY">);</span> <span class="w" id="mwDbc"> </span><span class="n" id="mwDbg">y</span><span class="w" id="mwDbk"> </span><span class="p" id="mwDbo">=</span><span class="w" id="mwDbs"> </span><span class="nb" id="mwDbw">zeros</span><span class="p" id="mwDb0">(</span><span class="nb" id="mwDb4">size</span><span class="p" id="mwDb8">(</span><span class="n" id="mwDcA">B</span><span class="p" id="mwDcE">));</span> <span class="w" id="mwDcI"> </span><span class="c" id="mwDcM">% find solution of Ly = B</span> <span class="w" id="mwDcQ"> </span><span class="k" id="mwDcU">for</span><span class="w" id="mwDcY"> </span><span class="nb" id="mwDcc">i</span><span class="w" id="mwDcg"> </span><span class="p" id="mwDck">=</span><span class="w" id="mwDco"> </span><span class="mi" id="mwDcs">1</span><span class="p" id="mwDcw">:</span><span class="n" id="mwDc0">n</span> <span class="w" id="mwDc4"> </span><span class="n" id="mwDc8">y</span><span class="p" id="mwDdA">(</span><span class="nb" id="mwDdE">i</span><span class="p" id="mwDdI">,:)</span><span class="w" id="mwDdM"> </span><span class="p" id="mwDdQ">=</span><span class="w" id="mwDdU"> </span><span class="n" id="mwDdY">B</span><span class="p" id="mwDdc">(</span><span class="nb" id="mwDdg">i</span><span class="p" id="mwDdk">,:)</span><span class="w" id="mwDdo"> </span><span class="o" id="mwDds">-</span><span class="w" id="mwDdw"> </span><span class="n" id="mwDd0">LU</span><span class="p" id="mwDd4">(</span><span class="nb" id="mwDd8">i</span><span class="p" id="mwDeA">,</span><span class="mi" id="mwDeE">1</span><span class="p" id="mwDeI">:</span><span class="nb" id="mwDeM">i</span><span class="p" id="mwDeQ">)</span><span class="o" id="mwDeU">*</span><span class="n" id="mwDeY">y</span><span class="p" id="mwDec">(</span><span class="mi" id="mwDeg">1</span><span class="p" id="mwDek">:</span><span class="nb" id="mwDeo">i</span><span class="p" id="mwDes">,:);</span> <span class="w" id="mwDew"> </span><span class="k" id="mwDe0">end</span> <span class="w" id="mwDe4"> </span><span class="c" id="mwDe8">% find solution of Ux = y</span> <span class="w" id="mwDfA"> </span><span class="n" id="mwDfE">x</span><span class="w" id="mwDfI"> </span><span class="p" id="mwDfM">=</span><span class="w" id="mwDfQ"> </span><span class="nb" id="mwDfU">zeros</span><span class="p" id="mwDfY">(</span><span class="nb" id="mwDfc">size</span><span class="p" id="mwDfg">(</span><span class="n" id="mwDfk">B</span><span class="p" id="mwDfo">));</span> <span class="w" id="mwDfs"> </span><span class="k" id="mwDfw">for</span><span class="w" id="mwDf0"> </span><span class="nb" id="mwDf4">i</span><span class="w" id="mwDf8"> </span><span class="p" id="mwDgA">=</span><span class="w" id="mwDgE"> </span><span class="n" id="mwDgI">n</span><span class="p" id="mwDgM">:(</span><span class="o" id="mwDgQ">-</span><span class="mi" id="mwDgU">1</span><span class="p" id="mwDgY">):</span><span class="mi" id="mwDgc">1</span> <span class="w" id="mwDgg"> </span><span class="n" id="mwDgk">x</span><span class="p" id="mwDgo">(</span><span class="nb" id="mwDgs">i</span><span class="p" id="mwDgw">,:)</span><span class="w" id="mwDg0"> </span><span class="p" id="mwDg4">=</span><span class="w" id="mwDg8"> </span><span class="p" id="mwDhA">(</span><span class="n" id="mwDhE">y</span><span class="p" id="mwDhI">(</span><span class="nb" id="mwDhM">i</span><span class="p" id="mwDhQ">,:)</span><span class="w" id="mwDhU"> </span><span class="o" id="mwDhY">-</span><span class="w" id="mwDhc"> </span><span class="n" id="mwDhg">LU</span><span class="p" id="mwDhk">(</span><span class="nb" id="mwDho">i</span><span class="p" id="mwDhs">,(</span><span class="nb" id="mwDhw">i</span><span class="w" id="mwDh0"> </span><span class="o" id="mwDh4">+</span><span class="w" id="mwDh8"> </span><span class="mi" id="mwDiA">1</span><span class="p" id="mwDiE">):</span><span class="n" id="mwDiI">n</span><span class="p" id="mwDiM">)</span><span class="o" id="mwDiQ">*</span><span class="n" id="mwDiU">x</span><span class="p" id="mwDiY">((</span><span class="nb" id="mwDic">i</span><span class="w" id="mwDig"> </span><span class="o" id="mwDik">+</span><span class="w" id="mwDio"> </span><span class="mi" id="mwDis">1</span><span class="p" id="mwDiw">):</span><span class="n" id="mwDi0">n</span><span class="p" id="mwDi4">,:))</span><span class="o" id="mwDi8">/</span><span class="n" id="mwDjA">LU</span><span class="p" id="mwDjE">(</span><span class="nb" id="mwDjI">i</span><span class="p" id="mwDjM">,</span><span class="w" id="mwDjQ"> </span><span class="nb" id="mwDjU">i</span><span class="p" id="mwDjY">);</span> <span class="w" id="mwDjc"> </span><span class="k" id="mwDjg">end</span><span class="w" id="mwDjk"> </span> <span class="k" id="mwDjo">end</span> <span class="n" id="mwDjs">A</span><span class="w" id="mwDjw"> </span><span class="p" id="mwDj0">=</span><span class="w" id="mwDj4"> </span><span class="p" id="mwDj8">[</span><span class="w" id="mwDkA"> </span><span class="mi" id="mwDkE">4</span><span class="w" id="mwDkI"> </span><span class="mi" id="mwDkM">3</span><span class="w" id="mwDkQ"> </span><span class="mi" id="mwDkU">3</span><span class="p" id="mwDkY">;</span><span class="w" id="mwDkc"> </span><span class="n" id="mwDkg">6</span><span class="w" id="mwDkk"> </span><span class="s" id="mwDko">3</span><span class="w" id="mwDks"> </span><span class="s" id="mwDkw">3</span><span class="p" id="mwDk0">;</span><span class="w" id="mwDk4"> </span><span class="n" id="mwDk8">3</span><span class="w" id="mwDlA"> </span><span class="s" id="mwDlE">4</span><span class="w" id="mwDlI"> </span><span class="s" id="mwDlM">3</span><span class="w" id="mwDlQ"> </span><span class="s" id="mwDlU">]</span> <span class="n" id="mwDlY">LU</span><span class="w" id="mwDlc"> </span><span class="p" id="mwDlg">=</span><span class="w" id="mwDlk"> </span><span class="n" id="mwDlo">LUDecompDoolittle</span><span class="p" id="mwDls">(</span><span class="n" id="mwDlw">A</span><span class="p" id="mwDl0">)</span> <span class="n" id="mwDl4">B</span><span class="w" id="mwDl8"> </span><span class="p" id="mwDmA">=</span><span class="w" id="mwDmE"> </span><span class="p" id="mwDmI">[</span><span class="w" id="mwDmM"> </span><span class="mi" id="mwDmQ">1</span><span class="w" id="mwDmU"> </span><span class="mi" id="mwDmY">2</span><span class="w" id="mwDmc"> </span><span class="mi" id="mwDmg">3</span><span class="p" id="mwDmk">;</span><span class="w" id="mwDmo"> </span><span class="n" id="mwDms">4</span><span class="w" id="mwDmw"> </span><span class="s" id="mwDm0">5</span><span class="w" id="mwDm4"> </span><span class="s" id="mwDm8">6</span><span class="p" id="mwDnA">;</span><span class="w" id="mwDnE"> </span><span class="n" id="mwDnI">7</span><span class="w" id="mwDnM"> </span><span class="s" id="mwDnQ">8</span><span class="w" id="mwDnU"> </span><span class="s" id="mwDnY">9</span><span class="p" id="mwDnc">;</span><span class="w" id="mwDng"> </span><span class="n" id="mwDnk">10</span><span class="w" id="mwDno"> </span><span class="s" id="mwDns">11</span><span class="w" id="mwDnw"> </span><span class="s" id="mwDn0">12</span><span class="w" id="mwDn4"> </span><span class="s" id="mwDn8">]</span><span class="o" id="mwDoA">'</span> <span class="n" id="mwDoE">x</span><span class="w" id="mwDoI"> </span><span class="p" id="mwDoM">=</span><span class="w" id="mwDoQ"> </span><span class="n" id="mwDoU">SolveLinearSystem</span><span class="p" id="mwDoY">(</span><span class="n" id="mwDoc">LU</span><span class="p" id="mwDog">,</span><span class="w" id="mwDok"> </span><span class="n" id="mwDoo">B</span><span class="p" id="mwDos">)</span> <span class="n" id="mwDow">A</span><span class="w" id="mwDo0"> </span><span class="o" id="mwDo4">*</span><span class="w" id="mwDo8"> </span><span class="n" id="mwDpA">x</span>