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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgesvx (f07ap)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgesvx (f07ap) uses the LU factorization to compute the solution to a complex system of linear equations
AX=B   or   ATX=B   or   AHX=B ,  
where A is an n by n matrix and X and B are n by r matrices. Error bounds on the solution and a condition estimate are also provided.

Syntax

[a, af, ipiv, equed, r, c, b, x, rcond, ferr, berr, rwork, info] = f07ap(fact, trans, a, af, ipiv, equed, r, c, b, 'n', n, 'nrhs_p', nrhs_p)
[a, af, ipiv, equed, r, c, b, x, rcond, ferr, berr, rwork, info] = nag_lapack_zgesvx(fact, trans, a, af, ipiv, equed, r, c, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zgesvx (f07ap) performs the following steps:
1. Equilibration
The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting fact='E'. In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems AX=B , ATX=B  and AHX=B  are
DR A DC DC-1X = DR B ,  
DR A DC T DR-1 X = DC B ,  
and
DR A DC H DR-1 X = DC B ,  
respectively, where DR  and DC  are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.
When equilibration is used, A will be overwritten by DR A DC  and B will be overwritten by DR B  (or DC B  when the solution of ATX=B  or AHX=B  is sought).
2. Factorization
The matrix A, or its scaled form, is copied and factored using the LU decomposition
A=PLU ,  
where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular.
This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to nag_lapack_zgesvx (f07ap) with the same matrix A.
3. Condition Number Estimation
The LU factorization of A determines whether a solution to the linear system exists. If some diagonal element of U is zero, then U is exactly singular, no solution exists and the function returns with a failure. Otherwise the factorized form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision then a warning code is returned on final exit.
4. Solution
The (equilibrated) system is solved for X ( DC-1X  or DR-1X ) using the factored form of A ( DRADC ).
5. Iterative Refinement
Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution.
6. Construct Solution Matrix X
If equilibration was used, the matrix X is premultiplied by DC  (if trans='N') or DR  (if trans='T' or 'C') so that it solves the original system before equilibration.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     fact – string (length ≥ 1)
Specifies whether or not the factorized form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factorized.
fact='F'
af and ipiv contain the factorized form of A. If equed'N', the matrix A has been equilibrated with scaling factors given by r and c. a, af and ipiv are not modified.
fact='N'
The matrix A will be copied to af and factorized.
fact='E'
The matrix A will be equilibrated if necessary, then copied to af and factorized.
Constraint: fact='F', 'N' or 'E'.
2:     trans – string (length ≥ 1)
Specifies the form of the system of equations.
trans='N'
AX=B (No transpose).
trans='T'
ATX=B (Transpose).
trans='C'
AHX=B (Conjugate transpose).
Constraint: trans='N', 'T' or 'C'.
3:     alda: – complex array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The n by n matrix A.
If fact='F' and equed'N', a must have been equilibrated by the scaling factors in r and/or c.
4:     afldaf: – complex array
The first dimension of the array af must be at least max1,n.
The second dimension of the array af must be at least max1,n.
If fact='F', af contains the factors L and U from the factorization A=PLU as computed by nag_lapack_zgetrf (f07ar). If equed'N', af is the factorized form of the equilibrated matrix A.
If fact='N' or 'E', af need not be set.
5:     ipiv: int64int32nag_int array
The dimension of the array ipiv must be at least max1,n
If fact='F', ipiv contains the pivot indices from the factorization A=PLU as computed by nag_lapack_zgetrf (f07ar); at the ith step row i of the matrix was interchanged with row ipivi. ipivi=i indicates a row interchange was not required.
If fact='N' or 'E', ipiv need not be set.
6:     equed – string (length ≥ 1)
If fact='N' or 'E', equed need not be set.
If fact='F', equed must specify the form of the equilibration that was performed as follows:
  • if equed='N', no equilibration;
  • if equed='R', row equilibration, i.e., A has been premultiplied by DR;
  • if equed='C', column equilibration, i.e., A has been postmultiplied by DC;
  • if equed='B', both row and column equilibration, i.e., A has been replaced by DRADC.
Constraint: if fact='F', equed='N', 'R', 'C' or 'B'.
7:     r: – double array
The dimension of the array r must be at least max1,n
If fact='N' or 'E', r need not be set.
If fact='F' and equed='R' or 'B', r must contain the row scale factors for A, DR; each element of r must be positive.
8:     c: – double array
The dimension of the array c must be at least max1,n
If fact='N' or 'E', c need not be set.
If fact='F' or equed='C' or 'B', c must contain the column scale factors for A, DC; each element of c must be positive.
9:     bldb: – complex array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,nrhs_p.
The n by r right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays a, af, b and the second dimension of the arrays a, af, ipiv, r, c.
n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs_p0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
If fact='F' or 'N', or if fact='E' and equed='N', a is not modified.
If fact='E' or equed'N', A is scaled as follows:
  • if equed='R', A=DRA;
  • if equed='C', A=ADC;
  • if equed='B', A=DRADC.
2:     afldaf: – complex array
The first dimension of the array af will be max1,n.
The second dimension of the array af will be max1,n.
If fact='N', af returns the factors L and U from the factorization A=PLU of the original matrix A.
If fact='E', af returns the factors L and U from the factorization A=PLU of the equilibrated matrix A (see the description of a for the form of the equilibrated matrix).
If fact='F', af is unchanged from entry.
3:     ipiv: int64int32nag_int array
The dimension of the array ipiv will be max1,n
If fact='N', ipiv contains the pivot indices from the factorization A=PLU of the original matrix A.
If fact='E', ipiv contains the pivot indices from the factorization A=PLU of the equilibrated matrix A.
If fact='F', ipiv is unchanged from entry.
4:     equed – string (length ≥ 1)
If fact='F', equed is unchanged from entry.
Otherwise, if no constraints are violated, equed specifies the form of equilibration that was performed as specified above.
5:     r: – double array
The dimension of the array r will be max1,n
If fact='F', r is unchanged from entry.
Otherwise, if no constraints are violated and equed='R' or 'B', r contains the row scale factors for A, DR, such that A is multiplied on the left by DR; each element of r is positive.
6:     c: – double array
The dimension of the array c will be max1,n
If fact='F', c is unchanged from entry.
Otherwise, if no constraints are violated and equed='C' or 'B', c contains the row scale factors for A, DC; each element of c is positive.
7:     bldb: – complex array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,nrhs_p.
If equed='N', b is not modified.
If trans='N' and equed='R' or 'B', b stores DRB.
If trans='T' or 'C' and equed='C' or 'B', b stores DCB.
8:     xldx: – complex array
The first dimension of the array x will be max1,n.
The second dimension of the array x will be max1,nrhs_p.
If info=0 or n+1, the n by r solution matrix X to the original system of equations. Note that the arrays A and B are modified on exit if equed'N', and the solution to the equilibrated system is DC-1X if trans='N' and equed='C' or 'B', or DR-1X if trans='T' or 'C' and equed='R' or 'B'.
9:     rcond – double scalar
If no constraints are violated, an estimate of the reciprocal condition number of the matrix A (after equilibration if that is performed), computed as rcond=1.0/A1 A-11 .
10:   ferrnrhs_p – double array
If info=0 or n+1, an estimate of the forward error bound for each computed solution vector, such that x^j-xj/xjferrj where x^j is the jth column of the computed solution returned in the array x and xj is the corresponding column of the exact solution X. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
11:   berrnrhs_p – double array
If info=0 or n+1, an estimate of the component-wise relative backward error of each computed solution vector x^j (i.e., the smallest relative change in any element of A or B that makes x^j an exact solution).
12:   rworkmax1,2×n – double array
rwork1 contains the reciprocal pivot growth factor A/U. The ‘max absolute element’ norm is used. If rwork1 is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution x, condition estimator rcond, and forward error bound ferr could be unreliable. If factorization fails with info>0andinfon, then rwork1 contains the reciprocal pivot growth factor for the leading info columns of A.
13:   info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0andinfon
Element _ of the diagonal is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. rcond=0.0 is returned.
W  info=n+1
U is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

Accuracy

For each right-hand side vector b, the computed solution x^ is the exact solution of a perturbed system of equations A+Ex^=b, where
EcnεPLU ,  
cn is a modest linear function of n, and ε is the machine precision. See Section 9.3 of Higham (2002) for further details.
If x is the true solution, then the computed solution x^ satisfies a forward error bound of the form
x-x^ x^ wc condA,x^,b  
where condA,x^,b = A-1 A x^ + b / x^ condA = A-1 A κ A. If x^  is the j th column of X , then wc  is returned in berrj  and a bound on x - x^ / x^  is returned in ferrj . See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The factorization of A  requires approximately 83 n3  floating-point operations.
Estimating the forward error involves solving a number of systems of linear equations of the form Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 8n2 operations.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The real analogue of this function is nag_lapack_dgesvx (f07ab).

Example

This example solves the equations
AX=B ,  
where A is the general matrix
A= -1.34+02.55i 0.28+3.17i -6.39-02.20i 0.72-00.92i -1.70-14.10i 33.10-1.50i -1.50+13.40i 12.90+13.80i -3.29-02.39i -1.91+4.42i -0.14-01.35i 1.72+01.35i 2.41+00.39i -0.56+1.47i -0.83-00.69i -1.96+00.67i  
and
B= 26.26+51.78i 31.32-06.70i 64.30-86.80i 158.60-14.20i -5.75+25.31i -2.15+30.19i 1.16+02.57i -2.56+07.55i .  
Error estimates for the solutions, information on scaling, an estimate of the reciprocal of the condition number of the scaled matrix A and an estimate of the reciprocal of the pivot growth factor for the factorization of A are also output.
function f07ap_example


fprintf('f07ap example results\n\n');

a = [ -1.34 +  2.55i,  0.28 + 3.17i,  -6.39 -  2.20i,  0.72 -  0.92i;
      -1.70 - 14.10i, 33.10 - 1.50i,  -1.50 + 13.40i, 12.90 + 13.80i;
      -3.29 -  2.39i, -1.91 + 4.42i,  -0.14 -  1.35i,  1.72 +  1.35i;
       2.41 +  0.39i, -0.56 + 1.47i,  -0.83 -  0.69i, -1.96 +  0.67i];
b = [ 26.26 + 51.78i,  31.32 -  6.70i;
      64.30 - 86.80i, 158.60 - 14.20i;
      -5.75 + 25.31i,  -2.15 + 30.19i;
       1.16 +  2.57i,  -2.56 +  7.55i];

n = size(a,1);

% Input parameter initialization
fact  = 'Equilibration';
trans = 'No transpose';
equed = 'N';
af   = a;
ipiv = zeros(n, 1, 'int64');
r    = zeros(n, 1);
c    = zeros(n, 1);

% Solve
[a, af, ipiv, equed, r, c, b, x, rcond, ferr, berr, rwork, info] = ...
  f07ap( ...
         fact, trans, a, af, ipiv, equed, r, c, b);

fprintf('Solution is x:\n');
disp(x);
fprintf('\nApproximate condition number = %8.3f\n',1/rcond);
fprintf('Approximate forward  errors  :\n');
fprintf('                               %11.3e\n',ferr);
fprintf('Approximate backward errors  :\n')
fprintf('                               %11.3e\n',berr);


f07ap example results

Solution is x:
   1.0000 + 1.0000i  -1.0000 - 2.0000i
   2.0000 - 3.0000i   5.0000 + 1.0000i
  -4.0000 - 5.0000i  -3.0000 + 4.0000i
   0.0000 + 6.0000i   2.0000 - 3.0000i


Approximate condition number =   96.228
Approximate forward  errors  :
                                 5.886e-14
                                 7.672e-14
Approximate backward errors  :
                                 8.022e-17
                                 1.100e-16

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