nag_zposvx (f07fpc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zposvx (f07fpc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zposvx (f07fpc) uses the Cholesky factorization
A=UHU   or   A=LLH
to compute the solution to a complex system of linear equations
AX=B ,
where A is an n by n Hermitian positive definite matrix and X and B are n by r matrices. Error bounds on the solution and a condition estimate are also provided.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zposvx (Nag_OrderType order, Nag_FactoredFormType fact, Nag_UploType uplo, Integer n, Integer nrhs, Complex a[], Integer pda, Complex af[], Integer pdaf, Nag_EquilibrationType *equed, double s[], Complex b[], Integer pdb, Complex x[], Integer pdx, double *rcond, double ferr[], double berr[], NagError *fail)

3  Description

nag_zposvx (f07fpc) performs the following steps:
  1. If fact=Nag_EquilibrateAndFactor, real diagonal scaling factors, DS , are computed to equilibrate the system:
    DS A DS DS-1 X = DS B .
    Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by DS A DS  and B by DS B.
  2. If fact=Nag_NotFactored or Nag_EquilibrateAndFactor, the Cholesky decomposition is used to factor the matrix A (after equilibration if fact=Nag_EquilibrateAndFactor) as A=UHU if uplo=Nag_Upper or A=LLH if uplo=Nag_Lower, where U is an upper triangular matrix and L is a lower triangular matrix.
  3. If the leading i by i principal minor of A is not positive definite, then the function returns with fail.errnum=i and fail.code= NE_MAT_NOT_POS_DEF. Otherwise, the factored 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, fail.code= NE_SINGULAR_WP is returned as a warning, but the function still goes on to solve for X and compute error bounds as described below.
  4. The system of equations is solved for X using the factored form of A.
  5. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
  6. If equilibration was used, the matrix X is premultiplied by DS  so that it solves the original system before equilibration.

4  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

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     factNag_FactoredFormTypeInput
On entry: 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=Nag_Factored
af contains the factorized form of A. If equed=Nag_Equilibrated, the matrix A has been equilibrated with scaling factors given by s. a and af will not be modified.
fact=Nag_NotFactored
The matrix A will be copied to af and factorized.
fact=Nag_EquilibrateAndFactor
The matrix A will be equilibrated if necessary, then copied to af and factorized.
Constraint: fact=Nag_Factored, Nag_NotFactored or Nag_EquilibrateAndFactor.
3:     uploNag_UploTypeInput
On entry: if uplo=Nag_Upper, the upper triangle of A is stored.
If uplo=Nag_Lower, the lower triangle of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     nIntegerInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
5:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
6:     a[dim]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: the n by n Hermitian matrix A.
If fact=Nag_Factored and equed=Nag_Equilibrated, a must have been equilibrated by the scaling factor in s as DSADS.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: if fact=Nag_Factored or Nag_NotFactored, or if fact=Nag_EquilibrateAndFactor and equed=Nag_NoEquilibration, a is not modified.
If fact=Nag_EquilibrateAndFactor and equed=Nag_Equilibrated, a is overwritten by DSADS.
7:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax1,n.
8:     af[dim]ComplexInput/Output
Note: the dimension, dim, of the array af must be at least max1,pdaf×n.
The i,jth element of the matrix is stored in
  • af[j-1×pdaf+i-1] when order=Nag_ColMajor;
  • af[i-1×pdaf+j-1] when order=Nag_RowMajor.
On entry: if fact=Nag_Factored, af contains the triangular factor U or L from the Cholesky factorization A=UHU or A=LLH, in the same storage format as a. If equedNag_NoEquilibration, af is the factorized form of the equilibrated matrix DSADS.
On exit: if fact=Nag_NotFactored, af returns the triangular factor U or L from the Cholesky factorization A=UHU or A=LLH of the original matrix A.
If fact=Nag_EquilibrateAndFactor, af returns the triangular factor U or L from the Cholesky factorization A=UHU or A=LLH of the equilibrated matrix A (see the description of a for the form of the equilibrated matrix).
9:     pdafIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array af.
Constraint: pdafmax1,n.
10:   equedNag_EquilibrationType*Input/Output
On entry: if fact=Nag_NotFactored or Nag_EquilibrateAndFactor, equed need not be set.
If fact=Nag_Factored, equed must specify the form of the equilibration that was performed as follows:
  • if equed=Nag_NoEquilibration, no equilibration;
  • if equed=Nag_Equilibrated, equilibration was performed, i.e., A has been replaced by DSADS.
On exit: if fact=Nag_Factored, equed is unchanged from entry.
Otherwise, if no constraints are violated, equed specifies the form of the equilibration that was performed as specified above.
Constraint: if fact=Nag_Factored, equed=Nag_NoEquilibration or Nag_Equilibrated.
11:   s[dim]doubleInput/Output
Note: the dimension, dim, of the array s must be at least max1,n.
On entry: if fact=Nag_NotFactored or Nag_EquilibrateAndFactor, s need not be set.
If fact=Nag_Factored and equed=Nag_Equilibrated, s must contain the scale factors, DS, for A; each element of s must be positive.
On exit: if fact=Nag_Factored, s is unchanged from entry.
Otherwise, if no constraints are violated and equed=Nag_Equilibrated, s contains the scale factors, DS, for A; each element of s is positive.
12:   b[dim]ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the n by r right-hand side matrix B.
On exit: if equed=Nag_NoEquilibration, b is not modified.
If equed=Nag_Equilibrated, b is overwritten by DSB.
13:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
14:   x[dim]ComplexOutput
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×nrhs when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
The i,jth element of the matrix X is stored in
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=Nag_RowMajor.
On exit: if fail.code= NE_NOERROR or NE_SINGULAR_WP, 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=Nag_Equilibrated, and the solution to the equilibrated system is DS-1X.
15:   pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxmax1,n;
  • if order=Nag_RowMajor, pdxmax1,nrhs.
16:   rconddouble *Output
On exit: 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 .
17:   ferr[nrhs]doubleOutput
On exit: if fail.code= NE_NOERROR or NE_SINGULAR_WP, an estimate of the forward error bound for each computed solution vector, such that x^j-xj/xjferr[j-1] 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.
18:   berr[nrhs]doubleOutput
On exit: if fail.code= NE_NOERROR or NE_SINGULAR_WP, 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).
19:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdaf=value.
Constraint: pdaf>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdaf=value and n=value.
Constraint: pdafmax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
On entry, pdx=value and n=value.
Constraint: pdxmax1,n.
On entry, pdx=value and nrhs=value.
Constraint: pdxmax1,nrhs.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_MAT_NOT_POS_DEF
The leading minor of order value of A is not positive definite, so the factorization could not be completed, and the solution has not been computed. rcond=0.0 is returned.
NE_SINGULAR_WP
U (or L) 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.

7  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 cn is a modest linear function of n, and ε is the machine precision. See Section 10.1 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 berr[j-1]  and a bound on x - x^ / x^  is returned in ferr[j-1] . See Section 4.4 of Anderson et al. (1999) for further details.

8  Further Comments

The factorization of A  requires approximately 43 n3  floating point operations.
For each right-hand side, computation of the backward error involves a minimum of 16n2  floating point operations. Each step of iterative refinement involves an additional 24n2  operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form Ax=b ; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 8n2  operations.
The real analogue of this function is nag_dposvx (f07fbc).

9  Example

This example solves the equations
AX=B ,
where A  is the Hermitian positive definite matrix
A = 3.23i+0.00 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58i+0.00 -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09i+0.00 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29i+0.00
and
B = 3.93-06.14i 1.48+06.58i 6.17+09.42i 4.64-04.75i -7.17-21.83i -4.91+02.29i 1.99-14.38i 7.64-10.79i .
Error estimates for the solutions, information on equilibration and an estimate of the reciprocal of the condition number of the scaled matrix A  are also output.

9.1  Program Text

Program Text (f07fpce.c)

9.2  Program Data

Program Data (f07fpce.d)

9.3  Program Results

Program Results (f07fpce.r)


nag_zposvx (f07fpc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012