nag_zspsvx (f07qpc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zspsvx (f07qpc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zspsvx (f07qpc) uses the diagonal pivoting factorization
A=UDUT   or   A=LDLT
to compute the solution to a complex system of linear equations
AX=B ,
where A is an n by n symmetric matrix stored in packed format 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_zspsvx (Nag_OrderType order, Nag_FactoredFormType fact, Nag_UploType uplo, Integer n, Integer nrhs, const Complex ap[], Complex afp[], Integer ipiv[], const Complex b[], Integer pdb, Complex x[], Integer pdx, double *rcond, double ferr[], double berr[], NagError *fail)

3  Description

nag_zspsvx (f07qpc) performs the following steps:
  1. If fact=Nag_NotFactored, the diagonal pivoting method is used to factor A as A=UDUT if uplo=Nag_Upper or A=LDLT if uplo=Nag_Lower, where U (or L) is a product of permutation and unit upper (lower) triangular matrices and D is symmetric and block diagonal with 1 by 1 and 2 by 2 diagonal blocks.
  2. If some dii=0, so that D is exactly singular, then the function returns with fail.errnum=i and fail.code= NE_SINGULAR. 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.
  3. The system of equations is solved for X using the factored form of A.
  4. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

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 has been supplied.
fact=Nag_Factored
afp and ipiv contain the factorized form of the matrix A. afp and ipiv will not be modified.
fact=Nag_NotFactored
The matrix A will be copied to afp and factorized.
Constraint: fact=Nag_Factored or Nag_NotFactored.
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:     ap[dim]const ComplexInput
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the n by n symmetric matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ap[j-1×j/2+i-1], for ij;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ap[i-1×i/2+j-1], for ij.
7:     afp[dim]ComplexInput/Output
Note: the dimension, dim, of the array afp must be at least max1,n×n+1/2.
On entry: if fact=Nag_Factored, afp contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A=UDUT or A=LDLT as computed by nag_zsptrf (f07qrc), stored as a packed triangular matrix in the same storage format as A.
On exit: if fact=Nag_NotFactored, afp contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A=UDUT or A=LDLT as computed by nag_zsptrf (f07qrc), stored as a packed triangular matrix in the same storage format as A.
8:     ipiv[n]IntegerInput/Output
On exit: details of the interchanges and the block structure of D. More precisely,
  • if ipiv[i-1]=k>0, dii is a 1 by 1 pivot block and the ith row and column of A were interchanged with the kth row and column;
  • if uplo=Nag_Upper and ipiv[i-2]=ipiv[i-1]=-l<0, di-1,i-1d-i,i-1 d-i,i-1dii  is a 2 by 2 pivot block and the i-1th row and column of A were interchanged with the lth row and column;
  • if uplo=Nag_Lower and ipiv[i-1]=ipiv[i]=-m<0, diidi+1,idi+1,idi+1,i+1 is a 2 by 2 pivot block and the i+1th row and column of A were interchanged with the mth row and column.
9:     b[dim]const ComplexInput
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.
10:   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.
11:   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.
12:   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.
13:   rconddouble *Output
On exit: the estimate of the reciprocal condition number of the matrix A. If rcond=0.0, the matrix may be exactly singular. This condition is indicated by fail.code= NE_SINGULAR. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by fail.code= NE_SINGULAR_WP.
14:   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.
15:   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).
16:   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, pdb=value.
Constraint: pdb>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
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_SINGULAR
Dvalue,value is exactly zero. The factorization has been completed, but the factor D is exactly singular, so the solution and error bounds could not be computed. rcond=0.0 is returned.
NE_SINGULAR_WP
D 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
E1 = Oε A1 ,
where ε is the machine precision. See Chapter 11 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_dspsvx (f07pbc). The complex Hermitian analogue of this function is nag_zhpsvx (f07ppc).

9  Example

This example solves the equations
AX=B ,
where A  is the complex symmetric matrix
A = -0.56+0.12i -1.54-2.86i 5.32-1.59i 3.80+0.92i -1.54-2.86i -2.83-0.03i -3.52+0.58i -7.86-2.96i 5.32-1.59i -3.52+0.58i 8.86+1.81i 5.14-0.64i 3.80+0.92i -7.86-2.96i 5.14-0.64i -0.39-0.71i
and
B = -6.43+19.24i -4.59-35.53i -0.49-01.47i 6.95+20.49i -48.18+66.00i -12.08-27.02i -55.64+41.22i -19.09-35.97i .
Error estimates for the solutions, and an estimate of the reciprocal of the condition number of the matrix A  are also output.

9.1  Program Text

Program Text (f07qpce.c)

9.2  Program Data

Program Data (f07qpce.d)

9.3  Program Results

Program Results (f07qpce.r)


nag_zspsvx (f07qpc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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