NAG CL Interface
f07qpc (zspsvx)

1 Purpose

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>
void  f07qpc (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)
The function may be called by the names: f07qpc, nag_lapacklin_zspsvx or nag_zspsvx.

3 Description

f07qpc performs the following steps:
  1. 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. 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. 3.The system of equations is solved for X using the factored form of A.
  4. 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 https://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: order Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: fact Nag_FactoredFormType Input
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: uplo Nag_UploType Input
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: n Integer Input
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint: n0.
5: nrhs Integer Input
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 Complex Input
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] Complex Input/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 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 f07qrc, stored as a packed triangular matrix in the same storage format as A.
8: ipiv[n] Integer Input/Output
On entry: if fact=Nag_Factored, ipiv contains details of the interchanges and the block structure of D, as determined by f07qrc.
  • 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.
On exit: if fact=Nag_NotFactored, ipiv contains details of the interchanges and the block structure of D, as determined by f07qrc, as described above.
9: b[dim] const Complex Input
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: pdb Integer Input
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] Complex Output
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: pdx Integer Input
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: rcond double * 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] double Output
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] double Output
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: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
Element value of the diagonal 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 Parallelism and Performance

f07qpc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07qpc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 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 f07pbc. The complex Hermitian analogue of this function is f07ppc.

10 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.

10.1 Program Text

Program Text (f07qpce.c)

10.2 Program Data

Program Data (f07qpce.d)

10.3 Program Results

Program Results (f07qpce.r)