zhpsvx(3P) Sun Performance Library zhpsvx(3P)NAMEzhpsvx - use the diagonal pivoting factorization A = U*D*U**H or A =
L*D*L**H to compute the solution to a complex system of linear equa‐
tions A * X = B, where A is an N-by-N Hermitian matrix stored in packed
format and X and B are N-by-NRHS matrices
SYNOPSIS
SUBROUTINE ZHPSVX(FACT, UPLO, N, NRHS, A, AF, IPIVOT, B, LDB, X, LDX,
RCOND, FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT, UPLO
DOUBLE COMPLEX A(*), AF(*), B(LDB,*), X(LDX,*), WORK(*)
INTEGER N, NRHS, LDB, LDX, INFO
INTEGER IPIVOT(*)
DOUBLE PRECISION RCOND
DOUBLE PRECISION FERR(*), BERR(*), WORK2(*)
SUBROUTINE ZHPSVX_64(FACT, UPLO, N, NRHS, A, AF, IPIVOT, B, LDB, X,
LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT, UPLO
DOUBLE COMPLEX A(*), AF(*), B(LDB,*), X(LDX,*), WORK(*)
INTEGER*8 N, NRHS, LDB, LDX, INFO
INTEGER*8 IPIVOT(*)
DOUBLE PRECISION RCOND
DOUBLE PRECISION FERR(*), BERR(*), WORK2(*)
F95 INTERFACE
SUBROUTINE HPSVX(FACT, UPLO, [N], [NRHS], A, AF, IPIVOT, B, [LDB], X,
[LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, UPLO
COMPLEX(8), DIMENSION(:) :: A, AF, WORK
COMPLEX(8), DIMENSION(:,:) :: B, X
INTEGER :: N, NRHS, LDB, LDX, INFO
INTEGER, DIMENSION(:) :: IPIVOT
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: FERR, BERR, WORK2
SUBROUTINE HPSVX_64(FACT, UPLO, [N], [NRHS], A, AF, IPIVOT, B, [LDB],
X, [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, UPLO
COMPLEX(8), DIMENSION(:) :: A, AF, WORK
COMPLEX(8), DIMENSION(:,:) :: B, X
INTEGER(8) :: N, NRHS, LDB, LDX, INFO
INTEGER(8), DIMENSION(:) :: IPIVOT
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: FERR, BERR, WORK2
C INTERFACE
#include <sunperf.h>
void zhpsvx(char fact, char uplo, int n, int nrhs, doublecomplex *a,
doublecomplex *af, int *ipivot, doublecomplex *b, int ldb,
doublecomplex *x, int ldx, double *rcond, double *ferr, dou‐
ble *berr, int *info);
void zhpsvx_64(char fact, char uplo, long n, long nrhs, doublecomplex
*a, doublecomplex *af, long *ipivot, doublecomplex *b, long
ldb, doublecomplex *x, long ldx, double *rcond, double *ferr,
double *berr, long *info);
PURPOSEzhpsvx uses the diagonal pivoting factorization A = U*D*U**H or A =
L*D*L**H to compute the solution to a complex system of linear equa‐
tions A * X = B, where A is an N-by-N Hermitian matrix stored in packed
format and X and B are N-by-NRHS matrices.
Error bounds on the solution and a condition estimate are also pro‐
vided.
The following steps are performed:
1. If FACT = 'N', the diagonal pivoting method is used to factor A as
A = U * D * U**H, if UPLO = 'U', or
A = L * D * L**H, if UPLO = 'L',
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices and D is Hermitian and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
2. If some D(i,i)=0, so that D is exactly singular, then the routine
returns with INFO = i. 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,
INFO = N+1 is returned as a warning, but the routine 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 calculate error bounds and backward error estimates
for it.
ARGUMENTS
FACT (input)
Specifies whether or not the factored form of A has been sup‐
plied on entry. = 'F': On entry, AF and IPIVOT contain the
factored form of A. AF and IPIVOT will not be modified. =
'N': The matrix A will be copied to AF and factored.
UPLO (input)
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) The number of linear equations, i.e., the order of the matrix
A. N >= 0.
NRHS (input)
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A (input) COMPLEX*16 array, dimension (N*(N+1)/2)
The upper or lower triangle of the Hermitian matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array A as follows: if UPLO = 'U', A(i + (j-1)*j/2) =
A(i,j) for 1<=i<=j; if UPLO = 'L', A(i + (j-1)*(2*n-j)/2) =
A(i,j) for j<=i<=n. See below for further details.
AF (input or output) COMPLEX*16 array, dimension (N*(N+1)/2)
If FACT = 'F', then AF is an input argument and on entry con‐
tains the block diagonal matrix D and the multipliers used to
obtain the factor U or L from the factorization A = U*D*U**H
or A = L*D*L**H as computed by ZHPTRF, stored as a packed
triangular matrix in the same storage format as A.
If FACT = 'N', then AF is an output argument and on exit con‐
tains the block diagonal matrix D and the multipliers used to
obtain the factor U or L from the factorization A = U*D*U**H
or A = L*D*L**H as computed by ZHPTRF, stored as a packed
triangular matrix in the same storage format as A.
IPIVOT (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIVOT is an input argument and on entry
contains details of the interchanges and the block structure
of D, as determined by ZHPTRF. If IPIVOT(k) > 0, then rows
and columns k and IPIVOT(k) were interchanged and D(k,k) is a
1-by-1 diagonal block. If UPLO = 'U' and IPIVOT(k) =
IPIVOT(k-1) < 0, then rows and columns k-1 and -IPIVOT(k)
were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal
block. If UPLO = 'L' and IPIVOT(k) = IPIVOT(k+1) < 0, then
rows and columns k+1 and -IPIVOT(k) were interchanged and
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
If FACT = 'N', then IPIVOT is an output argument and on exit
contains details of the interchanges and the block structure
of D, as determined by ZHPTRF.
B (input) COMPLEX*16 array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB (input)
The leading dimension of the array B. LDB >= max(1,N).
X (output) COMPLEX*16 array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
LDX (input)
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output)
The estimate of the reciprocal condition number of the matrix
A. If RCOND is less than the machine precision (in particu‐
lar, if RCOND = 0), the matrix is singular to working preci‐
sion. This condition is indicated by a return code of INFO >
0.
FERR (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X). If XTRUE is
the true solution corresponding to X(j), FERR(j) is an esti‐
mated upper bound for the magnitude of the largest element in
(X(j) - XTRUE) divided by the magnitude of the largest ele‐
ment in X(j). The estimate is as reliable as the estimate
for RCOND, and is almost always a slight overestimate of the
true error.
BERR (output) DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in any ele‐
ment of A or B that makes X(j) an exact solution).
WORK (workspace)
COMPLEX*16 array, dimension(2*N)
DOUBLE PRECISION array, WORK2 (workspace)
dimension(N)
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: D(i,i) is exactly zero. The factorization has been
completed but the factor D is exactly singular, so the solu‐
tion and error bounds could not be computed. RCOND = 0 is
returned. = N+1: 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.
FURTHER DETAILS
The packed storage scheme is illustrated by the following example when
N = 4, UPLO = 'U':
Two-dimensional storage of the Hermitian matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = conjg(aji))
a44
Packed storage of the upper triangle of A:
A = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
6 Mar 2009 zhpsvx(3P)
