Contents
zhpsvx - use the diagonal pivoting factorization A =
U*D*U**H or A = L*D*L**H to compute the solution to a com-
plex system of linear equations 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
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, doub-
lecomplex *a, doublecomplex *af, int *ipivot,
doublecomplex *b, int ldb, doublecomplex *x, int
ldx, double *rcond, double *ferr, double *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);
zhpsvx 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 equations A * X = B, where A is an N-by-N Hermi-
tian 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 provided.
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 fac-
tored form
of A.
4. Iterative refinement is applied to improve the computed
solution
matrix and calculate error bounds and backward error
estimates
for it.
FACT (input)
Specifies whether or not the factored form of A
has been supplied 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 contains 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 CHPTRF, 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 contains 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 CHPTRF, 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
CHPTRF. 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
CHPTRF.
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 particular, if RCOND = 0), the
matrix is singular to working precision. 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 solu-
tion 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 estimated
upper bound for the magnitude of the largest ele-
ment in (X(j) - XTRUE) divided by the magnitude of
the largest element 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 element 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 ille-
gal 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 solution 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.
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 ]