zptsvx(3P) Sun Performance Library zptsvx(3P)NAMEzptsvx - use the factorization 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 Her‐
mitian positive definite tridiagonal matrix and X and B are N-by-NRHS
matrices
SYNOPSIS
SUBROUTINE ZPTSVX(FACT, N, NRHS, D, E, DF, EF, B, LDB, X,
LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT
DOUBLE COMPLEX E(*), EF(*), B(LDB,*), X(LDX,*), WORK(*)
INTEGER N, NRHS, LDB, LDX, INFO
DOUBLE PRECISION RCOND
DOUBLE PRECISION D(*), DF(*), FERR(*), BERR(*), WORK2(*)
SUBROUTINE ZPTSVX_64(FACT, N, NRHS, D, E, DF, EF, B, LDB,
X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT
DOUBLE COMPLEX E(*), EF(*), B(LDB,*), X(LDX,*), WORK(*)
INTEGER*8 N, NRHS, LDB, LDX, INFO
DOUBLE PRECISION RCOND
DOUBLE PRECISION D(*), DF(*), FERR(*), BERR(*), WORK2(*)
F95 INTERFACE
SUBROUTINE PTSVX(FACT, [N], [NRHS], D, E, DF, EF, B, [LDB],
X, [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT
COMPLEX(8), DIMENSION(:) :: E, EF, WORK
COMPLEX(8), DIMENSION(:,:) :: B, X
INTEGER :: N, NRHS, LDB, LDX, INFO
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: D, DF, FERR, BERR, WORK2
SUBROUTINE PTSVX_64(FACT, [N], [NRHS], D, E, DF, EF, B,
[LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT
COMPLEX(8), DIMENSION(:) :: E, EF, WORK
COMPLEX(8), DIMENSION(:,:) :: B, X
INTEGER(8) :: N, NRHS, LDB, LDX, INFO
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: D, DF, FERR, BERR, WORK2
C INTERFACE
#include <sunperf.h>
void zptsvx(char fact, int n, int nrhs, double *d, doublecomplex *e,
double *df, doublecomplex *ef, doublecomplex *b, int ldb,
doublecomplex *x, int ldx, double *rcond, double *ferr, dou‐
ble *berr, int *info);
void zptsvx_64(char fact, long n, long nrhs, double *d, doublecomplex
*e, double *df, doublecomplex *ef, doublecomplex *b, long
ldb, doublecomplex *x, long ldx, double *rcond, double *ferr,
double *berr, long *info);
PURPOSEzptsvx uses the factorization 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 Her‐
mitian positive definite tridiagonal matrix 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 matrix A is factored as A = L*D*L**H, where L
is a unit lower bidiagonal matrix and D is diagonal. The
factorization can also be regarded as having the form
A = U**H*D*U.
2. If the leading i-by-i principal minor is not positive definite,
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 the matrix A is
supplied on entry. = 'F': On entry, DF and EF contain the
factored form of A. D, E, DF, and EF will not be modified.
= 'N': The matrix A will be copied to DF and EF and fac‐
tored.
N (input) 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.
D (input) The n diagonal elements of the tridiagonal matrix A.
E (input) The (n-1) subdiagonal elements of the tridiagonal matrix A.
DF (input or output)
If FACT = 'F', then DF is an input argument and on entry con‐
tains the n diagonal elements of the diagonal matrix D from
the L*D*L**H factorization of A. If FACT = 'N', then DF is
an output argument and on exit contains the n diagonal ele‐
ments of the diagonal matrix D from the L*D*L**H factoriza‐
tion of A.
EF (input or output)
If FACT = 'F', then EF is an input argument and on entry con‐
tains the (n-1) subdiagonal elements of the unit bidiagonal
factor L from the L*D*L**H factorization of A. If FACT =
'N', then EF is an output argument and on exit contains the
(n-1) subdiagonal elements of the unit bidiagonal factor L
from the L*D*L**H factorization of A.
B (input) On entry, the N-by-NRHS right hand side matrix B. Unchanged
on exit.
LDB (input)
The leading dimension of the array B. LDB >= max(1,N).
X (output)
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 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 condi‐
tion is indicated by a return code of INFO > 0.
FERR (output)
The 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 estimated upper
bound for the magnitude of the largest element in (X(j)-
XTRUE) divided by the magnitude of the largest element in
X(j).
BERR (output)
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)
dimension(N)
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: the leading minor of order i of A is not positive def‐
inite, so the factorization could not be completed, and the
solution has not been computed. RCOND = 0 is returned. =
N+1: U is nonsingular, but RCOND is less than machine preci‐
sion, meaning that the matrix is singular to working preci‐
sion. Nevertheless, the solution and error bounds are com‐
puted because there are a number of situations where the com‐
puted solution can be more accurate than the value of RCOND
would suggest.
6 Mar 2009 zptsvx(3P)
