sgtsvx(3P) Sun Performance Library sgtsvx(3P)NAMEsgtsvx - use the LU factorization to compute the solution to a real
system of linear equations A * X = B or A**T * X = B,
SYNOPSIS
SUBROUTINE SGTSVX(FACT, TRANSA, N, NRHS, LOW, D, UP, LOWF, DF,
UPF1, UPF2, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
WORK2, INFO)
CHARACTER * 1 FACT, TRANSA
INTEGER N, NRHS, LDB, LDX, INFO
INTEGER IPIVOT(*), WORK2(*)
REAL RCOND
REAL LOW(*), D(*), UP(*), LOWF(*), DF(*), UPF1(*), UPF2(*), B(LDB,*),
X(LDX,*), FERR(*), BERR(*), WORK(*)
SUBROUTINE SGTSVX_64(FACT, TRANSA, N, NRHS, LOW, D, UP, LOWF,
DF, UPF1, UPF2, IPIVOT, B, LDB, X, LDX, RCOND, FERR, BERR,
WORK, WORK2, INFO)
CHARACTER * 1 FACT, TRANSA
INTEGER*8 N, NRHS, LDB, LDX, INFO
INTEGER*8 IPIVOT(*), WORK2(*)
REAL RCOND
REAL LOW(*), D(*), UP(*), LOWF(*), DF(*), UPF1(*), UPF2(*), B(LDB,*),
X(LDX,*), FERR(*), BERR(*), WORK(*)
F95 INTERFACE
SUBROUTINE GTSVX(FACT, [TRANSA], [N], [NRHS], LOW, D, UP, LOWF,
DF, UPF1, UPF2, IPIVOT, B, [LDB], X, [LDX], RCOND, FERR, BERR,
[WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, TRANSA
INTEGER :: N, NRHS, LDB, LDX, INFO
INTEGER, DIMENSION(:) :: IPIVOT, WORK2
REAL :: RCOND
REAL, DIMENSION(:) :: LOW, D, UP, LOWF, DF, UPF1, UPF2, FERR, BERR,
WORK
REAL, DIMENSION(:,:) :: B, X
SUBROUTINE GTSVX_64(FACT, [TRANSA], [N], [NRHS], LOW, D, UP, LOWF,
DF, UPF1, UPF2, IPIVOT, B, [LDB], X, [LDX], RCOND, FERR, BERR,
[WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, TRANSA
INTEGER(8) :: N, NRHS, LDB, LDX, INFO
INTEGER(8), DIMENSION(:) :: IPIVOT, WORK2
REAL :: RCOND
REAL, DIMENSION(:) :: LOW, D, UP, LOWF, DF, UPF1, UPF2, FERR, BERR,
WORK
REAL, DIMENSION(:,:) :: B, X
C INTERFACE
#include <sunperf.h>
void sgtsvx(char fact, char transa, int n, int nrhs, float *low, float
*d, float *up, float *lowf, float *df, float *upf1, float
*upf2, int *ipivot, float *b, int ldb, float *x, int ldx,
float *rcond, float *ferr, float *berr, int *info);
void sgtsvx_64(char fact, char transa, long n, long nrhs, float *low,
float *d, float *up, float *lowf, float *df, float *upf1,
float *upf2, long *ipivot, float *b, long ldb, float *x, long
ldx, float *rcond, float *ferr, float *berr, long *info);
PURPOSEsgtsvx uses the LU factorization to compute the solution to a real sys‐
tem of linear equations A * X = B or A**T * X = B, where A is a tridi‐
agonal matrix of order N 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 LU decomposition is used to factor the matrix A
as A = L * U, where L is a product of permutation and unit lower
bidiagonal matrices and U is upper triangular with nonzeros in
only the main diagonal and first two superdiagonals.
2. If some U(i,i)=0, so that U 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': LOWF, DF, UPF1, UPF2, and IPIVOT
contain the factored form of A; LOW, D, UP, LOWF, DF, UPF1,
UPF2 and IPIVOT will not be modified. = 'N': The matrix
will be copied to LOWF, DF, and UPF1 and factored.
TRANSA (input)
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
TRANSA is defaulted to 'N' for F95 INTERFACE.
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 matrix B. NRHS >= 0.
LOW (input)
The (n-1) subdiagonal elements of A.
D (input) The n diagonal elements of A.
UP (input/output)
The (n-1) superdiagonal elements of A.
LOWF (input or output)
If FACT = 'F', then LOWF is an input argument and on entry
contains the (n-1) multipliers that define the matrix L from
the LU factorization of A as computed by SGTTRF.
If FACT = 'N', then LOWF is an output argument and on exit
contains the (n-1) multipliers that define the matrix L from
the LU factorization of 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 upper triangular matrix
U from the LU factorization of A.
If FACT = 'N', then DF is an output argument and on exit con‐
tains the n diagonal elements of the upper triangular matrix
U from the LU factorization of A.
UPF1 (input or output)
If FACT = 'F', then UPF1 is an input argument and on entry
contains the (n-1) elements of the first superdiagonal of U.
If FACT = 'N', then UPF1 is an output argument and on exit
contains the (n-1) elements of the first superdiagonal of U.
UPF2 (input or output)
If FACT = 'F', then UPF2 is an input argument and on entry
contains the (n-2) elements of the second superdiagonal of U.
If FACT = 'N', then UPF2 is an output argument and on exit
contains the (n-2) elements of the second superdiagonal of U.
IPIVOT (input/output)
If FACT = 'F', then IPIVOT is an input argument and on entry
contains the pivot indices from the LU factorization of A as
computed by SGTTRF.
If FACT = 'N', then IPIVOT is an output argument and on exit
contains the pivot indices from the LU factorization of A;
row i of the matrix was interchanged with row IPIVOT(i).
IPIVOT(i) will always be either i or i+1; IPIVOT(i) = i indi‐
cates a row interchange was not required.
B (input) The N-by-NRHS right hand side matrix B.
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 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)
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)
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(3*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: U(i,i) is exactly zero. The factorization has not
been completed unless i = N, but the factor U is exactly sin‐
gular, so the solution and error bounds could not be com‐
puted. RCOND = 0 is returned. = N+1: U 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.
6 Mar 2009 sgtsvx(3P)
