cpftrs - solve a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization computed by CPFTRF
SUBROUTINE CPFTRS(TRANSR, UPLO, N, NRHS, A, B, LDB, INFO) CHARACTER*1 TRANSR, UPLO INTEGER INFO, LDB, N, NRHS COMPLEX A(0:*), B(LDB,*) SUBROUTINE CPFTRS_64(TRANSR, UPLO, N, NRHS, A, B, LDB, INFO) CHARACTER*1 TRANSR, UPLO INTEGER*8 INFO, LDB, N, NRHS COMPLEX A(0:*), B(LDB,*) F95 INTERFACE SUBROUTINE PFTRS(TRANSR, UPLO, N, NRHS, A, B, LDB, INFO) INTEGER :: N, NRHS, LDB, INFO CHARACTER(LEN=1) :: TRANSR, UPLO COMPLEX, DIMENSION(:,:) :: B COMPLEX, DIMENSION(:) :: A SUBROUTINE PFTRS_64(TRANSR, UPLO, N, NRHS, A, B, LDB, INFO) INTEGER(8) :: N, NRHS, LDB, INFO CHARACTER(LEN=1) :: TRANSR, UPLO COMPLEX, DIMENSION(:,:) :: B COMPLEX, DIMENSION(:) :: A C INTERFACE #include <sunperf.h> void cpftrs (char transr, char uplo, int n, int nrhs, floatcomplex *a, floatcomplex *b, int ldb, int *info); void cpftrs_64 (char transr, char uplo, long n, long nrhs, floatcomplex *a, floatcomplex *b, long ldb, long *info);
Oracle Solaris Studio Performance Library cpftrs(3P)
NAME
cpftrs - solve a system of linear equations A*X = B with a Hermitian
positive definite matrix A using the Cholesky factorization computed by
CPFTRF
SYNOPSIS
SUBROUTINE CPFTRS(TRANSR, UPLO, N, NRHS, A, B, LDB, INFO)
CHARACTER*1 TRANSR, UPLO
INTEGER INFO, LDB, N, NRHS
COMPLEX A(0:*), B(LDB,*)
SUBROUTINE CPFTRS_64(TRANSR, UPLO, N, NRHS, A, B, LDB, INFO)
CHARACTER*1 TRANSR, UPLO
INTEGER*8 INFO, LDB, N, NRHS
COMPLEX A(0:*), B(LDB,*)
F95 INTERFACE
SUBROUTINE PFTRS(TRANSR, UPLO, N, NRHS, A, B, LDB, INFO)
INTEGER :: N, NRHS, LDB, INFO
CHARACTER(LEN=1) :: TRANSR, UPLO
COMPLEX, DIMENSION(:,:) :: B
COMPLEX, DIMENSION(:) :: A
SUBROUTINE PFTRS_64(TRANSR, UPLO, N, NRHS, A, B, LDB, INFO)
INTEGER(8) :: N, NRHS, LDB, INFO
CHARACTER(LEN=1) :: TRANSR, UPLO
COMPLEX, DIMENSION(:,:) :: B
COMPLEX, DIMENSION(:) :: A
C INTERFACE
#include <sunperf.h>
void cpftrs (char transr, char uplo, int n, int nrhs, floatcomplex *a,
floatcomplex *b, int ldb, int *info);
void cpftrs_64 (char transr, char uplo, long n, long nrhs, floatcomplex
*a, floatcomplex *b, long ldb, long *info);
PURPOSE
cpftrs solves a system of linear equations A*X = B with a Hermitian
positive definite matrix A using the Cholesky factorization A = U**H*U
or A = L*L**H computed by CPFTRF.
ARGUMENTS
TRANSR (input)
TRANSR is CHARACTER*1
= 'N': The Normal TRANSR of RFP A is stored;
= 'C': The Conjugate-transpose TRANSR of RFP A is stored.
UPLO (input)
UPLO is CHARACTER*1
= 'U': Upper triangle of RFP A is stored;
= 'L': Lower triangle of RFP A is stored.
N (input)
N is INTEGER
The order of the matrix A. N >= 0.
NRHS (input)
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A (input)
A is COMPLEX array, dimension ( N*(N+1)/2 );
The triangular factor U or L from the Cholesky factorization
of RFP A=U**H*U or RFP A=L*L**H, as computed by CPFTRF. See
note below for more details about RFP A.
B (input/output)
B is COMPLEX array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB (input)
LDB is INTEGER
The leading dimension of the array B.
LDB >= max(1,N).
INFO (output)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
FURTHER NOTES ON RFP FORMAT
We first consider Standard Packed Format when N is even.
We give an example where N = 6.
AP is Upper AP is Lower
00 01 02 03 04 05 00
11 12 13 14 15 10 11
22 23 24 25 20 21 22
33 34 35 30 31 32 33
44 45 40 41 42 43 44
55 50 51 52 53 54 55
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
conjugate-transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
conjugate-transpose of the last three columns of AP lower.
To denote conjugate we place -- above the element. This covers the case
N even and TRANSR = 'N'.
RFP A RFP A
-- -- --
03 04 05 33 43 53
-- --
13 14 15 00 44 54
--
23 24 25 10 11 55
33 34 35 20 21 22
--
00 44 45 30 31 32
-- --
01 11 55 40 41 42
-- -- --
02 12 22 50 51 52
Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:
RFP A RFP A
-- -- -- -- -- -- -- -- -- --
03 13 23 33 00 01 02 33 00 10 20 30 40 50
-- -- -- -- -- -- -- -- -- --
04 14 24 34 44 11 12 43 44 11 21 31 41 51
-- -- -- -- -- -- -- -- -- --
05 15 25 35 45 55 22 53 54 55 22 32 42 52
We next consider Standard Packed Format when N is odd.
We give an example where N = 5.
AP is Upper AP is Lower
00 01 02 03 04 00
11 12 13 14 10 11
22 23 24 20 21 22
33 34 30 31 32 33
44 40 41 42 43 44
Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
conjugate-transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
conjugate-transpose of the last two columns of AP lower.
To denote conjugate we place -- above the element. This covers the case
N odd and TRANSR = 'N'.
RFP A RFP A
-- --
02 03 04 00 33 43
--
12 13 14 10 11 44
22 23 24 20 21 22
--
00 33 34 30 31 32
-- --
01 11 44 40 41 42
Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:
RFP A RFP A
-- -- -- -- -- -- -- -- --
02 12 22 00 01 00 10 20 30 40 50
-- -- -- -- -- -- -- -- --
03 13 23 33 11 33 11 21 31 41 51
-- -- -- -- -- -- -- -- --
04 14 24 34 44 43 44 22 32 42 52
7 Nov 2015 cpftrs(3P)