cpftrs (l) - Linux Manuals

cpftrs: solves a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization A = U**HU or A = LL**H computed by CPFTRF

Command to display cpftrs manual in Linux: $ man l cpftrs

NAME

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

SYNOPSIS

SUBROUTINE CPFTRS(: TRANSR, UPLO, N, NRHS, A, B, LDB, INFO )
: CHARACTER TRANSR, UPLO
: INTEGER INFO, LDB, N, NRHS
: COMPLEX A( 0: * ), B( LDB, * )

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) CHARACTER: = aqNaq: The Normal TRANSR of RFP A is stored;
= aqCaq: The Conjugate-transpose TRANSR of RFP A is stored.
UPLO (input) CHARACTER: = aqUaq: Upper triangle of RFP A is stored;
= aqLaq: Lower triangle of RFP A is stored.
N (input) INTEGER: The order of the matrix A. N >= 0.
NRHS (input) INTEGER: The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
A (input) 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) COMPLEX array, dimension (LDB,NRHS): On entry, the right hand side matrix B. On exit, the solution matrix X.
LDB (input) INTEGER: The leading dimension of the array B. LDB >= max(1,N).
INFO (output) INTEGER: = 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

FURTHER DETAILS

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 = aqNaq. RFP holds AP as follows:
For UPLO = aqUaq 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 = aqLaq 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 = aqNaq.

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 = aqCaq. 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 = aqNaq. RFP holds AP as follows:
For UPLO = aqUaq 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 = aqLaq 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 = aqNaq.

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 = aqCaq. 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