cpftrs.l - Online Linux Manual Page

CPFTRS - Online Linux Manual PageSection : 1
Updated : November 2008
Source : LAPACK routine (version 3.2)
Note : LAPACK routine (version 3.2)

NAMECPFTRS - 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

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

PURPOSECPFTRS 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.

ARGUMENTSTRANSR (input) CHARACTER = 'N': The Normal TRANSR of RFP A is stored;
= 'C': The Conjugate-transpose TRANSR of RFP A is stored. UPLO (input) CHARACTER
= 'U': Upper triangle of RFP A is stored;
= 'L': 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 DETAILSWe 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
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