CPFTRF(1LAPACK routine (version 3.2)CPFTRF(1)NAME
CPFTRF - computes the Cholesky factorization of a complex Hermitian
positive definite matrix A
SYNOPSIS
SUBROUTINE CPFTRF( TRANSR, UPLO, N, A, INFO )
CHARACTER TRANSR, UPLO
INTEGER N, INFO
COMPLEX A( 0: * )
PURPOSE
CPFTRF computes the Cholesky factorization of a complex Hermitian posi‐
tive definite matrix A. The factorization has the form
A = U**H * U, if UPLO = 'U', or
A = L * L**H, if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular. This
is the block version of the algorithm, calling Level 3 BLAS.
ARGUMENTS
TRANSR (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.
A (input/output) COMPLEX array, dimension ( N*(N+1)/2 );
On entry, the Hermitian matrix A in RFP format. RFP format is
described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
the Conjugate-transpose of RFP A as defined when TRANSR = 'N'.
The contents of RFP A are defined by UPLO as follows: If UPLO =
'U' the RFP A contains the nt elements of upper packed A. If
UPLO = 'L' the RFP A contains the elements of lower packed A.
The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR is
'N' the LDA is N+1 when N is even and N is odd. See the Note
below for more details. On exit, if INFO = 0, the factor U or
L from the Cholesky factorization RFP A = U**H*U or RFP A =
L*L**H.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not positive
definite, and the factorization could not be completed. Fur‐
ther 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) con‐
sists 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) con‐
sists 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) con‐
sists 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) con‐
sists 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- trans‐
pose 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
LAPACK routine (version 3.2) November 2008 CPFTRF(1)