1CPFTRF(1)LAPACK routine (version 3.2) CPFTRF(1)
2
6 CPFTRF - computes the Cholesky factorization of a complex Hermitian
7 positive definite matrix A
8
10 SUBROUTINE CPFTRF( TRANSR, UPLO, N, A, INFO )
11 CHARACTER TRANSR, UPLO
13 INTEGER N, INFO
15 COMPLEX A( 0: * )
17
19 CPFTRF computes the Cholesky factorization of a complex Hermitian posi‐
20 tive definite matrix A. The factorization has the form
21 A = U**H * U, if UPLO = 'U', or
22 A = L * L**H, if UPLO = 'L',
23 where U is an upper triangular matrix and L is lower triangular. This
24 is the block version of the algorithm, calling Level 3 BLAS.
25
27 TRANSR (input) CHARACTER
28 = 'N': The Normal TRANSR of RFP A is stored;
29 = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
30 UPLO (input) CHARACTER
32 = 'U': Upper triangle of RFP A is stored;
33 = 'L': Lower triangle of RFP A is stored.
34 N (input) INTEGER
36 The order of the matrix A. N >= 0.
37 A (input/output) COMPLEX array, dimension ( N*(N+1)/2 );
39 On entry, the Hermitian matrix A in RFP format. RFP format is
40 described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
41 then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
42 (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
43 the Conjugate-transpose of RFP A as defined when TRANSR = 'N'.
44 The contents of RFP A are defined by UPLO as follows: If UPLO =
45 'U' the RFP A contains the nt elements of upper packed A. If
46 UPLO = 'L' the RFP A contains the elements of lower packed A.
47 The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR is
48 'N' the LDA is N+1 when N is even and N is odd. See the Note
49 below for more details. On exit, if INFO = 0, the factor U or
50 L from the Cholesky factorization RFP A = U**H*U or RFP A =
51 L*L**H.
52 INFO (output) INTEGER
54 = 0: successful exit
55 < 0: if INFO = -i, the i-th argument had an illegal value
56 > 0: if INFO = i, the leading minor of order i is not positive
57 definite, and the factorization could not be completed. Fur‐
58 ther Notes on RFP Format: ============================ We first
59 consider Standard Packed Format when N is even. We give an
60 example where N = 6. AP is Upper AP is Lower 00 01
61 02 03 04 05 00 11 12 13 14 15 10 11 22 23 24 25
62 20 21 22 33 34 35 30 31 32 33 44 45 40 41 42 43 44
63 55 50 51 52 53 54 55 Let TRANSR = 'N'. RFP holds AP as
64 follows: For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists
65 of the last
66 three columns of AP upper. The lower triangle A(4:6,0:2) con‐
67 sists of conjugate-transpose of the first three columns of AP
68 upper. For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists
69 of the first
70 three columns of AP lower. The upper triangle A(0:2,0:2) con‐
71 sists of conjugate-transpose of the last three columns of AP
72 lower. To denote conjugate we place -- above the element. This
73 covers the case N even and TRANSR = 'N'. RFP A
74 RFP A -- -- -- 03 04 05 33 43 53 -- -- 13 14 15
75 00 44 54 -- 23 24 25 10 11 55 33 34 35
76 20 21 22 -- 00 44 45 30 31 32 -- -- 01 11 55
77 40 41 42 -- -- -- 02 12 22 50 51 52 Now let
78 TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
79 transpose of RFP A above. One therefore gets: RFP A
80 RFP A -- -- -- -- -- -- -- -- -- -- 03 13 23 33
81 00 01 02 33 00 10 20 30 40 50 -- -- -- -- --
82 -- -- -- -- -- 04 14 24 34 44 11 12 43 44 11 21 31 41 51 --
83 -- -- -- -- -- -- -- -- -- 05 15 25 35 45 55 22
84 53 54 55 22 32 42 52 We next consider Standard Packed Format
85 when N is odd. We give an example where N = 5. AP is Upper
86 AP is Lower 00 01 02 03 04 00 11 12 13 14
87 10 11 22 23 24 20 21 22 33 34 30 31
88 32 33 44 40 41 42 43 44 Let TRANSR = 'N'. RFP
89 holds AP as follows: For UPLO = 'U' the upper trapezoid
90 A(0:4,0:2) consists of the last
91 three columns of AP upper. The lower triangle A(3:4,0:1) con‐
92 sists of conjugate-transpose of the first two columns of AP
93 upper. For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists
94 of the first
95 three columns of AP lower. The upper triangle A(0:1,1:2) con‐
96 sists of conjugate-transpose of the last two columns of AP
97 lower. To denote conjugate we place -- above the element. This
98 covers the case N odd and TRANSR = 'N'. RFP A
99 RFP A -- -- 02 03 04 00 33 43 -- 12 13 14
100 10 11 44 22 23 24 20 21 22 -- 00 33 34
101 30 31 32 -- -- 01 11 44 40 41 42 Now let TRANSR
102 = 'C'. RFP A in both UPLO cases is just the conjugate- trans‐
103 pose of RFP A above. One therefore gets: RFP A
104 RFP A -- -- -- -- -- -- -- -- -- 02 12 22 00
105 01 00 10 20 30 40 50 -- -- -- --
106 -- -- -- -- -- 03 13 23 33 11 33 11 21 31 41 51 --
107 -- -- -- -- -- -- -- -- 04 14 24 34 44
108 43 44 22 32 42 52
109 LAPACK routine (version 3.2) November 2008 CPFTRF(1)