1DPFTRF(1)LAPACK routine (version 3.2) DPFTRF(1)
2
6 DPFTRF - computes the Cholesky factorization of a real symmetric posi‐
7 tive definite matrix A
8
10 SUBROUTINE DPFTRF( TRANSR, UPLO, N, A, INFO )
11 CHARACTER TRANSR, UPLO
13 INTEGER N, INFO
15 DOUBLE PRECISION A( 0: * )
17
19 DPFTRF computes the Cholesky factorization of a real symmetric positive
20 definite matrix A. The factorization has the form
21 A = U**T * U, if UPLO = 'U', or
22 A = L * L**T, 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 = 'T': The 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) DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
39 On entry, the symmetric 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 = 'T' then RFP is
43 the transpose of RFP A as defined when TRANSR = 'N'. The con‐
44 tents of RFP A are defined by UPLO as follows: If UPLO = 'U'
45 the RFP A contains the NT elements of upper packed A. If UPLO =
46 'L' the RFP A contains the elements of lower packed A. The LDA
47 of RFP A is (N+1)/2 when TRANSR = 'T'. When TRANSR is 'N' the
48 LDA is N+1 when N is even and N is odd. See the Note below for
49 more details. On exit, if INFO = 0, the factor U or L from the
50 Cholesky factorization RFP A = U**T*U or RFP A = L*L**T.
51 INFO (output) INTEGER
53 = 0: successful exit
54 < 0: if INFO = -i, the i-th argument had an illegal value
55 > 0: if INFO = i, the leading minor of order i is not positive
56 definite, and the factorization could not be completed.
57
59 We first consider Rectangular Full Packed (RFP) Format when N is even.
60 We give an example where N = 6.
61 AP is Upper AP is Lower
62 00 01 02 03 04 05 00
63 11 12 13 14 15 10 11
64 22 23 24 25 20 21 22
65 33 34 35 30 31 32 33
66 44 45 40 41 42 43 44
67 55 50 51 52 53 54 55
68 Let TRANSR = 'N'. RFP holds AP as follows:
69 For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
70 three columns of AP upper. The lower triangle A(4:6,0:2) consists of
71 the transpose of the first three columns of AP upper.
72 For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
73 three columns of AP lower. The upper triangle A(0:2,0:2) consists of
74 the transpose of the last three columns of AP lower.
75 This covers the case N even and TRANSR = 'N'.
76 RFP A RFP A
77 03 04 05 33 43 53
78 13 14 15 00 44 54
79 23 24 25 10 11 55
80 33 34 35 20 21 22
81 00 44 45 30 31 32
82 01 11 55 40 41 42
83 02 12 22 50 51 52
84 Now let TRANSR = 'T'. RFP A in both UPLO cases is just the transpose of
85 RFP A above. One therefore gets:
86 RFP A RFP A
87 03 13 23 33 00 01 02 33 00 10 20 30 40 50
88 04 14 24 34 44 11 12 43 44 11 21 31 41 51
89 05 15 25 35 45 55 22 53 54 55 22 32 42 52
90 We first consider Rectangular Full Packed (RFP) Format when N is odd.
91 We give an example where N = 5.
92 AP is Upper AP is Lower
93 00 01 02 03 04 00
94 11 12 13 14 10 11
95 22 23 24 20 21 22
96 33 34 30 31 32 33
97 44 40 41 42 43 44
98 Let TRANSR = 'N'. RFP holds AP as follows:
99 For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
100 three columns of AP upper. The lower triangle A(3:4,0:1) consists of
101 the transpose of the first two columns of AP upper.
102 For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
103 three columns of AP lower. The upper triangle A(0:1,1:2) consists of
104 the transpose of the last two columns of AP lower.
105 This covers the case N odd and TRANSR = 'N'.
106 RFP A RFP A
107 02 03 04 00 33 43
108 12 13 14 10 11 44
109 22 23 24 20 21 22
110 00 33 34 30 31 32
111 01 11 44 40 41 42
112 Now let TRANSR = 'T'. RFP A in both UPLO cases is just the transpose of
113 RFP A above. One therefore gets:
114 RFP A RFP A
115 02 12 22 00 01 00 10 20 30 40 50
116 03 13 23 33 11 33 11 21 31 41 51
117 04 14 24 34 44 43 44 22 32 42 52
118 LAPACK routine (version 3.2) November 2008 DPFTRF(1)