1DTFTRI(1) LAPACK routine (version 3.2) DTFTRI(1)
2
6 DTFTRI - computes the inverse of a triangular matrix A stored in RFP
7 format
8
10 SUBROUTINE DTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
11 CHARACTER TRANSR, UPLO, DIAG
13 INTEGER INFO, N
15 DOUBLE PRECISION A( 0: * )
17
19 DTFTRI computes the inverse of a triangular matrix A stored in RFP for‐
20 mat. This is a Level 3 BLAS version of the algorithm.
21
23 TRANSR (input) CHARACTER
24 = 'N': The Normal TRANSR of RFP A is stored;
25 = 'T': The Transpose TRANSR of RFP A is stored.
26 UPLO (input) CHARACTER
28 = 'U': A is upper triangular;
29 = 'L': A is lower triangular.
30 DIAG (input) CHARACTER
32 = 'N': A is non-unit triangular;
33 = 'U': A is unit triangular.
34 N (input) INTEGER
36 The order of the matrix A. N >= 0.
37 A (input/output) DOUBLE PRECISION array, dimension (0:nt-1);
39 nt=N*(N+1)/2. On entry, the triangular factor of a Hermitian
40 Positive Definite matrix A in RFP format. RFP format is
41 described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
42 then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
43 (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
44 the transpose of RFP A as defined when TRANSR = 'N'. The con‐
45 tents of RFP A are defined by UPLO as follows: If UPLO = 'U'
46 the RFP A contains the nt elements of upper packed A; If UPLO =
47 'L' the RFP A contains the nt elements of lower packed A. The
48 LDA of RFP A is (N+1)/2 when TRANSR = 'T'. When TRANSR is 'N'
49 the LDA is N+1 when N is even and N is odd. See the Note below
50 for more details. On exit, the (triangular) inverse of the
51 original matrix, in the same storage format.
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, A(i,i) is exactly zero. The triangular
57 matrix is singular and its inverse can not be computed.
58
60 We first consider Rectangular Full Packed (RFP) Format when N is even.
61 We give an example where N = 6.
62 AP is Upper AP is Lower
63 00 01 02 03 04 05 00
64 11 12 13 14 15 10 11
65 22 23 24 25 20 21 22
66 33 34 35 30 31 32 33
67 44 45 40 41 42 43 44
68 55 50 51 52 53 54 55
69 Let TRANSR = 'N'. RFP holds AP as follows:
70 For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
71 three columns of AP upper. The lower triangle A(4:6,0:2) consists of
72 the transpose of the first three columns of AP upper.
73 For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
74 three columns of AP lower. The upper triangle A(0:2,0:2) consists of
75 the transpose of the last three columns of AP lower.
76 This covers the case N even and TRANSR = 'N'.
77 RFP A RFP A
78 03 04 05 33 43 53
79 13 14 15 00 44 54
80 23 24 25 10 11 55
81 33 34 35 20 21 22
82 00 44 45 30 31 32
83 01 11 55 40 41 42
84 02 12 22 50 51 52
85 Now let TRANSR = 'T'. RFP A in both UPLO cases is just the transpose of
86 RFP A above. One therefore gets:
87 RFP A RFP A
88 03 13 23 33 00 01 02 33 00 10 20 30 40 50
89 04 14 24 34 44 11 12 43 44 11 21 31 41 51
90 05 15 25 35 45 55 22 53 54 55 22 32 42 52
91 We first consider Rectangular Full Packed (RFP) Format when N is odd.
92 We give an example where N = 5.
93 AP is Upper AP is Lower
94 00 01 02 03 04 00
95 11 12 13 14 10 11
96 22 23 24 20 21 22
97 33 34 30 31 32 33
98 44 40 41 42 43 44
99 Let TRANSR = 'N'. RFP holds AP as follows:
100 For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
101 three columns of AP upper. The lower triangle A(3:4,0:1) consists of
102 the transpose of the first two columns of AP upper.
103 For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
104 three columns of AP lower. The upper triangle A(0:1,1:2) consists of
105 the transpose of the last two columns of AP lower.
106 This covers the case N odd and TRANSR = 'N'.
107 RFP A RFP A
108 02 03 04 00 33 43
109 12 13 14 10 11 44
110 22 23 24 20 21 22
111 00 33 34 30 31 32
112 01 11 44 40 41 42
113 Now let TRANSR = 'T'. RFP A in both UPLO cases is just the transpose of
114 RFP A above. One therefore gets:
115 RFP A RFP A
116 02 12 22 00 01 00 10 20 30 40 50
117 03 13 23 33 11 33 11 21 31 41 51
118 04 14 24 34 44 43 44 22 32 42 52
119 LAPACK routine (version 3.2) November 2008 DTFTRI(1)