1ZTFTRI(1)LAPACK routine (version 3.2) ZTFTRI(1)
2
6 ZTFTRI - computes the inverse of a triangular matrix A stored in RFP
7 format
8
10 SUBROUTINE ZTFTRI( TRANSR, UPLO, DIAG, N, A, INFO )
11 CHARACTER TRANSR, UPLO, DIAG
13 INTEGER INFO, N
15 COMPLEX*16 A( 0: * )
17
19 ZTFTRI 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 = 'C': The Conjugate-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) COMPLEX*16 array, dimension ( N*(N+1)/2 );
39 On entry, the triangular matrix A in RFP format. RFP format is
40 described by TRANSR, UPLO, and N as follows: If TRANSR =
41 'N' 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 nt elements of lower packed
47 A. The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR
48 is 'N' the LDA is N+1 when N is even and N is odd. See the Note
49 below for more details. On exit, the (triangular) inverse of
50 the original matrix, in the same storage format.
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, A(i,i) is exactly zero. The triangular
56 matrix is singular and its inverse can not be computed.
57
59 We first consider Standard Packed 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 conjugate-transpose of the first three columns of AP upper. For UPLO =
72 'L' the lower trapezoid A(1:6,0:2) consists of the first three columns
73 of AP lower. The upper triangle A(0:2,0:2) consists of conjugate-trans‐
74 pose of the last three columns of AP lower. To denote conjugate we
75 place -- above the element. This covers the case N even and TRANSR =
76 'N'.
77 RFP A RFP A
78 -- -- --
79 03 04 05 33 43 53
80 -- --
81 13 14 15 00 44 54
82 --
83 23 24 25 10 11 55
84 33 34 35 20 21 22
85 --
86 00 44 45 30 31 32
87 -- --
88 01 11 55 40 41 42
89 -- -- --
90 02 12 22 50 51 52
91 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
92 transpose of RFP A above. One therefore gets:
93 RFP A RFP A
94 -- -- -- -- -- -- -- -- -- --
95 03 13 23 33 00 01 02 33 00 10 20 30 40 50
96 -- -- -- -- -- -- -- -- -- --
97 04 14 24 34 44 11 12 43 44 11 21 31 41 51
98 -- -- -- -- -- -- -- -- -- --
99 05 15 25 35 45 55 22 53 54 55 22 32 42 52
100 We next consider Standard Packed Format when N is odd.
101 We give an example where N = 5.
102 AP is Upper AP is Lower
103 00 01 02 03 04 00
104 11 12 13 14 10 11
105 22 23 24 20 21 22
106 33 34 30 31 32 33
107 44 40 41 42 43 44
108 Let TRANSR = 'N'. RFP holds AP as follows:
109 For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
110 three columns of AP upper. The lower triangle A(3:4,0:1) consists of
111 conjugate-transpose of the first two columns of AP upper. For UPLO =
112 'L' the lower trapezoid A(0:4,0:2) consists of the first three columns
113 of AP lower. The upper triangle A(0:1,1:2) consists of conjugate-trans‐
114 pose of the last two columns of AP lower. To denote conjugate we
115 place -- above the element. This covers the case N odd and TRANSR =
116 'N'.
117 RFP A RFP A
118 -- --
119 02 03 04 00 33 43
120 --
121 12 13 14 10 11 44
122 22 23 24 20 21 22
123 --
124 00 33 34 30 31 32
125 -- --
126 01 11 44 40 41 42
127 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
128 transpose of RFP A above. One therefore gets:
129 RFP A RFP A
130 -- -- -- -- -- -- -- -- --
131 02 12 22 00 01 00 10 20 30 40 50
132 -- -- -- -- -- -- -- -- --
133 03 13 23 33 11 33 11 21 31 41 51
134 -- -- -- -- -- -- -- -- --
135 04 14 24 34 44 43 44 22 32 42 52
136 LAPACK routine (version 3.2) November 2008 ZTFTRI(1)