1ZPFTRI(1)LAPACK routine (version 3.2) ZPFTRI(1)
2
6 ZPFTRI - computes the inverse of a complex Hermitian positive definite
7 matrix A using the Cholesky factorization A = U**H*U or A = L*L**H com‐
8 puted by ZPFTRF
9
11 SUBROUTINE ZPFTRI( TRANSR, UPLO, N, A, INFO )
12 CHARACTER TRANSR, UPLO
14 INTEGER INFO, N
16 COMPLEX*16 A( 0: * )
18
20 ZPFTRI computes the inverse of a complex Hermitian positive definite
21 matrix A using the Cholesky factorization A = U**H*U or A = L*L**H com‐
22 puted by ZPFTRF.
23
25 TRANSR (input) CHARACTER
26 = 'N': The Normal TRANSR of RFP A is stored;
27 = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
28 UPLO (input) CHARACTER
30 = 'U': Upper triangle of A is stored;
31 = 'L': Lower triangle of A is stored.
32 N (input) INTEGER
34 The order of the matrix A. N >= 0.
35 A (input/output) COMPLEX*16 array, dimension ( N*(N+1)/2 );
37 On entry, the Hermitian matrix A in RFP format. RFP format is
38 described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
39 then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
40 (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
41 the Conjugate-transpose of RFP A as defined when TRANSR = 'N'.
42 The contents of RFP A are defined by UPLO as follows: If UPLO =
43 'U' the RFP A contains the nt elements of upper packed A. If
44 UPLO = 'L' the RFP A contains the elements of lower packed A.
45 The LDA of RFP A is (N+1)/2 when TRANSR = 'C'. When TRANSR is
46 'N' the LDA is N+1 when N is even and N is odd. See the Note
47 below for more details. On exit, the Hermitian inverse of the
48 original matrix, in the same storage format.
49 INFO (output) INTEGER
51 = 0: successful exit
52 < 0: if INFO = -i, the i-th argument had an illegal value
53 > 0: if INFO = i, the (i,i) element of the factor U or L is
54 zero, and the inverse could not be computed.
55
57 We first consider Standard Packed Format when N is even.
58 We give an example where N = 6.
59 AP is Upper AP is Lower
60 00 01 02 03 04 05 00
61 11 12 13 14 15 10 11
62 22 23 24 25 20 21 22
63 33 34 35 30 31 32 33
64 44 45 40 41 42 43 44
65 55 50 51 52 53 54 55
66 Let TRANSR = 'N'. RFP holds AP as follows:
67 For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
68 three columns of AP upper. The lower triangle A(4:6,0:2) consists of
69 conjugate-transpose of the first three columns of AP upper. For UPLO =
70 'L' the lower trapezoid A(1:6,0:2) consists of the first three columns
71 of AP lower. The upper triangle A(0:2,0:2) consists of conjugate-trans‐
72 pose of the last three columns of AP lower. To denote conjugate we
73 place -- above the element. This covers the case N even and TRANSR =
74 'N'.
75 RFP A RFP A
76 -- -- --
77 03 04 05 33 43 53
78 -- --
79 13 14 15 00 44 54
80 --
81 23 24 25 10 11 55
82 33 34 35 20 21 22
83 --
84 00 44 45 30 31 32
85 -- --
86 01 11 55 40 41 42
87 -- -- --
88 02 12 22 50 51 52
89 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
90 transpose of RFP A above. One therefore gets:
91 RFP A RFP A
92 -- -- -- -- -- -- -- -- -- --
93 03 13 23 33 00 01 02 33 00 10 20 30 40 50
94 -- -- -- -- -- -- -- -- -- --
95 04 14 24 34 44 11 12 43 44 11 21 31 41 51
96 -- -- -- -- -- -- -- -- -- --
97 05 15 25 35 45 55 22 53 54 55 22 32 42 52
98 We next consider Standard Packed Format when N is odd.
99 We give an example where N = 5.
100 AP is Upper AP is Lower
101 00 01 02 03 04 00
102 11 12 13 14 10 11
103 22 23 24 20 21 22
104 33 34 30 31 32 33
105 44 40 41 42 43 44
106 Let TRANSR = 'N'. RFP holds AP as follows:
107 For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
108 three columns of AP upper. The lower triangle A(3:4,0:1) consists of
109 conjugate-transpose of the first two columns of AP upper. For UPLO =
110 'L' the lower trapezoid A(0:4,0:2) consists of the first three columns
111 of AP lower. The upper triangle A(0:1,1:2) consists of conjugate-trans‐
112 pose of the last two columns of AP lower. To denote conjugate we
113 place -- above the element. This covers the case N odd and TRANSR =
114 'N'.
115 RFP A RFP A
116 -- --
117 02 03 04 00 33 43
118 --
119 12 13 14 10 11 44
120 22 23 24 20 21 22
121 --
122 00 33 34 30 31 32
123 -- --
124 01 11 44 40 41 42
125 Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
126 transpose of RFP A above. One therefore gets:
127 RFP A RFP A
128 -- -- -- -- -- -- -- -- --
129 02 12 22 00 01 00 10 20 30 40 50
130 -- -- -- -- -- -- -- -- --
131 03 13 23 33 11 33 11 21 31 41 51
132 -- -- -- -- -- -- -- -- --
133 04 14 24 34 44 43 44 22 32 42 52
134 LAPACK routine (version 3.2) November 2008 ZPFTRI(1)