1SPFTRI(1)LAPACK routine (version 3.2) SPFTRI(1)
2
6 SPFTRI - computes the inverse of a real (symmetric) positive definite
7 matrix A using the Cholesky factorization A = U**T*U or A = L*L**T com‐
8 puted by SPFTRF
9
11 SUBROUTINE SPFTRI( TRANSR, UPLO, N, A, INFO )
12 CHARACTER TRANSR, UPLO
14 INTEGER INFO, N
16 REAL A( 0: * )
18
20 SPFTRI computes the inverse of a real (symmetric) positive definite
21 matrix A using the Cholesky factorization A = U**T*U or A = L*L**T com‐
22 puted by SPFTRF.
23
25 TRANSR (input) CHARACTER
26 = 'N': The Normal TRANSR of RFP A is stored;
27 = 'T': The 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) REAL array, dimension ( N*(N+1)/2 )
37 On entry, the symmetric 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 = 'T' then RFP is
41 the transpose of RFP A as defined when TRANSR = 'N'. The con‐
42 tents of RFP A are defined by UPLO as follows: If UPLO = 'U'
43 the RFP A contains the nt elements of upper packed A. If UPLO =
44 'L' the RFP A contains the elements of lower packed A. The LDA
45 of RFP A is (N+1)/2 when TRANSR = 'T'. When TRANSR is 'N' the
46 LDA is N+1 when N is even and N is odd. See the Note below for
47 more details. On exit, the symmetric inverse of the original
48 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 Rectangular Full Packed (RFP) 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 the transpose of the first three columns of AP upper.
70 For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
71 three columns of AP lower. The upper triangle A(0:2,0:2) consists of
72 the transpose of the last three columns of AP lower.
73 This covers the case N even and TRANSR = 'N'.
74 RFP A RFP A
75 03 04 05 33 43 53
76 13 14 15 00 44 54
77 23 24 25 10 11 55
78 33 34 35 20 21 22
79 00 44 45 30 31 32
80 01 11 55 40 41 42
81 02 12 22 50 51 52
82 Now let TRANSR = 'T'. RFP A in both UPLO cases is just the transpose of
83 RFP A above. One therefore gets:
84 RFP A RFP A
85 03 13 23 33 00 01 02 33 00 10 20 30 40 50
86 04 14 24 34 44 11 12 43 44 11 21 31 41 51
87 05 15 25 35 45 55 22 53 54 55 22 32 42 52
88 We first consider Rectangular Full Packed (RFP) Format when N is odd.
89 We give an example where N = 5.
90 AP is Upper AP is Lower
91 00 01 02 03 04 00
92 11 12 13 14 10 11
93 22 23 24 20 21 22
94 33 34 30 31 32 33
95 44 40 41 42 43 44
96 Let TRANSR = 'N'. RFP holds AP as follows:
97 For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
98 three columns of AP upper. The lower triangle A(3:4,0:1) consists of
99 the transpose of the first two columns of AP upper.
100 For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
101 three columns of AP lower. The upper triangle A(0:1,1:2) consists of
102 the transpose of the last two columns of AP lower.
103 This covers the case N odd and TRANSR = 'N'.
104 RFP A RFP A
105 02 03 04 00 33 43
106 12 13 14 10 11 44
107 22 23 24 20 21 22
108 00 33 34 30 31 32
109 01 11 44 40 41 42
110 Now let TRANSR = 'T'. RFP A in both UPLO cases is just the transpose of
111 RFP A above. One therefore gets:
112 RFP A RFP A
113 02 12 22 00 01 00 10 20 30 40 50
114 03 13 23 33 11 33 11 21 31 41 51
115 04 14 24 34 44 43 44 22 32 42 52
116 LAPACK routine (version 3.2) November 2008 SPFTRI(1)