1SPFTRS(1)LAPACK routine (version 3.2) SPFTRS(1)
2
6 SPFTRS - solves a system of linear equations A*X = B with a symmetric
7 positive definite matrix A using the Cholesky factorization A = U**T*U
8 or A = L*L**T computed by SPFTRF
9
11 SUBROUTINE SPFTRS( TRANSR, UPLO, N, NRHS, A, B, LDB, INFO )
12 CHARACTER TRANSR, UPLO
14 INTEGER INFO, LDB, N, NRHS
16 REAL A( 0: * ), B( LDB, * )
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20 SPFTRS solves a system of linear equations A*X = B with a symmetric
21 positive definite matrix A using the Cholesky factorization A = U**T*U
22 or A = L*L**T computed 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 RFP A is stored;
31 = 'L': Lower triangle of RFP A is stored.
32 N (input) INTEGER
34 The order of the matrix A. N >= 0.
35 NRHS (input) INTEGER
37 The number of right hand sides, i.e., the number of columns of
38 the matrix B. NRHS >= 0.
39 A (input) REAL array, dimension ( N*(N+1)/2 )
41 The triangular factor U or L from the Cholesky factorization of
42 RFP A = U**H*U or RFP A = L*L**T, as computed by SPFTRF. See
43 note below for more details about RFP A.
44 B (input/output) REAL array, dimension (LDB,NRHS)
46 On entry, the right hand side matrix B. On exit, the solution
47 matrix X.
48 LDB (input) INTEGER
50 The leading dimension of the array B. LDB >= max(1,N).
51 INFO (output) INTEGER
53 = 0: successful exit
54 < 0: if INFO = -i, the i-th argument had an illegal value
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 SPFTRS(1)