1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
|
SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
* -- LAPACK driver routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* DPBSV computes the solution to a real system of linear equations
* A * X = B,
* where A is an N-by-N symmetric positive definite band matrix and X
* and B are N-by-NRHS matrices.
*
* The Cholesky decomposition is used to factor A as
* A = U**T * U, if UPLO = 'U', or
* A = L * L**T, if UPLO = 'L',
* where U is an upper triangular band matrix, and L is a lower
* triangular band matrix, with the same number of superdiagonals or
* subdiagonals as A. The factored form of A is then used to solve the
* system of equations A * X = B.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The number of linear equations, i.e., the order of the
* matrix A. N >= 0.
*
* KD (input) INTEGER
* The number of superdiagonals of the matrix A if UPLO = 'U',
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
* On entry, the upper or lower triangle of the symmetric band
* matrix A, stored in the first KD+1 rows of the array. The
* j-th column of A is stored in the j-th column of the array AB
* as follows:
* if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
* See below for further details.
*
* On exit, if INFO = 0, the triangular factor U or L from the
* Cholesky factorization A = U**T*U or A = L*L**T of the band
* matrix A, in the same storage format as A.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= KD+1.
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS right hand side matrix B.
* On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, the leading minor of order i of A is not
* positive definite, so the factorization could not be
* completed, and the solution has not been computed.
*
* Further Details
* ===============
*
* The band storage scheme is illustrated by the following example, when
* N = 6, KD = 2, and UPLO = 'U':
*
* On entry: On exit:
*
* * * a13 a24 a35 a46 * * u13 u24 u35 u46
* * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
* a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*
* Similarly, if UPLO = 'L' the format of A is as follows:
*
* On entry: On exit:
*
* a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
* a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
* a31 a42 a53 a64 * * l31 l42 l53 l64 * *
*
* Array elements marked * are not used by the routine.
*
* =====================================================================
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DPBTRF, DPBTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KD.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPBSV ', -INFO )
RETURN
END IF
*
* Compute the Cholesky factorization A = U'*U or A = L*L'.
*
CALL DPBTRF( UPLO, N, KD, AB, LDAB, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
CALL DPBTRS( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )
*
END IF
RETURN
*
* End of DPBSV
*
END
|
