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
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
|
---
:name: dgbrfs
:md5sum: b56549a2e7880c6fef2c12fa919a14d7
:category: :subroutine
:arguments:
- trans:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- kl:
:type: integer
:intent: input
- ku:
:type: integer
:intent: input
- nrhs:
:type: integer
:intent: input
- ab:
:type: doublereal
:intent: input
:dims:
- ldab
- n
- ldab:
:type: integer
:intent: input
- afb:
:type: doublereal
:intent: input
:dims:
- ldafb
- n
- ldafb:
:type: integer
:intent: input
- ipiv:
:type: integer
:intent: input
:dims:
- n
- b:
:type: doublereal
:intent: input
:dims:
- ldb
- nrhs
- ldb:
:type: integer
:intent: input
- x:
:type: doublereal
:intent: input/output
:dims:
- ldx
- nrhs
- ldx:
:type: integer
:intent: input
- ferr:
:type: doublereal
:intent: output
:dims:
- nrhs
- berr:
:type: doublereal
:intent: output
:dims:
- nrhs
- work:
:type: doublereal
:intent: workspace
:dims:
- 3*n
- iwork:
:type: integer
:intent: workspace
:dims:
- n
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* DGBRFS improves the computed solution to a system of linear\n\
* equations when the coefficient matrix is banded, and provides\n\
* error bounds and backward error estimates for the solution.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* TRANS (input) CHARACTER*1\n\
* Specifies the form of the system of equations:\n\
* = 'N': A * X = B (No transpose)\n\
* = 'T': A**T * X = B (Transpose)\n\
* = 'C': A**H * X = B (Conjugate transpose = Transpose)\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix A. N >= 0.\n\
*\n\
* KL (input) INTEGER\n\
* The number of subdiagonals within the band of A. KL >= 0.\n\
*\n\
* KU (input) INTEGER\n\
* The number of superdiagonals within the band of A. KU >= 0.\n\
*\n\
* NRHS (input) INTEGER\n\
* The number of right hand sides, i.e., the number of columns\n\
* of the matrices B and X. NRHS >= 0.\n\
*\n\
* AB (input) DOUBLE PRECISION array, dimension (LDAB,N)\n\
* The original band matrix A, stored in rows 1 to KL+KU+1.\n\
* The j-th column of A is stored in the j-th column of the\n\
* array AB as follows:\n\
* AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).\n\
*\n\
* LDAB (input) INTEGER\n\
* The leading dimension of the array AB. LDAB >= KL+KU+1.\n\
*\n\
* AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N)\n\
* Details of the LU factorization of the band matrix A, as\n\
* computed by DGBTRF. U is stored as an upper triangular band\n\
* matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and\n\
* the multipliers used during the factorization are stored in\n\
* rows KL+KU+2 to 2*KL+KU+1.\n\
*\n\
* LDAFB (input) INTEGER\n\
* The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.\n\
*\n\
* IPIV (input) INTEGER array, dimension (N)\n\
* The pivot indices from DGBTRF; for 1<=i<=N, row i of the\n\
* matrix was interchanged with row IPIV(i).\n\
*\n\
* B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)\n\
* The right hand side matrix B.\n\
*\n\
* LDB (input) INTEGER\n\
* The leading dimension of the array B. LDB >= max(1,N).\n\
*\n\
* X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)\n\
* On entry, the solution matrix X, as computed by DGBTRS.\n\
* On exit, the improved solution matrix X.\n\
*\n\
* LDX (input) INTEGER\n\
* The leading dimension of the array X. LDX >= max(1,N).\n\
*\n\
* FERR (output) DOUBLE PRECISION array, dimension (NRHS)\n\
* The estimated forward error bound for each solution vector\n\
* X(j) (the j-th column of the solution matrix X).\n\
* If XTRUE is the true solution corresponding to X(j), FERR(j)\n\
* is an estimated upper bound for the magnitude of the largest\n\
* element in (X(j) - XTRUE) divided by the magnitude of the\n\
* largest element in X(j). The estimate is as reliable as\n\
* the estimate for RCOND, and is almost always a slight\n\
* overestimate of the true error.\n\
*\n\
* BERR (output) DOUBLE PRECISION array, dimension (NRHS)\n\
* The componentwise relative backward error of each solution\n\
* vector X(j) (i.e., the smallest relative change in\n\
* any element of A or B that makes X(j) an exact solution).\n\
*\n\
* WORK (workspace) DOUBLE PRECISION array, dimension (3*N)\n\
*\n\
* IWORK (workspace) INTEGER array, dimension (N)\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* < 0: if INFO = -i, the i-th argument had an illegal value\n\
*\n\
* Internal Parameters\n\
* ===================\n\
*\n\
* ITMAX is the maximum number of steps of iterative refinement.\n\
*\n\n\
* =====================================================================\n\
*\n"
|
