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
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
|
---
:name: cspsvx
:md5sum: 301c107c780567b960fdeac5c4653ffd
:category: :subroutine
:arguments:
- fact:
:type: char
:intent: input
- uplo:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- nrhs:
:type: integer
:intent: input
- ap:
:type: complex
:intent: input
:dims:
- n*(n+1)/2
- afp:
:type: complex
:intent: input/output
:dims:
- n*(n+1)/2
- ipiv:
:type: integer
:intent: input/output
:dims:
- n
- b:
:type: complex
:intent: input
:dims:
- ldb
- nrhs
- ldb:
:type: integer
:intent: input
- x:
:type: complex
:intent: output
:dims:
- ldx
- nrhs
- ldx:
:type: integer
:intent: input
- rcond:
:type: real
:intent: output
- ferr:
:type: real
:intent: output
:dims:
- nrhs
- berr:
:type: real
:intent: output
:dims:
- nrhs
- work:
:type: complex
:intent: workspace
:dims:
- 2*n
- rwork:
:type: real
:intent: workspace
:dims:
- n
- info:
:type: integer
:intent: output
:substitutions:
ldx: MAX(1,n)
:fortran_help: " SUBROUTINE CSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* CSPSVX uses the diagonal pivoting factorization A = U*D*U**T or\n\
* A = L*D*L**T to compute the solution to a complex system of linear\n\
* equations A * X = B, where A is an N-by-N symmetric matrix stored\n\
* in packed format and X and B are N-by-NRHS matrices.\n\
*\n\
* Error bounds on the solution and a condition estimate are also\n\
* provided.\n\
*\n\
* Description\n\
* ===========\n\
*\n\
* The following steps are performed:\n\
*\n\
* 1. If FACT = 'N', the diagonal pivoting method is used to factor A as\n\
* A = U * D * U**T, if UPLO = 'U', or\n\
* A = L * D * L**T, if UPLO = 'L',\n\
* where U (or L) is a product of permutation and unit upper (lower)\n\
* triangular matrices and D is symmetric and block diagonal with\n\
* 1-by-1 and 2-by-2 diagonal blocks.\n\
*\n\
* 2. If some D(i,i)=0, so that D is exactly singular, then the routine\n\
* returns with INFO = i. Otherwise, the factored form of A is used\n\
* to estimate the condition number of the matrix A. If the\n\
* reciprocal of the condition number is less than machine precision,\n\
* INFO = N+1 is returned as a warning, but the routine still goes on\n\
* to solve for X and compute error bounds as described below.\n\
*\n\
* 3. The system of equations is solved for X using the factored form\n\
* of A.\n\
*\n\
* 4. Iterative refinement is applied to improve the computed solution\n\
* matrix and calculate error bounds and backward error estimates\n\
* for it.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* FACT (input) CHARACTER*1\n\
* Specifies whether or not the factored form of A has been\n\
* supplied on entry.\n\
* = 'F': On entry, AFP and IPIV contain the factored form\n\
* of A. AP, AFP and IPIV will not be modified.\n\
* = 'N': The matrix A will be copied to AFP and factored.\n\
*\n\
* UPLO (input) CHARACTER*1\n\
* = 'U': Upper triangle of A is stored;\n\
* = 'L': Lower triangle of A is stored.\n\
*\n\
* N (input) INTEGER\n\
* The number of linear equations, i.e., the order of the\n\
* matrix A. N >= 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\
* AP (input) COMPLEX array, dimension (N*(N+1)/2)\n\
* The upper or lower triangle of the symmetric matrix A, packed\n\
* columnwise in a linear array. The j-th column of A is stored\n\
* in the array AP as follows:\n\
* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;\n\
* if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.\n\
* See below for further details.\n\
*\n\
* AFP (input or output) COMPLEX array, dimension (N*(N+1)/2)\n\
* If FACT = 'F', then AFP is an input argument and on entry\n\
* contains the block diagonal matrix D and the multipliers used\n\
* to obtain the factor U or L from the factorization\n\
* A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as\n\
* a packed triangular matrix in the same storage format as A.\n\
*\n\
* If FACT = 'N', then AFP is an output argument and on exit\n\
* contains the block diagonal matrix D and the multipliers used\n\
* to obtain the factor U or L from the factorization\n\
* A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as\n\
* a packed triangular matrix in the same storage format as A.\n\
*\n\
* IPIV (input or output) INTEGER array, dimension (N)\n\
* If FACT = 'F', then IPIV is an input argument and on entry\n\
* contains details of the interchanges and the block structure\n\
* of D, as determined by CSPTRF.\n\
* If IPIV(k) > 0, then rows and columns k and IPIV(k) were\n\
* interchanged and D(k,k) is a 1-by-1 diagonal block.\n\
* If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and\n\
* columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)\n\
* is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =\n\
* IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were\n\
* interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.\n\
*\n\
* If FACT = 'N', then IPIV is an output argument and on exit\n\
* contains details of the interchanges and the block structure\n\
* of D, as determined by CSPTRF.\n\
*\n\
* B (input) COMPLEX array, dimension (LDB,NRHS)\n\
* The N-by-NRHS 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 (output) COMPLEX array, dimension (LDX,NRHS)\n\
* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.\n\
*\n\
* LDX (input) INTEGER\n\
* The leading dimension of the array X. LDX >= max(1,N).\n\
*\n\
* RCOND (output) REAL\n\
* The estimate of the reciprocal condition number of the matrix\n\
* A. If RCOND is less than the machine precision (in\n\
* particular, if RCOND = 0), the matrix is singular to working\n\
* precision. This condition is indicated by a return code of\n\
* INFO > 0.\n\
*\n\
* FERR (output) REAL 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) REAL 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) COMPLEX array, dimension (2*N)\n\
*\n\
* RWORK (workspace) REAL 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\
* > 0: if INFO = i, and i is\n\
* <= N: D(i,i) is exactly zero. The factorization\n\
* has been completed but the factor D is exactly\n\
* singular, so the solution and error bounds could\n\
* not be computed. RCOND = 0 is returned.\n\
* = N+1: D is nonsingular, but RCOND is less than machine\n\
* precision, meaning that the matrix is singular\n\
* to working precision. Nevertheless, the\n\
* solution and error bounds are computed because\n\
* there are a number of situations where the\n\
* computed solution can be more accurate than the\n\
* value of RCOND would suggest.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* The packed storage scheme is illustrated by the following example\n\
* when N = 4, UPLO = 'U':\n\
*\n\
* Two-dimensional storage of the symmetric matrix A:\n\
*\n\
* a11 a12 a13 a14\n\
* a22 a23 a24\n\
* a33 a34 (aij = aji)\n\
* a44\n\
*\n\
* Packed storage of the upper triangle of A:\n\
*\n\
* AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]\n\
*\n\
* =====================================================================\n\
*\n"
|
