LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
sppsvx.f
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1 *> \brief <b> SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
22 * X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER EQUED, FACT, UPLO
26 * INTEGER INFO, LDB, LDX, N, NRHS
27 * REAL RCOND
28 * ..
29 * .. Array Arguments ..
30 * INTEGER IWORK( * )
31 * REAL AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
32 * \$ FERR( * ), S( * ), WORK( * ), X( LDX, * )
33 * ..
34 *
35 *
36 *> \par Purpose:
37 * =============
38 *>
39 *> \verbatim
40 *>
41 *> SPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
42 *> compute the solution to a real system of linear equations
43 *> A * X = B,
44 *> where A is an N-by-N symmetric positive definite matrix stored in
45 *> packed format and X and B are N-by-NRHS matrices.
46 *>
47 *> Error bounds on the solution and a condition estimate are also
48 *> provided.
49 *> \endverbatim
50 *
51 *> \par Description:
52 * =================
53 *>
54 *> \verbatim
55 *>
56 *> The following steps are performed:
57 *>
58 *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
59 *> the system:
60 *> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
61 *> Whether or not the system will be equilibrated depends on the
62 *> scaling of the matrix A, but if equilibration is used, A is
63 *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
64 *>
65 *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
66 *> factor the matrix A (after equilibration if FACT = 'E') as
67 *> A = U**T* U, if UPLO = 'U', or
68 *> A = L * L**T, if UPLO = 'L',
69 *> where U is an upper triangular matrix and L is a lower triangular
70 *> matrix.
71 *>
72 *> 3. If the leading i-by-i principal minor is not positive definite,
73 *> then the routine returns with INFO = i. Otherwise, the factored
74 *> form of A is used to estimate the condition number of the matrix
75 *> A. If the reciprocal of the condition number is less than machine
76 *> precision, INFO = N+1 is returned as a warning, but the routine
77 *> still goes on to solve for X and compute error bounds as
78 *> described below.
79 *>
80 *> 4. The system of equations is solved for X using the factored form
81 *> of A.
82 *>
83 *> 5. Iterative refinement is applied to improve the computed solution
84 *> matrix and calculate error bounds and backward error estimates
85 *> for it.
86 *>
87 *> 6. If equilibration was used, the matrix X is premultiplied by
88 *> diag(S) so that it solves the original system before
89 *> equilibration.
90 *> \endverbatim
91 *
92 * Arguments:
93 * ==========
94 *
95 *> \param[in] FACT
96 *> \verbatim
97 *> FACT is CHARACTER*1
98 *> Specifies whether or not the factored form of the matrix A is
99 *> supplied on entry, and if not, whether the matrix A should be
100 *> equilibrated before it is factored.
101 *> = 'F': On entry, AFP contains the factored form of A.
102 *> If EQUED = 'Y', the matrix A has been equilibrated
103 *> with scaling factors given by S. AP and AFP will not
104 *> be modified.
105 *> = 'N': The matrix A will be copied to AFP and factored.
106 *> = 'E': The matrix A will be equilibrated if necessary, then
107 *> copied to AFP and factored.
108 *> \endverbatim
109 *>
110 *> \param[in] UPLO
111 *> \verbatim
112 *> UPLO is CHARACTER*1
113 *> = 'U': Upper triangle of A is stored;
114 *> = 'L': Lower triangle of A is stored.
115 *> \endverbatim
116 *>
117 *> \param[in] N
118 *> \verbatim
119 *> N is INTEGER
120 *> The number of linear equations, i.e., the order of the
121 *> matrix A. N >= 0.
122 *> \endverbatim
123 *>
124 *> \param[in] NRHS
125 *> \verbatim
126 *> NRHS is INTEGER
127 *> The number of right hand sides, i.e., the number of columns
128 *> of the matrices B and X. NRHS >= 0.
129 *> \endverbatim
130 *>
131 *> \param[in,out] AP
132 *> \verbatim
133 *> AP is REAL array, dimension (N*(N+1)/2)
134 *> On entry, the upper or lower triangle of the symmetric matrix
135 *> A, packed columnwise in a linear array, except if FACT = 'F'
136 *> and EQUED = 'Y', then A must contain the equilibrated matrix
137 *> diag(S)*A*diag(S). The j-th column of A is stored in the
138 *> array AP as follows:
139 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
140 *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
141 *> See below for further details. A is not modified if
142 *> FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
143 *>
144 *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
145 *> diag(S)*A*diag(S).
146 *> \endverbatim
147 *>
148 *> \param[in,out] AFP
149 *> \verbatim
150 *> AFP is REAL array, dimension
151 *> (N*(N+1)/2)
152 *> If FACT = 'F', then AFP is an input argument and on entry
153 *> contains the triangular factor U or L from the Cholesky
154 *> factorization A = U**T*U or A = L*L**T, in the same storage
155 *> format as A. If EQUED .ne. 'N', then AFP is the factored
156 *> form of the equilibrated matrix A.
157 *>
158 *> If FACT = 'N', then AFP is an output argument and on exit
159 *> returns the triangular factor U or L from the Cholesky
160 *> factorization A = U**T * U or A = L * L**T of the original
161 *> matrix A.
162 *>
163 *> If FACT = 'E', then AFP is an output argument and on exit
164 *> returns the triangular factor U or L from the Cholesky
165 *> factorization A = U**T * U or A = L * L**T of the equilibrated
166 *> matrix A (see the description of AP for the form of the
167 *> equilibrated matrix).
168 *> \endverbatim
169 *>
170 *> \param[in,out] EQUED
171 *> \verbatim
172 *> EQUED is CHARACTER*1
173 *> Specifies the form of equilibration that was done.
174 *> = 'N': No equilibration (always true if FACT = 'N').
175 *> = 'Y': Equilibration was done, i.e., A has been replaced by
176 *> diag(S) * A * diag(S).
177 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
178 *> output argument.
179 *> \endverbatim
180 *>
181 *> \param[in,out] S
182 *> \verbatim
183 *> S is REAL array, dimension (N)
184 *> The scale factors for A; not accessed if EQUED = 'N'. S is
185 *> an input argument if FACT = 'F'; otherwise, S is an output
186 *> argument. If FACT = 'F' and EQUED = 'Y', each element of S
187 *> must be positive.
188 *> \endverbatim
189 *>
190 *> \param[in,out] B
191 *> \verbatim
192 *> B is REAL array, dimension (LDB,NRHS)
193 *> On entry, the N-by-NRHS right hand side matrix B.
194 *> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
195 *> B is overwritten by diag(S) * B.
196 *> \endverbatim
197 *>
198 *> \param[in] LDB
199 *> \verbatim
200 *> LDB is INTEGER
201 *> The leading dimension of the array B. LDB >= max(1,N).
202 *> \endverbatim
203 *>
204 *> \param[out] X
205 *> \verbatim
206 *> X is REAL array, dimension (LDX,NRHS)
207 *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
208 *> the original system of equations. Note that if EQUED = 'Y',
209 *> A and B are modified on exit, and the solution to the
210 *> equilibrated system is inv(diag(S))*X.
211 *> \endverbatim
212 *>
213 *> \param[in] LDX
214 *> \verbatim
215 *> LDX is INTEGER
216 *> The leading dimension of the array X. LDX >= max(1,N).
217 *> \endverbatim
218 *>
219 *> \param[out] RCOND
220 *> \verbatim
221 *> RCOND is REAL
222 *> The estimate of the reciprocal condition number of the matrix
223 *> A after equilibration (if done). If RCOND is less than the
224 *> machine precision (in particular, if RCOND = 0), the matrix
225 *> is singular to working precision. This condition is
226 *> indicated by a return code of INFO > 0.
227 *> \endverbatim
228 *>
229 *> \param[out] FERR
230 *> \verbatim
231 *> FERR is REAL array, dimension (NRHS)
232 *> The estimated forward error bound for each solution vector
233 *> X(j) (the j-th column of the solution matrix X).
234 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
235 *> is an estimated upper bound for the magnitude of the largest
236 *> element in (X(j) - XTRUE) divided by the magnitude of the
237 *> largest element in X(j). The estimate is as reliable as
238 *> the estimate for RCOND, and is almost always a slight
239 *> overestimate of the true error.
240 *> \endverbatim
241 *>
242 *> \param[out] BERR
243 *> \verbatim
244 *> BERR is REAL array, dimension (NRHS)
245 *> The componentwise relative backward error of each solution
246 *> vector X(j) (i.e., the smallest relative change in
247 *> any element of A or B that makes X(j) an exact solution).
248 *> \endverbatim
249 *>
250 *> \param[out] WORK
251 *> \verbatim
252 *> WORK is REAL array, dimension (3*N)
253 *> \endverbatim
254 *>
255 *> \param[out] IWORK
256 *> \verbatim
257 *> IWORK is INTEGER array, dimension (N)
258 *> \endverbatim
259 *>
260 *> \param[out] INFO
261 *> \verbatim
262 *> INFO is INTEGER
263 *> = 0: successful exit
264 *> < 0: if INFO = -i, the i-th argument had an illegal value
265 *> > 0: if INFO = i, and i is
266 *> <= N: the leading minor of order i of A is
267 *> not positive definite, so the factorization
268 *> could not be completed, and the solution has not
269 *> been computed. RCOND = 0 is returned.
270 *> = N+1: U is nonsingular, but RCOND is less than machine
271 *> precision, meaning that the matrix is singular
272 *> to working precision. Nevertheless, the
273 *> solution and error bounds are computed because
274 *> there are a number of situations where the
275 *> computed solution can be more accurate than the
276 *> value of RCOND would suggest.
277 *> \endverbatim
278 *
279 * Authors:
280 * ========
281 *
282 *> \author Univ. of Tennessee
283 *> \author Univ. of California Berkeley
284 *> \author Univ. of Colorado Denver
285 *> \author NAG Ltd.
286 *
287 *> \date April 2012
288 *
289 *> \ingroup realOTHERsolve
290 *
291 *> \par Further Details:
292 * =====================
293 *>
294 *> \verbatim
295 *>
296 *> The packed storage scheme is illustrated by the following example
297 *> when N = 4, UPLO = 'U':
298 *>
299 *> Two-dimensional storage of the symmetric matrix A:
300 *>
301 *> a11 a12 a13 a14
302 *> a22 a23 a24
303 *> a33 a34 (aij = conjg(aji))
304 *> a44
305 *>
306 *> Packed storage of the upper triangle of A:
307 *>
308 *> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
309 *> \endverbatim
310 *>
311 * =====================================================================
312  SUBROUTINE sppsvx( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
313  \$ x, ldx, rcond, ferr, berr, work, iwork, info )
314 *
315 * -- LAPACK driver routine (version 3.4.1) --
316 * -- LAPACK is a software package provided by Univ. of Tennessee, --
317 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
318 * April 2012
319 *
320 * .. Scalar Arguments ..
321  CHARACTER EQUED, FACT, UPLO
322  INTEGER INFO, LDB, LDX, N, NRHS
323  REAL RCOND
324 * ..
325 * .. Array Arguments ..
326  INTEGER IWORK( * )
327  REAL AFP( * ), AP( * ), B( ldb, * ), BERR( * ),
328  \$ ferr( * ), s( * ), work( * ), x( ldx, * )
329 * ..
330 *
331 * =====================================================================
332 *
333 * .. Parameters ..
334  REAL ZERO, ONE
335  parameter ( zero = 0.0e+0, one = 1.0e+0 )
336 * ..
337 * .. Local Scalars ..
338  LOGICAL EQUIL, NOFACT, RCEQU
339  INTEGER I, INFEQU, J
340  REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
341 * ..
342 * .. External Functions ..
343  LOGICAL LSAME
344  REAL SLAMCH, SLANSP
345  EXTERNAL lsame, slamch, slansp
346 * ..
347 * .. External Subroutines ..
348  EXTERNAL scopy, slacpy, slaqsp, sppcon, sppequ, spprfs,
349  \$ spptrf, spptrs, xerbla
350 * ..
351 * .. Intrinsic Functions ..
352  INTRINSIC max, min
353 * ..
354 * .. Executable Statements ..
355 *
356  info = 0
357  nofact = lsame( fact, 'N' )
358  equil = lsame( fact, 'E' )
359  IF( nofact .OR. equil ) THEN
360  equed = 'N'
361  rcequ = .false.
362  ELSE
363  rcequ = lsame( equed, 'Y' )
364  smlnum = slamch( 'Safe minimum' )
365  bignum = one / smlnum
366  END IF
367 *
368 * Test the input parameters.
369 *
370  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
371  \$ THEN
372  info = -1
373  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
374  \$ THEN
375  info = -2
376  ELSE IF( n.LT.0 ) THEN
377  info = -3
378  ELSE IF( nrhs.LT.0 ) THEN
379  info = -4
380  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
381  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
382  info = -7
383  ELSE
384  IF( rcequ ) THEN
385  smin = bignum
386  smax = zero
387  DO 10 j = 1, n
388  smin = min( smin, s( j ) )
389  smax = max( smax, s( j ) )
390  10 CONTINUE
391  IF( smin.LE.zero ) THEN
392  info = -8
393  ELSE IF( n.GT.0 ) THEN
394  scond = max( smin, smlnum ) / min( smax, bignum )
395  ELSE
396  scond = one
397  END IF
398  END IF
399  IF( info.EQ.0 ) THEN
400  IF( ldb.LT.max( 1, n ) ) THEN
401  info = -10
402  ELSE IF( ldx.LT.max( 1, n ) ) THEN
403  info = -12
404  END IF
405  END IF
406  END IF
407 *
408  IF( info.NE.0 ) THEN
409  CALL xerbla( 'SPPSVX', -info )
410  RETURN
411  END IF
412 *
413  IF( equil ) THEN
414 *
415 * Compute row and column scalings to equilibrate the matrix A.
416 *
417  CALL sppequ( uplo, n, ap, s, scond, amax, infequ )
418  IF( infequ.EQ.0 ) THEN
419 *
420 * Equilibrate the matrix.
421 *
422  CALL slaqsp( uplo, n, ap, s, scond, amax, equed )
423  rcequ = lsame( equed, 'Y' )
424  END IF
425  END IF
426 *
427 * Scale the right-hand side.
428 *
429  IF( rcequ ) THEN
430  DO 30 j = 1, nrhs
431  DO 20 i = 1, n
432  b( i, j ) = s( i )*b( i, j )
433  20 CONTINUE
434  30 CONTINUE
435  END IF
436 *
437  IF( nofact .OR. equil ) THEN
438 *
439 * Compute the Cholesky factorization A = U**T * U or A = L * L**T.
440 *
441  CALL scopy( n*( n+1 ) / 2, ap, 1, afp, 1 )
442  CALL spptrf( uplo, n, afp, info )
443 *
444 * Return if INFO is non-zero.
445 *
446  IF( info.GT.0 )THEN
447  rcond = zero
448  RETURN
449  END IF
450  END IF
451 *
452 * Compute the norm of the matrix A.
453 *
454  anorm = slansp( 'I', uplo, n, ap, work )
455 *
456 * Compute the reciprocal of the condition number of A.
457 *
458  CALL sppcon( uplo, n, afp, anorm, rcond, work, iwork, info )
459 *
460 * Compute the solution matrix X.
461 *
462  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
463  CALL spptrs( uplo, n, nrhs, afp, x, ldx, info )
464 *
465 * Use iterative refinement to improve the computed solution and
466 * compute error bounds and backward error estimates for it.
467 *
468  CALL spprfs( uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr,
469  \$ work, iwork, info )
470 *
471 * Transform the solution matrix X to a solution of the original
472 * system.
473 *
474  IF( rcequ ) THEN
475  DO 50 j = 1, nrhs
476  DO 40 i = 1, n
477  x( i, j ) = s( i )*x( i, j )
478  40 CONTINUE
479  50 CONTINUE
480  DO 60 j = 1, nrhs
481  ferr( j ) = ferr( j ) / scond
482  60 CONTINUE
483  END IF
484 *
485 * Set INFO = N+1 if the matrix is singular to working precision.
486 *
487  IF( rcond.LT.slamch( 'Epsilon' ) )
488  \$ info = n + 1
489 *
490  RETURN
491 *
492 * End of SPPSVX
493 *
494  END
subroutine sppcon(UPLO, N, AP, ANORM, RCOND, WORK, IWORK, INFO)
SPPCON
Definition: sppcon.f:120
subroutine spptrs(UPLO, N, NRHS, AP, B, LDB, INFO)
SPPTRS
Definition: spptrs.f:110
subroutine slaqsp(UPLO, N, AP, S, SCOND, AMAX, EQUED)
SLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppeq...
Definition: slaqsp.f:127
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine sppsvx(FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices ...
Definition: sppsvx.f:314
subroutine spptrf(UPLO, N, AP, INFO)
SPPTRF
Definition: spptrf.f:121
subroutine spprfs(UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SPPRFS
Definition: spprfs.f:173
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53
subroutine sppequ(UPLO, N, AP, S, SCOND, AMAX, INFO)
SPPEQU
Definition: sppequ.f:118