1 *> \brief <b> ZSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices</b>
3 * =========== DOCUMENTATION ===========
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21 * SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
22 * LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
24 * .. Scalar Arguments ..
25 * CHARACTER FACT, UPLO
26 * INTEGER INFO, LDB, LDX, N, NRHS
27 * DOUBLE PRECISION RCOND
29 * .. Array Arguments ..
31 * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
32 * COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
42 *> ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
43 *> A = L*D*L**T to compute the solution to a complex system of linear
44 *> equations A * X = B, where A is an N-by-N symmetric matrix stored
45 *> in packed format and X and B are N-by-NRHS matrices.
47 *> Error bounds on the solution and a condition estimate are also
56 *> The following steps are performed:
58 *> 1. If FACT = 'N', the diagonal pivoting method is used to factor A as
59 *> A = U * D * U**T, if UPLO = 'U', or
60 *> A = L * D * L**T, if UPLO = 'L',
61 *> where U (or L) is a product of permutation and unit upper (lower)
62 *> triangular matrices and D is symmetric and block diagonal with
63 *> 1-by-1 and 2-by-2 diagonal blocks.
65 *> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
66 *> returns with INFO = i. Otherwise, the factored form of A is used
67 *> to estimate the condition number of the matrix A. If the
68 *> reciprocal of the condition number is less than machine precision,
69 *> INFO = N+1 is returned as a warning, but the routine still goes on
70 *> to solve for X and compute error bounds as described below.
72 *> 3. The system of equations is solved for X using the factored form
75 *> 4. Iterative refinement is applied to improve the computed solution
76 *> matrix and calculate error bounds and backward error estimates
85 *> FACT is CHARACTER*1
86 *> Specifies whether or not the factored form of A has been
88 *> = 'F': On entry, AFP and IPIV contain the factored form
89 *> of A. AP, AFP and IPIV will not be modified.
90 *> = 'N': The matrix A will be copied to AFP and factored.
95 *> UPLO is CHARACTER*1
96 *> = 'U': Upper triangle of A is stored;
97 *> = 'L': Lower triangle of A is stored.
103 *> The number of linear equations, i.e., the order of the
110 *> The number of right hand sides, i.e., the number of columns
111 *> of the matrices B and X. NRHS >= 0.
116 *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
117 *> The upper or lower triangle of the symmetric matrix A, packed
118 *> columnwise in a linear array. The j-th column of A is stored
119 *> in the array AP as follows:
120 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
121 *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
122 *> See below for further details.
125 *> \param[in,out] AFP
127 *> AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
128 *> If FACT = 'F', then AFP is an input argument and on entry
129 *> contains the block diagonal matrix D and the multipliers used
130 *> to obtain the factor U or L from the factorization
131 *> A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as
132 *> a packed triangular matrix in the same storage format as A.
134 *> If FACT = 'N', then AFP is an output argument and on exit
135 *> contains the block diagonal matrix D and the multipliers used
136 *> to obtain the factor U or L from the factorization
137 *> A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as
138 *> a packed triangular matrix in the same storage format as A.
141 *> \param[in,out] IPIV
143 *> IPIV is INTEGER array, dimension (N)
144 *> If FACT = 'F', then IPIV is an input argument and on entry
145 *> contains details of the interchanges and the block structure
146 *> of D, as determined by ZSPTRF.
147 *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
148 *> interchanged and D(k,k) is a 1-by-1 diagonal block.
149 *> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
150 *> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
151 *> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
152 *> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
153 *> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
155 *> If FACT = 'N', then IPIV is an output argument and on exit
156 *> contains details of the interchanges and the block structure
157 *> of D, as determined by ZSPTRF.
162 *> B is COMPLEX*16 array, dimension (LDB,NRHS)
163 *> The N-by-NRHS right hand side matrix B.
169 *> The leading dimension of the array B. LDB >= max(1,N).
174 *> X is COMPLEX*16 array, dimension (LDX,NRHS)
175 *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
181 *> The leading dimension of the array X. LDX >= max(1,N).
186 *> RCOND is DOUBLE PRECISION
187 *> The estimate of the reciprocal condition number of the matrix
188 *> A. If RCOND is less than the machine precision (in
189 *> particular, if RCOND = 0), the matrix is singular to working
190 *> precision. This condition is indicated by a return code of
196 *> FERR is DOUBLE PRECISION array, dimension (NRHS)
197 *> The estimated forward error bound for each solution vector
198 *> X(j) (the j-th column of the solution matrix X).
199 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
200 *> is an estimated upper bound for the magnitude of the largest
201 *> element in (X(j) - XTRUE) divided by the magnitude of the
202 *> largest element in X(j). The estimate is as reliable as
203 *> the estimate for RCOND, and is almost always a slight
204 *> overestimate of the true error.
209 *> BERR is DOUBLE PRECISION array, dimension (NRHS)
210 *> The componentwise relative backward error of each solution
211 *> vector X(j) (i.e., the smallest relative change in
212 *> any element of A or B that makes X(j) an exact solution).
217 *> WORK is COMPLEX*16 array, dimension (2*N)
222 *> RWORK is DOUBLE PRECISION array, dimension (N)
228 *> = 0: successful exit
229 *> < 0: if INFO = -i, the i-th argument had an illegal value
230 *> > 0: if INFO = i, and i is
231 *> <= N: D(i,i) is exactly zero. The factorization
232 *> has been completed but the factor D is exactly
233 *> singular, so the solution and error bounds could
234 *> not be computed. RCOND = 0 is returned.
235 *> = N+1: D is nonsingular, but RCOND is less than machine
236 *> precision, meaning that the matrix is singular
237 *> to working precision. Nevertheless, the
238 *> solution and error bounds are computed because
239 *> there are a number of situations where the
240 *> computed solution can be more accurate than the
241 *> value of RCOND would suggest.
247 *> \author Univ. of Tennessee
248 *> \author Univ. of California Berkeley
249 *> \author Univ. of Colorado Denver
254 *> \ingroup complex16OTHERsolve
256 *> \par Further Details:
257 * =====================
261 *> The packed storage scheme is illustrated by the following example
262 *> when N = 4, UPLO = 'U':
264 *> Two-dimensional storage of the symmetric matrix A:
268 *> a33 a34 (aij = aji)
271 *> Packed storage of the upper triangle of A:
273 *> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
276 * =====================================================================
277 SUBROUTINE ZSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
278 $ LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
280 * -- LAPACK driver routine (version 3.4.1) --
281 * -- LAPACK is a software package provided by Univ. of Tennessee, --
282 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
285 * .. Scalar Arguments ..
287 INTEGER INFO, LDB, LDX, N, NRHS
288 DOUBLE PRECISION RCOND
290 * .. Array Arguments ..
292 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
293 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
297 * =====================================================================
300 DOUBLE PRECISION ZERO
301 PARAMETER ( ZERO = 0.0D+0 )
303 * .. Local Scalars ..
305 DOUBLE PRECISION ANORM
307 * .. External Functions ..
309 DOUBLE PRECISION DLAMCH, ZLANSP
310 EXTERNAL LSAME, DLAMCH, ZLANSP
312 * .. External Subroutines ..
313 EXTERNAL XERBLA, ZCOPY, ZLACPY, ZSPCON, ZSPRFS, ZSPTRF,
316 * .. Intrinsic Functions ..
319 * .. Executable Statements ..
321 * Test the input parameters.
324 NOFACT = LSAME( FACT, 'N' )
325 IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
327 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
330 ELSE IF( N.LT.0 ) THEN
332 ELSE IF( NRHS.LT.0 ) THEN
334 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
336 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
340 CALL XERBLA( 'ZSPSVX', -INFO )
346 * Compute the factorization A = U*D*U**T or A = L*D*L**T.
348 CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
349 CALL ZSPTRF( UPLO, N, AFP, IPIV, INFO )
351 * Return if INFO is non-zero.
359 * Compute the norm of the matrix A.
361 ANORM = ZLANSP( 'I', UPLO, N, AP, RWORK )
363 * Compute the reciprocal of the condition number of A.
365 CALL ZSPCON( UPLO, N, AFP, IPIV, ANORM, RCOND, WORK, INFO )
367 * Compute the solution vectors X.
369 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
370 CALL ZSPTRS( UPLO, N, NRHS, AFP, IPIV, X, LDX, INFO )
372 * Use iterative refinement to improve the computed solutions and
373 * compute error bounds and backward error estimates for them.
375 CALL ZSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR,
376 $ BERR, WORK, RWORK, INFO )
378 * Set INFO = N+1 if the matrix is singular to working precision.
380 IF( RCOND.LT.DLAMCH( 'Epsilon' ) )