1*> \brief <b> ZHESVX computes the solution to system of linear equations A * X = B for HE matrices</b> 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download ZHESVX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhesvx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhesvx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhesvx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE ZHESVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, 22* LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, 23* RWORK, INFO ) 24* 25* .. Scalar Arguments .. 26* CHARACTER FACT, UPLO 27* INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS 28* DOUBLE PRECISION RCOND 29* .. 30* .. Array Arguments .. 31* INTEGER IPIV( * ) 32* DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 33* COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 34* $ WORK( * ), X( LDX, * ) 35* .. 36* 37* 38*> \par Purpose: 39* ============= 40*> 41*> \verbatim 42*> 43*> ZHESVX uses the diagonal pivoting factorization to compute the 44*> solution to a complex system of linear equations A * X = B, 45*> where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS 46*> matrices. 47*> 48*> Error bounds on the solution and a condition estimate are also 49*> provided. 50*> \endverbatim 51* 52*> \par Description: 53* ================= 54*> 55*> \verbatim 56*> 57*> The following steps are performed: 58*> 59*> 1. If FACT = 'N', the diagonal pivoting method is used to factor A. 60*> The form of the factorization is 61*> A = U * D * U**H, if UPLO = 'U', or 62*> A = L * D * L**H, if UPLO = 'L', 63*> where U (or L) is a product of permutation and unit upper (lower) 64*> triangular matrices, and D is Hermitian and block diagonal with 65*> 1-by-1 and 2-by-2 diagonal blocks. 66*> 67*> 2. If some D(i,i)=0, so that D is exactly singular, then the routine 68*> returns with INFO = i. Otherwise, the factored form of A is used 69*> to estimate the condition number of the matrix A. If the 70*> reciprocal of the condition number is less than machine precision, 71*> INFO = N+1 is returned as a warning, but the routine still goes on 72*> to solve for X and compute error bounds as described below. 73*> 74*> 3. The system of equations is solved for X using the factored form 75*> of A. 76*> 77*> 4. Iterative refinement is applied to improve the computed solution 78*> matrix and calculate error bounds and backward error estimates 79*> for it. 80*> \endverbatim 81* 82* Arguments: 83* ========== 84* 85*> \param[in] FACT 86*> \verbatim 87*> FACT is CHARACTER*1 88*> Specifies whether or not the factored form of A has been 89*> supplied on entry. 90*> = 'F': On entry, AF and IPIV contain the factored form 91*> of A. A, AF and IPIV will not be modified. 92*> = 'N': The matrix A will be copied to AF and factored. 93*> \endverbatim 94*> 95*> \param[in] UPLO 96*> \verbatim 97*> UPLO is CHARACTER*1 98*> = 'U': Upper triangle of A is stored; 99*> = 'L': Lower triangle of A is stored. 100*> \endverbatim 101*> 102*> \param[in] N 103*> \verbatim 104*> N is INTEGER 105*> The number of linear equations, i.e., the order of the 106*> matrix A. N >= 0. 107*> \endverbatim 108*> 109*> \param[in] NRHS 110*> \verbatim 111*> NRHS is INTEGER 112*> The number of right hand sides, i.e., the number of columns 113*> of the matrices B and X. NRHS >= 0. 114*> \endverbatim 115*> 116*> \param[in] A 117*> \verbatim 118*> A is COMPLEX*16 array, dimension (LDA,N) 119*> The Hermitian matrix A. If UPLO = 'U', the leading N-by-N 120*> upper triangular part of A contains the upper triangular part 121*> of the matrix A, and the strictly lower triangular part of A 122*> is not referenced. If UPLO = 'L', the leading N-by-N lower 123*> triangular part of A contains the lower triangular part of 124*> the matrix A, and the strictly upper triangular part of A is 125*> not referenced. 126*> \endverbatim 127*> 128*> \param[in] LDA 129*> \verbatim 130*> LDA is INTEGER 131*> The leading dimension of the array A. LDA >= max(1,N). 132*> \endverbatim 133*> 134*> \param[in,out] AF 135*> \verbatim 136*> AF is COMPLEX*16 array, dimension (LDAF,N) 137*> If FACT = 'F', then AF is an input argument and on entry 138*> contains the block diagonal matrix D and the multipliers used 139*> to obtain the factor U or L from the factorization 140*> A = U*D*U**H or A = L*D*L**H as computed by ZHETRF. 141*> 142*> If FACT = 'N', then AF is an output argument and on exit 143*> returns the block diagonal matrix D and the multipliers used 144*> to obtain the factor U or L from the factorization 145*> A = U*D*U**H or A = L*D*L**H. 146*> \endverbatim 147*> 148*> \param[in] LDAF 149*> \verbatim 150*> LDAF is INTEGER 151*> The leading dimension of the array AF. LDAF >= max(1,N). 152*> \endverbatim 153*> 154*> \param[in,out] IPIV 155*> \verbatim 156*> IPIV is INTEGER array, dimension (N) 157*> If FACT = 'F', then IPIV is an input argument and on entry 158*> contains details of the interchanges and the block structure 159*> of D, as determined by ZHETRF. 160*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were 161*> interchanged and D(k,k) is a 1-by-1 diagonal block. 162*> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and 163*> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) 164*> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = 165*> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were 166*> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 167*> 168*> If FACT = 'N', then IPIV is an output argument and on exit 169*> contains details of the interchanges and the block structure 170*> of D, as determined by ZHETRF. 171*> \endverbatim 172*> 173*> \param[in] B 174*> \verbatim 175*> B is COMPLEX*16 array, dimension (LDB,NRHS) 176*> The N-by-NRHS right hand side matrix B. 177*> \endverbatim 178*> 179*> \param[in] LDB 180*> \verbatim 181*> LDB is INTEGER 182*> The leading dimension of the array B. LDB >= max(1,N). 183*> \endverbatim 184*> 185*> \param[out] X 186*> \verbatim 187*> X is COMPLEX*16 array, dimension (LDX,NRHS) 188*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. 189*> \endverbatim 190*> 191*> \param[in] LDX 192*> \verbatim 193*> LDX is INTEGER 194*> The leading dimension of the array X. LDX >= max(1,N). 195*> \endverbatim 196*> 197*> \param[out] RCOND 198*> \verbatim 199*> RCOND is DOUBLE PRECISION 200*> The estimate of the reciprocal condition number of the matrix 201*> A. If RCOND is less than the machine precision (in 202*> particular, if RCOND = 0), the matrix is singular to working 203*> precision. This condition is indicated by a return code of 204*> INFO > 0. 205*> \endverbatim 206*> 207*> \param[out] FERR 208*> \verbatim 209*> FERR is DOUBLE PRECISION array, dimension (NRHS) 210*> The estimated forward error bound for each solution vector 211*> X(j) (the j-th column of the solution matrix X). 212*> If XTRUE is the true solution corresponding to X(j), FERR(j) 213*> is an estimated upper bound for the magnitude of the largest 214*> element in (X(j) - XTRUE) divided by the magnitude of the 215*> largest element in X(j). The estimate is as reliable as 216*> the estimate for RCOND, and is almost always a slight 217*> overestimate of the true error. 218*> \endverbatim 219*> 220*> \param[out] BERR 221*> \verbatim 222*> BERR is DOUBLE PRECISION array, dimension (NRHS) 223*> The componentwise relative backward error of each solution 224*> vector X(j) (i.e., the smallest relative change in 225*> any element of A or B that makes X(j) an exact solution). 226*> \endverbatim 227*> 228*> \param[out] WORK 229*> \verbatim 230*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) 231*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 232*> \endverbatim 233*> 234*> \param[in] LWORK 235*> \verbatim 236*> LWORK is INTEGER 237*> The length of WORK. LWORK >= max(1,2*N), and for best 238*> performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where 239*> NB is the optimal blocksize for ZHETRF. 240*> 241*> If LWORK = -1, then a workspace query is assumed; the routine 242*> only calculates the optimal size of the WORK array, returns 243*> this value as the first entry of the WORK array, and no error 244*> message related to LWORK is issued by XERBLA. 245*> \endverbatim 246*> 247*> \param[out] RWORK 248*> \verbatim 249*> RWORK is DOUBLE PRECISION array, dimension (N) 250*> \endverbatim 251*> 252*> \param[out] INFO 253*> \verbatim 254*> INFO is INTEGER 255*> = 0: successful exit 256*> < 0: if INFO = -i, the i-th argument had an illegal value 257*> > 0: if INFO = i, and i is 258*> <= N: D(i,i) is exactly zero. The factorization 259*> has been completed but the factor D is exactly 260*> singular, so the solution and error bounds could 261*> not be computed. RCOND = 0 is returned. 262*> = N+1: D is nonsingular, but RCOND is less than machine 263*> precision, meaning that the matrix is singular 264*> to working precision. Nevertheless, the 265*> solution and error bounds are computed because 266*> there are a number of situations where the 267*> computed solution can be more accurate than the 268*> value of RCOND would suggest. 269*> \endverbatim 270* 271* Authors: 272* ======== 273* 274*> \author Univ. of Tennessee 275*> \author Univ. of California Berkeley 276*> \author Univ. of Colorado Denver 277*> \author NAG Ltd. 278* 279*> \ingroup complex16HEsolve 280* 281* ===================================================================== 282 SUBROUTINE ZHESVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, 283 $ LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, 284 $ RWORK, INFO ) 285* 286* -- LAPACK driver routine -- 287* -- LAPACK is a software package provided by Univ. of Tennessee, -- 288* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 289* 290* .. Scalar Arguments .. 291 CHARACTER FACT, UPLO 292 INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS 293 DOUBLE PRECISION RCOND 294* .. 295* .. Array Arguments .. 296 INTEGER IPIV( * ) 297 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 298 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 299 $ WORK( * ), X( LDX, * ) 300* .. 301* 302* ===================================================================== 303* 304* .. Parameters .. 305 DOUBLE PRECISION ZERO 306 PARAMETER ( ZERO = 0.0D+0 ) 307* .. 308* .. Local Scalars .. 309 LOGICAL LQUERY, NOFACT 310 INTEGER LWKOPT, NB 311 DOUBLE PRECISION ANORM 312* .. 313* .. External Functions .. 314 LOGICAL LSAME 315 INTEGER ILAENV 316 DOUBLE PRECISION DLAMCH, ZLANHE 317 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANHE 318* .. 319* .. External Subroutines .. 320 EXTERNAL XERBLA, ZHECON, ZHERFS, ZHETRF, ZHETRS, ZLACPY 321* .. 322* .. Intrinsic Functions .. 323 INTRINSIC MAX 324* .. 325* .. Executable Statements .. 326* 327* Test the input parameters. 328* 329 INFO = 0 330 NOFACT = LSAME( FACT, 'N' ) 331 LQUERY = ( LWORK.EQ.-1 ) 332 IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN 333 INFO = -1 334 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) 335 $ THEN 336 INFO = -2 337 ELSE IF( N.LT.0 ) THEN 338 INFO = -3 339 ELSE IF( NRHS.LT.0 ) THEN 340 INFO = -4 341 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 342 INFO = -6 343 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 344 INFO = -8 345 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 346 INFO = -11 347 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 348 INFO = -13 349 ELSE IF( LWORK.LT.MAX( 1, 2*N ) .AND. .NOT.LQUERY ) THEN 350 INFO = -18 351 END IF 352* 353 IF( INFO.EQ.0 ) THEN 354 LWKOPT = MAX( 1, 2*N ) 355 IF( NOFACT ) THEN 356 NB = ILAENV( 1, 'ZHETRF', UPLO, N, -1, -1, -1 ) 357 LWKOPT = MAX( LWKOPT, N*NB ) 358 END IF 359 WORK( 1 ) = LWKOPT 360 END IF 361* 362 IF( INFO.NE.0 ) THEN 363 CALL XERBLA( 'ZHESVX', -INFO ) 364 RETURN 365 ELSE IF( LQUERY ) THEN 366 RETURN 367 END IF 368* 369 IF( NOFACT ) THEN 370* 371* Compute the factorization A = U*D*U**H or A = L*D*L**H. 372* 373 CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 374 CALL ZHETRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO ) 375* 376* Return if INFO is non-zero. 377* 378 IF( INFO.GT.0 )THEN 379 RCOND = ZERO 380 RETURN 381 END IF 382 END IF 383* 384* Compute the norm of the matrix A. 385* 386 ANORM = ZLANHE( 'I', UPLO, N, A, LDA, RWORK ) 387* 388* Compute the reciprocal of the condition number of A. 389* 390 CALL ZHECON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, INFO ) 391* 392* Compute the solution vectors X. 393* 394 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 395 CALL ZHETRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) 396* 397* Use iterative refinement to improve the computed solutions and 398* compute error bounds and backward error estimates for them. 399* 400 CALL ZHERFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, 401 $ LDX, FERR, BERR, WORK, RWORK, INFO ) 402* 403* Set INFO = N+1 if the matrix is singular to working precision. 404* 405 IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 406 $ INFO = N + 1 407* 408 WORK( 1 ) = LWKOPT 409* 410 RETURN 411* 412* End of ZHESVX 413* 414 END 415