1*> \brief <b> CSYSVXX computes the solution to system of linear equations A * X = B for SY 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 CSYSVXX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/csysvxx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/csysvxx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/csysvxx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE CSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, 22* EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, 23* N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, 24* NPARAMS, PARAMS, WORK, RWORK, INFO ) 25* 26* .. Scalar Arguments .. 27* CHARACTER EQUED, FACT, UPLO 28* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 29* $ N_ERR_BNDS 30* REAL RCOND, RPVGRW 31* .. 32* .. Array Arguments .. 33* INTEGER IPIV( * ) 34* COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 35* $ X( LDX, * ), WORK( * ) 36* REAL S( * ), PARAMS( * ), BERR( * ), 37* $ ERR_BNDS_NORM( NRHS, * ), 38* $ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) 39* .. 40* 41* 42*> \par Purpose: 43* ============= 44*> 45*> \verbatim 46*> 47*> CSYSVXX uses the diagonal pivoting factorization to compute the 48*> solution to a complex system of linear equations A * X = B, where 49*> A is an N-by-N symmetric matrix and X and B are N-by-NRHS 50*> matrices. 51*> 52*> If requested, both normwise and maximum componentwise error bounds 53*> are returned. CSYSVXX will return a solution with a tiny 54*> guaranteed error (O(eps) where eps is the working machine 55*> precision) unless the matrix is very ill-conditioned, in which 56*> case a warning is returned. Relevant condition numbers also are 57*> calculated and returned. 58*> 59*> CSYSVXX accepts user-provided factorizations and equilibration 60*> factors; see the definitions of the FACT and EQUED options. 61*> Solving with refinement and using a factorization from a previous 62*> CSYSVXX call will also produce a solution with either O(eps) 63*> errors or warnings, but we cannot make that claim for general 64*> user-provided factorizations and equilibration factors if they 65*> differ from what CSYSVXX would itself produce. 66*> \endverbatim 67* 68*> \par Description: 69* ================= 70*> 71*> \verbatim 72*> 73*> The following steps are performed: 74*> 75*> 1. If FACT = 'E', real scaling factors are computed to equilibrate 76*> the system: 77*> 78*> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B 79*> 80*> Whether or not the system will be equilibrated depends on the 81*> scaling of the matrix A, but if equilibration is used, A is 82*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 83*> 84*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor 85*> the matrix A (after equilibration if FACT = 'E') as 86*> 87*> A = U * D * U**T, if UPLO = 'U', or 88*> A = L * D * L**T, if UPLO = 'L', 89*> 90*> where U (or L) is a product of permutation and unit upper (lower) 91*> triangular matrices, and D is symmetric and block diagonal with 92*> 1-by-1 and 2-by-2 diagonal blocks. 93*> 94*> 3. If some D(i,i)=0, so that D is exactly singular, then the 95*> routine returns with INFO = i. Otherwise, the factored form of A 96*> is used to estimate the condition number of the matrix A (see 97*> argument RCOND). If the reciprocal of the condition number is 98*> less than machine precision, the routine still goes on to solve 99*> for X and compute error bounds as described below. 100*> 101*> 4. The system of equations is solved for X using the factored form 102*> of A. 103*> 104*> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), 105*> the routine will use iterative refinement to try to get a small 106*> error and error bounds. Refinement calculates the residual to at 107*> least twice the working precision. 108*> 109*> 6. If equilibration was used, the matrix X is premultiplied by 110*> diag(R) so that it solves the original system before 111*> equilibration. 112*> \endverbatim 113* 114* Arguments: 115* ========== 116* 117*> \verbatim 118*> Some optional parameters are bundled in the PARAMS array. These 119*> settings determine how refinement is performed, but often the 120*> defaults are acceptable. If the defaults are acceptable, users 121*> can pass NPARAMS = 0 which prevents the source code from accessing 122*> the PARAMS argument. 123*> \endverbatim 124*> 125*> \param[in] FACT 126*> \verbatim 127*> FACT is CHARACTER*1 128*> Specifies whether or not the factored form of the matrix A is 129*> supplied on entry, and if not, whether the matrix A should be 130*> equilibrated before it is factored. 131*> = 'F': On entry, AF and IPIV contain the factored form of A. 132*> If EQUED is not 'N', the matrix A has been 133*> equilibrated with scaling factors given by S. 134*> A, AF, and IPIV are not modified. 135*> = 'N': The matrix A will be copied to AF and factored. 136*> = 'E': The matrix A will be equilibrated if necessary, then 137*> copied to AF and factored. 138*> \endverbatim 139*> 140*> \param[in] UPLO 141*> \verbatim 142*> UPLO is CHARACTER*1 143*> = 'U': Upper triangle of A is stored; 144*> = 'L': Lower triangle of A is stored. 145*> \endverbatim 146*> 147*> \param[in] N 148*> \verbatim 149*> N is INTEGER 150*> The number of linear equations, i.e., the order of the 151*> matrix A. N >= 0. 152*> \endverbatim 153*> 154*> \param[in] NRHS 155*> \verbatim 156*> NRHS is INTEGER 157*> The number of right hand sides, i.e., the number of columns 158*> of the matrices B and X. NRHS >= 0. 159*> \endverbatim 160*> 161*> \param[in,out] A 162*> \verbatim 163*> A is COMPLEX array, dimension (LDA,N) 164*> The symmetric matrix A. If UPLO = 'U', the leading N-by-N 165*> upper triangular part of A contains the upper triangular 166*> part of the matrix A, and the strictly lower triangular 167*> part of A is not referenced. If UPLO = 'L', the leading 168*> N-by-N lower triangular part of A contains the lower 169*> triangular part of the matrix A, and the strictly upper 170*> triangular part of A is not referenced. 171*> 172*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 173*> diag(S)*A*diag(S). 174*> \endverbatim 175*> 176*> \param[in] LDA 177*> \verbatim 178*> LDA is INTEGER 179*> The leading dimension of the array A. LDA >= max(1,N). 180*> \endverbatim 181*> 182*> \param[in,out] AF 183*> \verbatim 184*> AF is COMPLEX array, dimension (LDAF,N) 185*> If FACT = 'F', then AF is an input argument and on entry 186*> contains the block diagonal matrix D and the multipliers 187*> used to obtain the factor U or L from the factorization A = 188*> U*D*U**T or A = L*D*L**T as computed by SSYTRF. 189*> 190*> If FACT = 'N', then AF is an output argument and on exit 191*> returns the block diagonal matrix D and the multipliers 192*> used to obtain the factor U or L from the factorization A = 193*> U*D*U**T or A = L*D*L**T. 194*> \endverbatim 195*> 196*> \param[in] LDAF 197*> \verbatim 198*> LDAF is INTEGER 199*> The leading dimension of the array AF. LDAF >= max(1,N). 200*> \endverbatim 201*> 202*> \param[in,out] IPIV 203*> \verbatim 204*> IPIV is INTEGER array, dimension (N) 205*> If FACT = 'F', then IPIV is an input argument and on entry 206*> contains details of the interchanges and the block 207*> structure of D, as determined by SSYTRF. If IPIV(k) > 0, 208*> then rows and columns k and IPIV(k) were interchanged and 209*> D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and 210*> IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and 211*> -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 212*> diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, 213*> then rows and columns k+1 and -IPIV(k) were interchanged 214*> and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 215*> 216*> If FACT = 'N', then IPIV is an output argument and on exit 217*> contains details of the interchanges and the block 218*> structure of D, as determined by SSYTRF. 219*> \endverbatim 220*> 221*> \param[in,out] EQUED 222*> \verbatim 223*> EQUED is CHARACTER*1 224*> Specifies the form of equilibration that was done. 225*> = 'N': No equilibration (always true if FACT = 'N'). 226*> = 'Y': Both row and column equilibration, i.e., A has been 227*> replaced by diag(S) * A * diag(S). 228*> EQUED is an input argument if FACT = 'F'; otherwise, it is an 229*> output argument. 230*> \endverbatim 231*> 232*> \param[in,out] S 233*> \verbatim 234*> S is REAL array, dimension (N) 235*> The scale factors for A. If EQUED = 'Y', A is multiplied on 236*> the left and right by diag(S). S is an input argument if FACT = 237*> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED 238*> = 'Y', each element of S must be positive. If S is output, each 239*> element of S is a power of the radix. If S is input, each element 240*> of S should be a power of the radix to ensure a reliable solution 241*> and error estimates. Scaling by powers of the radix does not cause 242*> rounding errors unless the result underflows or overflows. 243*> Rounding errors during scaling lead to refining with a matrix that 244*> is not equivalent to the input matrix, producing error estimates 245*> that may not be reliable. 246*> \endverbatim 247*> 248*> \param[in,out] B 249*> \verbatim 250*> B is COMPLEX array, dimension (LDB,NRHS) 251*> On entry, the N-by-NRHS right hand side matrix B. 252*> On exit, 253*> if EQUED = 'N', B is not modified; 254*> if EQUED = 'Y', B is overwritten by diag(S)*B; 255*> \endverbatim 256*> 257*> \param[in] LDB 258*> \verbatim 259*> LDB is INTEGER 260*> The leading dimension of the array B. LDB >= max(1,N). 261*> \endverbatim 262*> 263*> \param[out] X 264*> \verbatim 265*> X is COMPLEX array, dimension (LDX,NRHS) 266*> If INFO = 0, the N-by-NRHS solution matrix X to the original 267*> system of equations. Note that A and B are modified on exit if 268*> EQUED .ne. 'N', and the solution to the equilibrated system is 269*> inv(diag(S))*X. 270*> \endverbatim 271*> 272*> \param[in] LDX 273*> \verbatim 274*> LDX is INTEGER 275*> The leading dimension of the array X. LDX >= max(1,N). 276*> \endverbatim 277*> 278*> \param[out] RCOND 279*> \verbatim 280*> RCOND is REAL 281*> Reciprocal scaled condition number. This is an estimate of the 282*> reciprocal Skeel condition number of the matrix A after 283*> equilibration (if done). If this is less than the machine 284*> precision (in particular, if it is zero), the matrix is singular 285*> to working precision. Note that the error may still be small even 286*> if this number is very small and the matrix appears ill- 287*> conditioned. 288*> \endverbatim 289*> 290*> \param[out] RPVGRW 291*> \verbatim 292*> RPVGRW is REAL 293*> Reciprocal pivot growth. On exit, this contains the reciprocal 294*> pivot growth factor norm(A)/norm(U). The "max absolute element" 295*> norm is used. If this is much less than 1, then the stability of 296*> the LU factorization of the (equilibrated) matrix A could be poor. 297*> This also means that the solution X, estimated condition numbers, 298*> and error bounds could be unreliable. If factorization fails with 299*> 0<INFO<=N, then this contains the reciprocal pivot growth factor 300*> for the leading INFO columns of A. 301*> \endverbatim 302*> 303*> \param[out] BERR 304*> \verbatim 305*> BERR is REAL array, dimension (NRHS) 306*> Componentwise relative backward error. This is the 307*> componentwise relative backward error of each solution vector X(j) 308*> (i.e., the smallest relative change in any element of A or B that 309*> makes X(j) an exact solution). 310*> \endverbatim 311*> 312*> \param[in] N_ERR_BNDS 313*> \verbatim 314*> N_ERR_BNDS is INTEGER 315*> Number of error bounds to return for each right hand side 316*> and each type (normwise or componentwise). See ERR_BNDS_NORM and 317*> ERR_BNDS_COMP below. 318*> \endverbatim 319*> 320*> \param[out] ERR_BNDS_NORM 321*> \verbatim 322*> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) 323*> For each right-hand side, this array contains information about 324*> various error bounds and condition numbers corresponding to the 325*> normwise relative error, which is defined as follows: 326*> 327*> Normwise relative error in the ith solution vector: 328*> max_j (abs(XTRUE(j,i) - X(j,i))) 329*> ------------------------------ 330*> max_j abs(X(j,i)) 331*> 332*> The array is indexed by the type of error information as described 333*> below. There currently are up to three pieces of information 334*> returned. 335*> 336*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 337*> right-hand side. 338*> 339*> The second index in ERR_BNDS_NORM(:,err) contains the following 340*> three fields: 341*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 342*> reciprocal condition number is less than the threshold 343*> sqrt(n) * slamch('Epsilon'). 344*> 345*> err = 2 "Guaranteed" error bound: The estimated forward error, 346*> almost certainly within a factor of 10 of the true error 347*> so long as the next entry is greater than the threshold 348*> sqrt(n) * slamch('Epsilon'). This error bound should only 349*> be trusted if the previous boolean is true. 350*> 351*> err = 3 Reciprocal condition number: Estimated normwise 352*> reciprocal condition number. Compared with the threshold 353*> sqrt(n) * slamch('Epsilon') to determine if the error 354*> estimate is "guaranteed". These reciprocal condition 355*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 356*> appropriately scaled matrix Z. 357*> Let Z = S*A, where S scales each row by a power of the 358*> radix so all absolute row sums of Z are approximately 1. 359*> 360*> See Lapack Working Note 165 for further details and extra 361*> cautions. 362*> \endverbatim 363*> 364*> \param[out] ERR_BNDS_COMP 365*> \verbatim 366*> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) 367*> For each right-hand side, this array contains information about 368*> various error bounds and condition numbers corresponding to the 369*> componentwise relative error, which is defined as follows: 370*> 371*> Componentwise relative error in the ith solution vector: 372*> abs(XTRUE(j,i) - X(j,i)) 373*> max_j ---------------------- 374*> abs(X(j,i)) 375*> 376*> The array is indexed by the right-hand side i (on which the 377*> componentwise relative error depends), and the type of error 378*> information as described below. There currently are up to three 379*> pieces of information returned for each right-hand side. If 380*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 381*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most 382*> the first (:,N_ERR_BNDS) entries are returned. 383*> 384*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 385*> right-hand side. 386*> 387*> The second index in ERR_BNDS_COMP(:,err) contains the following 388*> three fields: 389*> err = 1 "Trust/don't trust" boolean. Trust the answer if the 390*> reciprocal condition number is less than the threshold 391*> sqrt(n) * slamch('Epsilon'). 392*> 393*> err = 2 "Guaranteed" error bound: The estimated forward error, 394*> almost certainly within a factor of 10 of the true error 395*> so long as the next entry is greater than the threshold 396*> sqrt(n) * slamch('Epsilon'). This error bound should only 397*> be trusted if the previous boolean is true. 398*> 399*> err = 3 Reciprocal condition number: Estimated componentwise 400*> reciprocal condition number. Compared with the threshold 401*> sqrt(n) * slamch('Epsilon') to determine if the error 402*> estimate is "guaranteed". These reciprocal condition 403*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 404*> appropriately scaled matrix Z. 405*> Let Z = S*(A*diag(x)), where x is the solution for the 406*> current right-hand side and S scales each row of 407*> A*diag(x) by a power of the radix so all absolute row 408*> sums of Z are approximately 1. 409*> 410*> See Lapack Working Note 165 for further details and extra 411*> cautions. 412*> \endverbatim 413*> 414*> \param[in] NPARAMS 415*> \verbatim 416*> NPARAMS is INTEGER 417*> Specifies the number of parameters set in PARAMS. If <= 0, the 418*> PARAMS array is never referenced and default values are used. 419*> \endverbatim 420*> 421*> \param[in,out] PARAMS 422*> \verbatim 423*> PARAMS is REAL array, dimension NPARAMS 424*> Specifies algorithm parameters. If an entry is < 0.0, then 425*> that entry will be filled with default value used for that 426*> parameter. Only positions up to NPARAMS are accessed; defaults 427*> are used for higher-numbered parameters. 428*> 429*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 430*> refinement or not. 431*> Default: 1.0 432*> = 0.0: No refinement is performed, and no error bounds are 433*> computed. 434*> = 1.0: Use the double-precision refinement algorithm, 435*> possibly with doubled-single computations if the 436*> compilation environment does not support DOUBLE 437*> PRECISION. 438*> (other values are reserved for future use) 439*> 440*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 441*> computations allowed for refinement. 442*> Default: 10 443*> Aggressive: Set to 100 to permit convergence using approximate 444*> factorizations or factorizations other than LU. If 445*> the factorization uses a technique other than 446*> Gaussian elimination, the guarantees in 447*> err_bnds_norm and err_bnds_comp may no longer be 448*> trustworthy. 449*> 450*> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 451*> will attempt to find a solution with small componentwise 452*> relative error in the double-precision algorithm. Positive 453*> is true, 0.0 is false. 454*> Default: 1.0 (attempt componentwise convergence) 455*> \endverbatim 456*> 457*> \param[out] WORK 458*> \verbatim 459*> WORK is COMPLEX array, dimension (2*N) 460*> \endverbatim 461*> 462*> \param[out] RWORK 463*> \verbatim 464*> RWORK is REAL array, dimension (2*N) 465*> \endverbatim 466*> 467*> \param[out] INFO 468*> \verbatim 469*> INFO is INTEGER 470*> = 0: Successful exit. The solution to every right-hand side is 471*> guaranteed. 472*> < 0: If INFO = -i, the i-th argument had an illegal value 473*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 474*> has been completed, but the factor U is exactly singular, so 475*> the solution and error bounds could not be computed. RCOND = 0 476*> is returned. 477*> = N+J: The solution corresponding to the Jth right-hand side is 478*> not guaranteed. The solutions corresponding to other right- 479*> hand sides K with K > J may not be guaranteed as well, but 480*> only the first such right-hand side is reported. If a small 481*> componentwise error is not requested (PARAMS(3) = 0.0) then 482*> the Jth right-hand side is the first with a normwise error 483*> bound that is not guaranteed (the smallest J such 484*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 485*> the Jth right-hand side is the first with either a normwise or 486*> componentwise error bound that is not guaranteed (the smallest 487*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or 488*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of 489*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 490*> about all of the right-hand sides check ERR_BNDS_NORM or 491*> ERR_BNDS_COMP. 492*> \endverbatim 493* 494* Authors: 495* ======== 496* 497*> \author Univ. of Tennessee 498*> \author Univ. of California Berkeley 499*> \author Univ. of Colorado Denver 500*> \author NAG Ltd. 501* 502*> \date April 2012 503* 504*> \ingroup complexSYsolve 505* 506* ===================================================================== 507 SUBROUTINE CSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, 508 $ EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, 509 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, 510 $ NPARAMS, PARAMS, WORK, RWORK, INFO ) 511* 512* -- LAPACK driver routine (version 3.7.0) -- 513* -- LAPACK is a software package provided by Univ. of Tennessee, -- 514* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 515* April 2012 516* 517* .. Scalar Arguments .. 518 CHARACTER EQUED, FACT, UPLO 519 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 520 $ N_ERR_BNDS 521 REAL RCOND, RPVGRW 522* .. 523* .. Array Arguments .. 524 INTEGER IPIV( * ) 525 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 526 $ X( LDX, * ), WORK( * ) 527 REAL S( * ), PARAMS( * ), BERR( * ), 528 $ ERR_BNDS_NORM( NRHS, * ), 529 $ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) 530* .. 531* 532* ================================================================== 533* 534* .. Parameters .. 535 REAL ZERO, ONE 536 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 537 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 538 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 539 INTEGER CMP_ERR_I, PIV_GROWTH_I 540 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 541 $ BERR_I = 3 ) 542 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 543 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 544 $ PIV_GROWTH_I = 9 ) 545* .. 546* .. Local Scalars .. 547 LOGICAL EQUIL, NOFACT, RCEQU 548 INTEGER INFEQU, J 549 REAL AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM 550* .. 551* .. External Functions .. 552 EXTERNAL LSAME, SLAMCH, CLA_SYRPVGRW 553 LOGICAL LSAME 554 REAL SLAMCH, CLA_SYRPVGRW 555* .. 556* .. External Subroutines .. 557 EXTERNAL CSYEQUB, CSYTRF, CSYTRS, CLACPY, 558 $ CLAQSY, XERBLA, CLASCL2, CSYRFSX 559* .. 560* .. Intrinsic Functions .. 561 INTRINSIC MAX, MIN 562* .. 563* .. Executable Statements .. 564* 565 INFO = 0 566 NOFACT = LSAME( FACT, 'N' ) 567 EQUIL = LSAME( FACT, 'E' ) 568 SMLNUM = SLAMCH( 'Safe minimum' ) 569 BIGNUM = ONE / SMLNUM 570 IF( NOFACT .OR. EQUIL ) THEN 571 EQUED = 'N' 572 RCEQU = .FALSE. 573 ELSE 574 RCEQU = LSAME( EQUED, 'Y' ) 575 ENDIF 576* 577* Default is failure. If an input parameter is wrong or 578* factorization fails, make everything look horrible. Only the 579* pivot growth is set here, the rest is initialized in CSYRFSX. 580* 581 RPVGRW = ZERO 582* 583* Test the input parameters. PARAMS is not tested until CSYRFSX. 584* 585 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 586 $ LSAME( FACT, 'F' ) ) THEN 587 INFO = -1 588 ELSE IF( .NOT.LSAME(UPLO, 'U') .AND. 589 $ .NOT.LSAME(UPLO, 'L') ) THEN 590 INFO = -2 591 ELSE IF( N.LT.0 ) THEN 592 INFO = -3 593 ELSE IF( NRHS.LT.0 ) THEN 594 INFO = -4 595 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 596 INFO = -6 597 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 598 INFO = -8 599 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 600 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 601 INFO = -10 602 ELSE 603 IF ( RCEQU ) THEN 604 SMIN = BIGNUM 605 SMAX = ZERO 606 DO 10 J = 1, N 607 SMIN = MIN( SMIN, S( J ) ) 608 SMAX = MAX( SMAX, S( J ) ) 609 10 CONTINUE 610 IF( SMIN.LE.ZERO ) THEN 611 INFO = -11 612 ELSE IF( N.GT.0 ) THEN 613 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 614 ELSE 615 SCOND = ONE 616 END IF 617 END IF 618 IF( INFO.EQ.0 ) THEN 619 IF( LDB.LT.MAX( 1, N ) ) THEN 620 INFO = -13 621 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 622 INFO = -15 623 END IF 624 END IF 625 END IF 626* 627 IF( INFO.NE.0 ) THEN 628 CALL XERBLA( 'CSYSVXX', -INFO ) 629 RETURN 630 END IF 631* 632 IF( EQUIL ) THEN 633* 634* Compute row and column scalings to equilibrate the matrix A. 635* 636 CALL CSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU ) 637 IF( INFEQU.EQ.0 ) THEN 638* 639* Equilibrate the matrix. 640* 641 CALL CLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 642 RCEQU = LSAME( EQUED, 'Y' ) 643 END IF 644 645 END IF 646* 647* Scale the right hand-side. 648* 649 IF( RCEQU ) CALL CLASCL2( N, NRHS, S, B, LDB ) 650* 651 IF( NOFACT .OR. EQUIL ) THEN 652* 653* Compute the LDL^T or UDU^T factorization of A. 654* 655 CALL CLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 656 CALL CSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO ) 657* 658* Return if INFO is non-zero. 659* 660 IF( INFO.GT.0 ) THEN 661* 662* Pivot in column INFO is exactly 0 663* Compute the reciprocal pivot growth factor of the 664* leading rank-deficient INFO columns of A. 665* 666 IF ( N.GT.0 ) 667 $ RPVGRW = CLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, 668 $ LDAF, IPIV, RWORK ) 669 RETURN 670 END IF 671 END IF 672* 673* Compute the reciprocal pivot growth factor RPVGRW. 674* 675 IF ( N.GT.0 ) 676 $ RPVGRW = CLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, 677 $ IPIV, RWORK ) 678* 679* Compute the solution matrix X. 680* 681 CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 682 CALL CSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) 683* 684* Use iterative refinement to improve the computed solution and 685* compute error bounds and backward error estimates for it. 686* 687 CALL CSYRFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, 688 $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, 689 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO ) 690* 691* Scale solutions. 692* 693 IF ( RCEQU ) THEN 694 CALL CLASCL2 (N, NRHS, S, X, LDX ) 695 END IF 696* 697 RETURN 698* 699* End of CSYSVXX 700* 701 END 702