Annotation of rpl/lapack/lapack/zsysvxx.f, revision 1.5
1.5 ! bertrand 1: *> \brief <b> ZSYSVXX 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 ZSYSVXX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zsysvxx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zsysvxx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zsysvxx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZSYSVXX( 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: * DOUBLE PRECISION RCOND, RPVGRW
! 31: * ..
! 32: * .. Array Arguments ..
! 33: * INTEGER IPIV( * )
! 34: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
! 35: * $ X( LDX, * ), WORK( * )
! 36: * DOUBLE PRECISION 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: *> ZSYSVXX uses the diagonal pivoting factorization to compute the
! 48: *> solution to a complex*16 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. ZSYSVXX 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: *> ZSYSVXX 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: *> ZSYSVXX 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 ZSYSVXX 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', double precision 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*16 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 or output) COMPLEX*16 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 DSYTRF.
! 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 or output) 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 DSYTRF. 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 DSYTRF.
! 219: *> \endverbatim
! 220: *>
! 221: *> \param[in,out] EQUED
! 222: *> \verbatim
! 223: *> EQUED is or output) 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 or output) DOUBLE PRECISION 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*16 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*16 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 DOUBLE PRECISION
! 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 DOUBLE PRECISION
! 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION 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 .LT. 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) * dlamch('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) * dlamch('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) * dlamch('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 .LE. 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 / output) DOUBLE PRECISION array, dimension NPARAMS
! 424: *> Specifies algorithm parameters. If an entry is .LT. 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: 1.0D+0
! 432: *> = 0.0 : No refinement is performed, and no error bounds are
! 433: *> computed.
! 434: *> = 1.0 : Use the extra-precise refinement algorithm.
! 435: *> (other values are reserved for future use)
! 436: *>
! 437: *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
! 438: *> computations allowed for refinement.
! 439: *> Default: 10
! 440: *> Aggressive: Set to 100 to permit convergence using approximate
! 441: *> factorizations or factorizations other than LU. If
! 442: *> the factorization uses a technique other than
! 443: *> Gaussian elimination, the guarantees in
! 444: *> err_bnds_norm and err_bnds_comp may no longer be
! 445: *> trustworthy.
! 446: *>
! 447: *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
! 448: *> will attempt to find a solution with small componentwise
! 449: *> relative error in the double-precision algorithm. Positive
! 450: *> is true, 0.0 is false.
! 451: *> Default: 1.0 (attempt componentwise convergence)
! 452: *> \endverbatim
! 453: *>
! 454: *> \param[out] WORK
! 455: *> \verbatim
! 456: *> WORK is COMPLEX*16 array, dimension (2*N)
! 457: *> \endverbatim
! 458: *>
! 459: *> \param[out] RWORK
! 460: *> \verbatim
! 461: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
! 462: *> \endverbatim
! 463: *>
! 464: *> \param[out] INFO
! 465: *> \verbatim
! 466: *> INFO is INTEGER
! 467: *> = 0: Successful exit. The solution to every right-hand side is
! 468: *> guaranteed.
! 469: *> < 0: If INFO = -i, the i-th argument had an illegal value
! 470: *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
! 471: *> has been completed, but the factor U is exactly singular, so
! 472: *> the solution and error bounds could not be computed. RCOND = 0
! 473: *> is returned.
! 474: *> = N+J: The solution corresponding to the Jth right-hand side is
! 475: *> not guaranteed. The solutions corresponding to other right-
! 476: *> hand sides K with K > J may not be guaranteed as well, but
! 477: *> only the first such right-hand side is reported. If a small
! 478: *> componentwise error is not requested (PARAMS(3) = 0.0) then
! 479: *> the Jth right-hand side is the first with a normwise error
! 480: *> bound that is not guaranteed (the smallest J such
! 481: *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
! 482: *> the Jth right-hand side is the first with either a normwise or
! 483: *> componentwise error bound that is not guaranteed (the smallest
! 484: *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
! 485: *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
! 486: *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
! 487: *> about all of the right-hand sides check ERR_BNDS_NORM or
! 488: *> ERR_BNDS_COMP.
! 489: *> \endverbatim
! 490: *
! 491: * Authors:
! 492: * ========
! 493: *
! 494: *> \author Univ. of Tennessee
! 495: *> \author Univ. of California Berkeley
! 496: *> \author Univ. of Colorado Denver
! 497: *> \author NAG Ltd.
! 498: *
! 499: *> \date November 2011
! 500: *
! 501: *> \ingroup complex16SYsolve
! 502: *
! 503: * =====================================================================
1.1 bertrand 504: SUBROUTINE ZSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
505: $ EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
506: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
507: $ NPARAMS, PARAMS, WORK, RWORK, INFO )
508: *
1.5 ! bertrand 509: * -- LAPACK driver routine (version 3.4.0) --
! 510: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 511: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 512: * November 2011
1.1 bertrand 513: *
514: * .. Scalar Arguments ..
515: CHARACTER EQUED, FACT, UPLO
516: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
517: $ N_ERR_BNDS
518: DOUBLE PRECISION RCOND, RPVGRW
519: * ..
520: * .. Array Arguments ..
521: INTEGER IPIV( * )
522: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
523: $ X( LDX, * ), WORK( * )
524: DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
525: $ ERR_BNDS_NORM( NRHS, * ),
526: $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
527: * ..
528: *
1.5 ! bertrand 529: * ==================================================================
1.1 bertrand 530: *
531: * .. Parameters ..
532: DOUBLE PRECISION ZERO, ONE
533: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
534: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
535: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
536: INTEGER CMP_ERR_I, PIV_GROWTH_I
537: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
538: $ BERR_I = 3 )
539: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
540: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
541: $ PIV_GROWTH_I = 9 )
542: * ..
543: * .. Local Scalars ..
544: LOGICAL EQUIL, NOFACT, RCEQU
545: INTEGER INFEQU, J
546: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
547: * ..
548: * .. External Functions ..
549: EXTERNAL LSAME, DLAMCH, ZLA_SYRPVGRW
550: LOGICAL LSAME
551: DOUBLE PRECISION DLAMCH, ZLA_SYRPVGRW
552: * ..
553: * .. External Subroutines ..
554: EXTERNAL ZSYCON, ZSYEQUB, ZSYTRF, ZSYTRS, ZLACPY,
555: $ ZLAQSY, XERBLA, ZLASCL2, ZSYRFSX
556: * ..
557: * .. Intrinsic Functions ..
558: INTRINSIC MAX, MIN
559: * ..
560: * .. Executable Statements ..
561: *
562: INFO = 0
563: NOFACT = LSAME( FACT, 'N' )
564: EQUIL = LSAME( FACT, 'E' )
565: SMLNUM = DLAMCH( 'Safe minimum' )
566: BIGNUM = ONE / SMLNUM
567: IF( NOFACT .OR. EQUIL ) THEN
568: EQUED = 'N'
569: RCEQU = .FALSE.
570: ELSE
571: RCEQU = LSAME( EQUED, 'Y' )
572: ENDIF
573: *
574: * Default is failure. If an input parameter is wrong or
575: * factorization fails, make everything look horrible. Only the
576: * pivot growth is set here, the rest is initialized in ZSYRFSX.
577: *
578: RPVGRW = ZERO
579: *
580: * Test the input parameters. PARAMS is not tested until ZSYRFSX.
581: *
582: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
583: $ LSAME( FACT, 'F' ) ) THEN
584: INFO = -1
585: ELSE IF( .NOT.LSAME(UPLO, 'U') .AND.
586: $ .NOT.LSAME(UPLO, 'L') ) THEN
587: INFO = -2
588: ELSE IF( N.LT.0 ) THEN
589: INFO = -3
590: ELSE IF( NRHS.LT.0 ) THEN
591: INFO = -4
592: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
593: INFO = -6
594: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
595: INFO = -8
596: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
597: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
598: INFO = -9
599: ELSE
600: IF ( RCEQU ) THEN
601: SMIN = BIGNUM
602: SMAX = ZERO
603: DO 10 J = 1, N
604: SMIN = MIN( SMIN, S( J ) )
605: SMAX = MAX( SMAX, S( J ) )
606: 10 CONTINUE
607: IF( SMIN.LE.ZERO ) THEN
608: INFO = -10
609: ELSE IF( N.GT.0 ) THEN
610: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
611: ELSE
612: SCOND = ONE
613: END IF
614: END IF
615: IF( INFO.EQ.0 ) THEN
616: IF( LDB.LT.MAX( 1, N ) ) THEN
617: INFO = -12
618: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
619: INFO = -14
620: END IF
621: END IF
622: END IF
623: *
624: IF( INFO.NE.0 ) THEN
625: CALL XERBLA( 'ZSYSVXX', -INFO )
626: RETURN
627: END IF
628: *
629: IF( EQUIL ) THEN
630: *
631: * Compute row and column scalings to equilibrate the matrix A.
632: *
633: CALL ZSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU )
634: IF( INFEQU.EQ.0 ) THEN
635: *
636: * Equilibrate the matrix.
637: *
638: CALL ZLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
639: RCEQU = LSAME( EQUED, 'Y' )
640: END IF
641:
642: END IF
643: *
644: * Scale the right hand-side.
645: *
646: IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB )
647: *
648: IF( NOFACT .OR. EQUIL ) THEN
649: *
650: * Compute the LDL^T or UDU^T factorization of A.
651: *
652: CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
653: CALL ZSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO )
654: *
655: * Return if INFO is non-zero.
656: *
657: IF( INFO.GT.0 ) THEN
658: *
659: * Pivot in column INFO is exactly 0
660: * Compute the reciprocal pivot growth factor of the
661: * leading rank-deficient INFO columns of A.
662: *
663: IF ( N.GT.0 )
664: $ RPVGRW = ZLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF,
665: $ LDAF, IPIV, RWORK )
666: RETURN
667: END IF
668: END IF
669: *
670: * Compute the reciprocal pivot growth factor RPVGRW.
671: *
672: IF ( N.GT.0 )
673: $ RPVGRW = ZLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF,
674: $ IPIV, RWORK )
675: *
676: * Compute the solution matrix X.
677: *
678: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
679: CALL ZSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
680: *
681: * Use iterative refinement to improve the computed solution and
682: * compute error bounds and backward error estimates for it.
683: *
684: CALL ZSYRFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
685: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
686: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
687: *
688: * Scale solutions.
689: *
690: IF ( RCEQU ) THEN
691: CALL ZLASCL2 (N, NRHS, S, X, LDX )
692: END IF
693: *
694: RETURN
695: *
696: * End of ZSYSVXX
697: *
698: END
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