Annotation of rpl/lapack/lapack/zgbrfsx.f, revision 1.5
1.5 ! bertrand 1: *> \brief \b ZGBRFSX
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZGBRFSX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbrfsx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgbrfsx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbrfsx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB,
! 22: * LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND,
! 23: * BERR, N_ERR_BNDS, ERR_BNDS_NORM,
! 24: * ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
! 25: * INFO )
! 26: *
! 27: * .. Scalar Arguments ..
! 28: * CHARACTER TRANS, EQUED
! 29: * INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS,
! 30: * $ NPARAMS, N_ERR_BNDS
! 31: * DOUBLE PRECISION RCOND
! 32: * ..
! 33: * .. Array Arguments ..
! 34: * INTEGER IPIV( * )
! 35: * COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
! 36: * $ X( LDX , * ),WORK( * )
! 37: * DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
! 38: * $ ERR_BNDS_NORM( NRHS, * ),
! 39: * $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
! 40: * ..
! 41: *
! 42: *
! 43: *> \par Purpose:
! 44: * =============
! 45: *>
! 46: *> \verbatim
! 47: *>
! 48: *> ZGBRFSX improves the computed solution to a system of linear
! 49: *> equations and provides error bounds and backward error estimates
! 50: *> for the solution. In addition to normwise error bound, the code
! 51: *> provides maximum componentwise error bound if possible. See
! 52: *> comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
! 53: *> error bounds.
! 54: *>
! 55: *> The original system of linear equations may have been equilibrated
! 56: *> before calling this routine, as described by arguments EQUED, R
! 57: *> and C below. In this case, the solution and error bounds returned
! 58: *> are for the original unequilibrated system.
! 59: *> \endverbatim
! 60: *
! 61: * Arguments:
! 62: * ==========
! 63: *
! 64: *> \verbatim
! 65: *> Some optional parameters are bundled in the PARAMS array. These
! 66: *> settings determine how refinement is performed, but often the
! 67: *> defaults are acceptable. If the defaults are acceptable, users
! 68: *> can pass NPARAMS = 0 which prevents the source code from accessing
! 69: *> the PARAMS argument.
! 70: *> \endverbatim
! 71: *>
! 72: *> \param[in] TRANS
! 73: *> \verbatim
! 74: *> TRANS is CHARACTER*1
! 75: *> Specifies the form of the system of equations:
! 76: *> = 'N': A * X = B (No transpose)
! 77: *> = 'T': A**T * X = B (Transpose)
! 78: *> = 'C': A**H * X = B (Conjugate transpose = Transpose)
! 79: *> \endverbatim
! 80: *>
! 81: *> \param[in] EQUED
! 82: *> \verbatim
! 83: *> EQUED is CHARACTER*1
! 84: *> Specifies the form of equilibration that was done to A
! 85: *> before calling this routine. This is needed to compute
! 86: *> the solution and error bounds correctly.
! 87: *> = 'N': No equilibration
! 88: *> = 'R': Row equilibration, i.e., A has been premultiplied by
! 89: *> diag(R).
! 90: *> = 'C': Column equilibration, i.e., A has been postmultiplied
! 91: *> by diag(C).
! 92: *> = 'B': Both row and column equilibration, i.e., A has been
! 93: *> replaced by diag(R) * A * diag(C).
! 94: *> The right hand side B has been changed accordingly.
! 95: *> \endverbatim
! 96: *>
! 97: *> \param[in] N
! 98: *> \verbatim
! 99: *> N is INTEGER
! 100: *> The order of the matrix A. N >= 0.
! 101: *> \endverbatim
! 102: *>
! 103: *> \param[in] KL
! 104: *> \verbatim
! 105: *> KL is INTEGER
! 106: *> The number of subdiagonals within the band of A. KL >= 0.
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[in] KU
! 110: *> \verbatim
! 111: *> KU is INTEGER
! 112: *> The number of superdiagonals within the band of A. KU >= 0.
! 113: *> \endverbatim
! 114: *>
! 115: *> \param[in] NRHS
! 116: *> \verbatim
! 117: *> NRHS is INTEGER
! 118: *> The number of right hand sides, i.e., the number of columns
! 119: *> of the matrices B and X. NRHS >= 0.
! 120: *> \endverbatim
! 121: *>
! 122: *> \param[in] AB
! 123: *> \verbatim
! 124: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
! 125: *> The original band matrix A, stored in rows 1 to KL+KU+1.
! 126: *> The j-th column of A is stored in the j-th column of the
! 127: *> array AB as follows:
! 128: *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
! 129: *> \endverbatim
! 130: *>
! 131: *> \param[in] LDAB
! 132: *> \verbatim
! 133: *> LDAB is INTEGER
! 134: *> The leading dimension of the array AB. LDAB >= KL+KU+1.
! 135: *> \endverbatim
! 136: *>
! 137: *> \param[in] AFB
! 138: *> \verbatim
! 139: *> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
! 140: *> Details of the LU factorization of the band matrix A, as
! 141: *> computed by DGBTRF. U is stored as an upper triangular band
! 142: *> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
! 143: *> the multipliers used during the factorization are stored in
! 144: *> rows KL+KU+2 to 2*KL+KU+1.
! 145: *> \endverbatim
! 146: *>
! 147: *> \param[in] LDAFB
! 148: *> \verbatim
! 149: *> LDAFB is INTEGER
! 150: *> The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
! 151: *> \endverbatim
! 152: *>
! 153: *> \param[in] IPIV
! 154: *> \verbatim
! 155: *> IPIV is INTEGER array, dimension (N)
! 156: *> The pivot indices from DGETRF; for 1<=i<=N, row i of the
! 157: *> matrix was interchanged with row IPIV(i).
! 158: *> \endverbatim
! 159: *>
! 160: *> \param[in,out] R
! 161: *> \verbatim
! 162: *> R is or output) DOUBLE PRECISION array, dimension (N)
! 163: *> The row scale factors for A. If EQUED = 'R' or 'B', A is
! 164: *> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
! 165: *> is not accessed. R is an input argument if FACT = 'F';
! 166: *> otherwise, R is an output argument. If FACT = 'F' and
! 167: *> EQUED = 'R' or 'B', each element of R must be positive.
! 168: *> If R is output, each element of R is a power of the radix.
! 169: *> If R is input, each element of R should be a power of the radix
! 170: *> to ensure a reliable solution and error estimates. Scaling by
! 171: *> powers of the radix does not cause rounding errors unless the
! 172: *> result underflows or overflows. Rounding errors during scaling
! 173: *> lead to refining with a matrix that is not equivalent to the
! 174: *> input matrix, producing error estimates that may not be
! 175: *> reliable.
! 176: *> \endverbatim
! 177: *>
! 178: *> \param[in,out] C
! 179: *> \verbatim
! 180: *> C is or output) DOUBLE PRECISION array, dimension (N)
! 181: *> The column scale factors for A. If EQUED = 'C' or 'B', A is
! 182: *> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
! 183: *> is not accessed. C is an input argument if FACT = 'F';
! 184: *> otherwise, C is an output argument. If FACT = 'F' and
! 185: *> EQUED = 'C' or 'B', each element of C must be positive.
! 186: *> If C is output, each element of C is a power of the radix.
! 187: *> If C is input, each element of C should be a power of the radix
! 188: *> to ensure a reliable solution and error estimates. Scaling by
! 189: *> powers of the radix does not cause rounding errors unless the
! 190: *> result underflows or overflows. Rounding errors during scaling
! 191: *> lead to refining with a matrix that is not equivalent to the
! 192: *> input matrix, producing error estimates that may not be
! 193: *> reliable.
! 194: *> \endverbatim
! 195: *>
! 196: *> \param[in] B
! 197: *> \verbatim
! 198: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
! 199: *> The right hand side matrix B.
! 200: *> \endverbatim
! 201: *>
! 202: *> \param[in] LDB
! 203: *> \verbatim
! 204: *> LDB is INTEGER
! 205: *> The leading dimension of the array B. LDB >= max(1,N).
! 206: *> \endverbatim
! 207: *>
! 208: *> \param[in,out] X
! 209: *> \verbatim
! 210: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
! 211: *> On entry, the solution matrix X, as computed by DGETRS.
! 212: *> On exit, the improved solution matrix X.
! 213: *> \endverbatim
! 214: *>
! 215: *> \param[in] LDX
! 216: *> \verbatim
! 217: *> LDX is INTEGER
! 218: *> The leading dimension of the array X. LDX >= max(1,N).
! 219: *> \endverbatim
! 220: *>
! 221: *> \param[out] RCOND
! 222: *> \verbatim
! 223: *> RCOND is DOUBLE PRECISION
! 224: *> Reciprocal scaled condition number. This is an estimate of the
! 225: *> reciprocal Skeel condition number of the matrix A after
! 226: *> equilibration (if done). If this is less than the machine
! 227: *> precision (in particular, if it is zero), the matrix is singular
! 228: *> to working precision. Note that the error may still be small even
! 229: *> if this number is very small and the matrix appears ill-
! 230: *> conditioned.
! 231: *> \endverbatim
! 232: *>
! 233: *> \param[out] BERR
! 234: *> \verbatim
! 235: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 236: *> Componentwise relative backward error. This is the
! 237: *> componentwise relative backward error of each solution vector X(j)
! 238: *> (i.e., the smallest relative change in any element of A or B that
! 239: *> makes X(j) an exact solution).
! 240: *> \endverbatim
! 241: *>
! 242: *> \param[in] N_ERR_BNDS
! 243: *> \verbatim
! 244: *> N_ERR_BNDS is INTEGER
! 245: *> Number of error bounds to return for each right hand side
! 246: *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
! 247: *> ERR_BNDS_COMP below.
! 248: *> \endverbatim
! 249: *>
! 250: *> \param[out] ERR_BNDS_NORM
! 251: *> \verbatim
! 252: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
! 253: *> For each right-hand side, this array contains information about
! 254: *> various error bounds and condition numbers corresponding to the
! 255: *> normwise relative error, which is defined as follows:
! 256: *>
! 257: *> Normwise relative error in the ith solution vector:
! 258: *> max_j (abs(XTRUE(j,i) - X(j,i)))
! 259: *> ------------------------------
! 260: *> max_j abs(X(j,i))
! 261: *>
! 262: *> The array is indexed by the type of error information as described
! 263: *> below. There currently are up to three pieces of information
! 264: *> returned.
! 265: *>
! 266: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
! 267: *> right-hand side.
! 268: *>
! 269: *> The second index in ERR_BNDS_NORM(:,err) contains the following
! 270: *> three fields:
! 271: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 272: *> reciprocal condition number is less than the threshold
! 273: *> sqrt(n) * dlamch('Epsilon').
! 274: *>
! 275: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 276: *> almost certainly within a factor of 10 of the true error
! 277: *> so long as the next entry is greater than the threshold
! 278: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
! 279: *> be trusted if the previous boolean is true.
! 280: *>
! 281: *> err = 3 Reciprocal condition number: Estimated normwise
! 282: *> reciprocal condition number. Compared with the threshold
! 283: *> sqrt(n) * dlamch('Epsilon') to determine if the error
! 284: *> estimate is "guaranteed". These reciprocal condition
! 285: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 286: *> appropriately scaled matrix Z.
! 287: *> Let Z = S*A, where S scales each row by a power of the
! 288: *> radix so all absolute row sums of Z are approximately 1.
! 289: *>
! 290: *> See Lapack Working Note 165 for further details and extra
! 291: *> cautions.
! 292: *> \endverbatim
! 293: *>
! 294: *> \param[out] ERR_BNDS_COMP
! 295: *> \verbatim
! 296: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
! 297: *> For each right-hand side, this array contains information about
! 298: *> various error bounds and condition numbers corresponding to the
! 299: *> componentwise relative error, which is defined as follows:
! 300: *>
! 301: *> Componentwise relative error in the ith solution vector:
! 302: *> abs(XTRUE(j,i) - X(j,i))
! 303: *> max_j ----------------------
! 304: *> abs(X(j,i))
! 305: *>
! 306: *> The array is indexed by the right-hand side i (on which the
! 307: *> componentwise relative error depends), and the type of error
! 308: *> information as described below. There currently are up to three
! 309: *> pieces of information returned for each right-hand side. If
! 310: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
! 311: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
! 312: *> the first (:,N_ERR_BNDS) entries are returned.
! 313: *>
! 314: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
! 315: *> right-hand side.
! 316: *>
! 317: *> The second index in ERR_BNDS_COMP(:,err) contains the following
! 318: *> three fields:
! 319: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 320: *> reciprocal condition number is less than the threshold
! 321: *> sqrt(n) * dlamch('Epsilon').
! 322: *>
! 323: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 324: *> almost certainly within a factor of 10 of the true error
! 325: *> so long as the next entry is greater than the threshold
! 326: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
! 327: *> be trusted if the previous boolean is true.
! 328: *>
! 329: *> err = 3 Reciprocal condition number: Estimated componentwise
! 330: *> reciprocal condition number. Compared with the threshold
! 331: *> sqrt(n) * dlamch('Epsilon') to determine if the error
! 332: *> estimate is "guaranteed". These reciprocal condition
! 333: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 334: *> appropriately scaled matrix Z.
! 335: *> Let Z = S*(A*diag(x)), where x is the solution for the
! 336: *> current right-hand side and S scales each row of
! 337: *> A*diag(x) by a power of the radix so all absolute row
! 338: *> sums of Z are approximately 1.
! 339: *>
! 340: *> See Lapack Working Note 165 for further details and extra
! 341: *> cautions.
! 342: *> \endverbatim
! 343: *>
! 344: *> \param[in] NPARAMS
! 345: *> \verbatim
! 346: *> NPARAMS is INTEGER
! 347: *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
! 348: *> PARAMS array is never referenced and default values are used.
! 349: *> \endverbatim
! 350: *>
! 351: *> \param[in,out] PARAMS
! 352: *> \verbatim
! 353: *> PARAMS is / output) DOUBLE PRECISION array, dimension NPARAMS
! 354: *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
! 355: *> that entry will be filled with default value used for that
! 356: *> parameter. Only positions up to NPARAMS are accessed; defaults
! 357: *> are used for higher-numbered parameters.
! 358: *>
! 359: *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
! 360: *> refinement or not.
! 361: *> Default: 1.0D+0
! 362: *> = 0.0 : No refinement is performed, and no error bounds are
! 363: *> computed.
! 364: *> = 1.0 : Use the double-precision refinement algorithm,
! 365: *> possibly with doubled-single computations if the
! 366: *> compilation environment does not support DOUBLE
! 367: *> PRECISION.
! 368: *> (other values are reserved for future use)
! 369: *>
! 370: *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
! 371: *> computations allowed for refinement.
! 372: *> Default: 10
! 373: *> Aggressive: Set to 100 to permit convergence using approximate
! 374: *> factorizations or factorizations other than LU. If
! 375: *> the factorization uses a technique other than
! 376: *> Gaussian elimination, the guarantees in
! 377: *> err_bnds_norm and err_bnds_comp may no longer be
! 378: *> trustworthy.
! 379: *>
! 380: *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
! 381: *> will attempt to find a solution with small componentwise
! 382: *> relative error in the double-precision algorithm. Positive
! 383: *> is true, 0.0 is false.
! 384: *> Default: 1.0 (attempt componentwise convergence)
! 385: *> \endverbatim
! 386: *>
! 387: *> \param[out] WORK
! 388: *> \verbatim
! 389: *> WORK is COMPLEX*16 array, dimension (2*N)
! 390: *> \endverbatim
! 391: *>
! 392: *> \param[out] RWORK
! 393: *> \verbatim
! 394: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
! 395: *> \endverbatim
! 396: *>
! 397: *> \param[out] INFO
! 398: *> \verbatim
! 399: *> INFO is INTEGER
! 400: *> = 0: Successful exit. The solution to every right-hand side is
! 401: *> guaranteed.
! 402: *> < 0: If INFO = -i, the i-th argument had an illegal value
! 403: *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
! 404: *> has been completed, but the factor U is exactly singular, so
! 405: *> the solution and error bounds could not be computed. RCOND = 0
! 406: *> is returned.
! 407: *> = N+J: The solution corresponding to the Jth right-hand side is
! 408: *> not guaranteed. The solutions corresponding to other right-
! 409: *> hand sides K with K > J may not be guaranteed as well, but
! 410: *> only the first such right-hand side is reported. If a small
! 411: *> componentwise error is not requested (PARAMS(3) = 0.0) then
! 412: *> the Jth right-hand side is the first with a normwise error
! 413: *> bound that is not guaranteed (the smallest J such
! 414: *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
! 415: *> the Jth right-hand side is the first with either a normwise or
! 416: *> componentwise error bound that is not guaranteed (the smallest
! 417: *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
! 418: *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
! 419: *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
! 420: *> about all of the right-hand sides check ERR_BNDS_NORM or
! 421: *> ERR_BNDS_COMP.
! 422: *> \endverbatim
! 423: *
! 424: * Authors:
! 425: * ========
! 426: *
! 427: *> \author Univ. of Tennessee
! 428: *> \author Univ. of California Berkeley
! 429: *> \author Univ. of Colorado Denver
! 430: *> \author NAG Ltd.
! 431: *
! 432: *> \date November 2011
! 433: *
! 434: *> \ingroup complex16GBcomputational
! 435: *
! 436: * =====================================================================
1.1 bertrand 437: SUBROUTINE ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB,
438: $ LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND,
439: $ BERR, N_ERR_BNDS, ERR_BNDS_NORM,
440: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
441: $ INFO )
442: *
1.5 ! bertrand 443: * -- LAPACK computational routine (version 3.4.0) --
! 444: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 445: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 446: * November 2011
1.1 bertrand 447: *
448: * .. Scalar Arguments ..
449: CHARACTER TRANS, EQUED
450: INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS,
451: $ NPARAMS, N_ERR_BNDS
452: DOUBLE PRECISION RCOND
453: * ..
454: * .. Array Arguments ..
455: INTEGER IPIV( * )
456: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
457: $ X( LDX , * ),WORK( * )
458: DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
459: $ ERR_BNDS_NORM( NRHS, * ),
460: $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
461: * ..
462: *
1.5 ! bertrand 463: * ==================================================================
1.1 bertrand 464: *
465: * .. Parameters ..
466: DOUBLE PRECISION ZERO, ONE
467: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
468: DOUBLE PRECISION ITREF_DEFAULT, ITHRESH_DEFAULT
469: DOUBLE PRECISION COMPONENTWISE_DEFAULT, RTHRESH_DEFAULT
470: DOUBLE PRECISION DZTHRESH_DEFAULT
471: PARAMETER ( ITREF_DEFAULT = 1.0D+0 )
472: PARAMETER ( ITHRESH_DEFAULT = 10.0D+0 )
473: PARAMETER ( COMPONENTWISE_DEFAULT = 1.0D+0 )
474: PARAMETER ( RTHRESH_DEFAULT = 0.5D+0 )
475: PARAMETER ( DZTHRESH_DEFAULT = 0.25D+0 )
476: INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
477: $ LA_LINRX_CWISE_I
478: PARAMETER ( LA_LINRX_ITREF_I = 1,
479: $ LA_LINRX_ITHRESH_I = 2 )
480: PARAMETER ( LA_LINRX_CWISE_I = 3 )
481: INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
482: $ LA_LINRX_RCOND_I
483: PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
484: PARAMETER ( LA_LINRX_RCOND_I = 3 )
485: * ..
486: * .. Local Scalars ..
487: CHARACTER(1) NORM
488: LOGICAL ROWEQU, COLEQU, NOTRAN, IGNORE_CWISE
489: INTEGER J, TRANS_TYPE, PREC_TYPE, REF_TYPE, N_NORMS,
490: $ ITHRESH
491: DOUBLE PRECISION ANORM, RCOND_TMP, ILLRCOND_THRESH, ERR_LBND,
492: $ CWISE_WRONG, RTHRESH, UNSTABLE_THRESH
493: * ..
494: * .. External Subroutines ..
495: EXTERNAL XERBLA, ZGBCON, ZLA_GBRFSX_EXTENDED
496: * ..
497: * .. Intrinsic Functions ..
498: INTRINSIC MAX, SQRT, TRANSFER
499: * ..
500: * .. External Functions ..
501: EXTERNAL LSAME, BLAS_FPINFO_X, ILATRANS, ILAPREC
502: EXTERNAL DLAMCH, ZLANGB, ZLA_GBRCOND_X, ZLA_GBRCOND_C
503: DOUBLE PRECISION DLAMCH, ZLANGB, ZLA_GBRCOND_X, ZLA_GBRCOND_C
504: LOGICAL LSAME
505: INTEGER BLAS_FPINFO_X
506: INTEGER ILATRANS, ILAPREC
507: * ..
508: * .. Executable Statements ..
509: *
510: * Check the input parameters.
511: *
512: INFO = 0
513: TRANS_TYPE = ILATRANS( TRANS )
514: REF_TYPE = INT( ITREF_DEFAULT )
515: IF ( NPARAMS .GE. LA_LINRX_ITREF_I ) THEN
516: IF ( PARAMS( LA_LINRX_ITREF_I ) .LT. 0.0D+0 ) THEN
517: PARAMS( LA_LINRX_ITREF_I ) = ITREF_DEFAULT
518: ELSE
519: REF_TYPE = PARAMS( LA_LINRX_ITREF_I )
520: END IF
521: END IF
522: *
523: * Set default parameters.
524: *
525: ILLRCOND_THRESH = DBLE( N ) * DLAMCH( 'Epsilon' )
526: ITHRESH = INT( ITHRESH_DEFAULT )
527: RTHRESH = RTHRESH_DEFAULT
528: UNSTABLE_THRESH = DZTHRESH_DEFAULT
529: IGNORE_CWISE = COMPONENTWISE_DEFAULT .EQ. 0.0D+0
530: *
531: IF ( NPARAMS.GE.LA_LINRX_ITHRESH_I ) THEN
532: IF ( PARAMS( LA_LINRX_ITHRESH_I ).LT.0.0D+0 ) THEN
533: PARAMS( LA_LINRX_ITHRESH_I ) = ITHRESH
534: ELSE
535: ITHRESH = INT( PARAMS( LA_LINRX_ITHRESH_I ) )
536: END IF
537: END IF
538: IF ( NPARAMS.GE.LA_LINRX_CWISE_I ) THEN
539: IF ( PARAMS( LA_LINRX_CWISE_I ).LT.0.0D+0 ) THEN
540: IF ( IGNORE_CWISE ) THEN
541: PARAMS( LA_LINRX_CWISE_I ) = 0.0D+0
542: ELSE
543: PARAMS( LA_LINRX_CWISE_I ) = 1.0D+0
544: END IF
545: ELSE
546: IGNORE_CWISE = PARAMS( LA_LINRX_CWISE_I ) .EQ. 0.0D+0
547: END IF
548: END IF
549: IF ( REF_TYPE .EQ. 0 .OR. N_ERR_BNDS .EQ. 0 ) THEN
550: N_NORMS = 0
551: ELSE IF ( IGNORE_CWISE ) THEN
552: N_NORMS = 1
553: ELSE
554: N_NORMS = 2
555: END IF
556: *
557: NOTRAN = LSAME( TRANS, 'N' )
558: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
559: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
560: *
561: * Test input parameters.
562: *
563: IF( TRANS_TYPE.EQ.-1 ) THEN
564: INFO = -1
565: ELSE IF( .NOT.ROWEQU .AND. .NOT.COLEQU .AND.
566: $ .NOT.LSAME( EQUED, 'N' ) ) THEN
567: INFO = -2
568: ELSE IF( N.LT.0 ) THEN
569: INFO = -3
570: ELSE IF( KL.LT.0 ) THEN
571: INFO = -4
572: ELSE IF( KU.LT.0 ) THEN
573: INFO = -5
574: ELSE IF( NRHS.LT.0 ) THEN
575: INFO = -6
576: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
577: INFO = -8
578: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
579: INFO = -10
580: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
581: INFO = -13
582: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
583: INFO = -15
584: END IF
585: IF( INFO.NE.0 ) THEN
586: CALL XERBLA( 'ZGBRFSX', -INFO )
587: RETURN
588: END IF
589: *
590: * Quick return if possible.
591: *
592: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
593: RCOND = 1.0D+0
594: DO J = 1, NRHS
595: BERR( J ) = 0.0D+0
596: IF ( N_ERR_BNDS .GE. 1 ) THEN
597: ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0
598: ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0
599: END IF
600: IF ( N_ERR_BNDS .GE. 2 ) THEN
601: ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 0.0D+0
602: ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 0.0D+0
603: END IF
604: IF ( N_ERR_BNDS .GE. 3 ) THEN
605: ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 1.0D+0
606: ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 1.0D+0
607: END IF
608: END DO
609: RETURN
610: END IF
611: *
612: * Default to failure.
613: *
614: RCOND = 0.0D+0
615: DO J = 1, NRHS
616: BERR( J ) = 1.0D+0
617: IF ( N_ERR_BNDS .GE. 1 ) THEN
618: ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0
619: ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0
620: END IF
621: IF ( N_ERR_BNDS .GE. 2 ) THEN
622: ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0
623: ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0
624: END IF
625: IF ( N_ERR_BNDS .GE. 3 ) THEN
626: ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 0.0D+0
627: ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 0.0D+0
628: END IF
629: END DO
630: *
631: * Compute the norm of A and the reciprocal of the condition
632: * number of A.
633: *
634: IF( NOTRAN ) THEN
635: NORM = 'I'
636: ELSE
637: NORM = '1'
638: END IF
639: ANORM = ZLANGB( NORM, N, KL, KU, AB, LDAB, RWORK )
640: CALL ZGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
641: $ WORK, RWORK, INFO )
642: *
643: * Perform refinement on each right-hand side
644: *
645: IF ( REF_TYPE .NE. 0 ) THEN
646:
647: PREC_TYPE = ILAPREC( 'E' )
648:
649: IF ( NOTRAN ) THEN
650: CALL ZLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
651: $ NRHS, AB, LDAB, AFB, LDAFB, IPIV, COLEQU, C, B,
652: $ LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM,
653: $ ERR_BNDS_COMP, WORK, RWORK, WORK(N+1),
654: $ TRANSFER (RWORK(1:2*N), (/ (ZERO, ZERO) /), N),
655: $ RCOND, ITHRESH, RTHRESH, UNSTABLE_THRESH, IGNORE_CWISE,
656: $ INFO )
657: ELSE
658: CALL ZLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
659: $ NRHS, AB, LDAB, AFB, LDAFB, IPIV, ROWEQU, R, B,
660: $ LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM,
661: $ ERR_BNDS_COMP, WORK, RWORK, WORK(N+1),
662: $ TRANSFER (RWORK(1:2*N), (/ (ZERO, ZERO) /), N),
663: $ RCOND, ITHRESH, RTHRESH, UNSTABLE_THRESH, IGNORE_CWISE,
664: $ INFO )
665: END IF
666: END IF
667:
668: ERR_LBND = MAX( 10.0D+0, SQRT( DBLE( N ) ) ) * DLAMCH( 'Epsilon' )
669: IF (N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 1) THEN
670: *
671: * Compute scaled normwise condition number cond(A*C).
672: *
673: IF ( COLEQU .AND. NOTRAN ) THEN
674: RCOND_TMP = ZLA_GBRCOND_C( TRANS, N, KL, KU, AB, LDAB, AFB,
675: $ LDAFB, IPIV, C, .TRUE., INFO, WORK, RWORK )
676: ELSE IF ( ROWEQU .AND. .NOT. NOTRAN ) THEN
677: RCOND_TMP = ZLA_GBRCOND_C( TRANS, N, KL, KU, AB, LDAB, AFB,
678: $ LDAFB, IPIV, R, .TRUE., INFO, WORK, RWORK )
679: ELSE
680: RCOND_TMP = ZLA_GBRCOND_C( TRANS, N, KL, KU, AB, LDAB, AFB,
681: $ LDAFB, IPIV, C, .FALSE., INFO, WORK, RWORK )
682: END IF
683: DO J = 1, NRHS
684: *
685: * Cap the error at 1.0.
686: *
687: IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I
688: $ .AND. ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .GT. 1.0D+0)
689: $ ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0
690: *
691: * Threshold the error (see LAWN).
692: *
693: IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN
694: ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0
695: ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 0.0D+0
696: IF ( INFO .LE. N ) INFO = N + J
697: ELSE IF ( ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .LT. ERR_LBND )
698: $ THEN
699: ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = ERR_LBND
700: ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0
701: END IF
702: *
703: * Save the condition number.
704: *
705: IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN
706: ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = RCOND_TMP
707: END IF
708:
709: END DO
710: END IF
711:
712: IF (N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 2) THEN
713: *
714: * Compute componentwise condition number cond(A*diag(Y(:,J))) for
715: * each right-hand side using the current solution as an estimate of
716: * the true solution. If the componentwise error estimate is too
717: * large, then the solution is a lousy estimate of truth and the
718: * estimated RCOND may be too optimistic. To avoid misleading users,
719: * the inverse condition number is set to 0.0 when the estimated
720: * cwise error is at least CWISE_WRONG.
721: *
722: CWISE_WRONG = SQRT( DLAMCH( 'Epsilon' ) )
723: DO J = 1, NRHS
724: IF (ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .LT. CWISE_WRONG )
725: $ THEN
726: RCOND_TMP = ZLA_GBRCOND_X( TRANS, N, KL, KU, AB, LDAB,
727: $ AFB, LDAFB, IPIV, X( 1, J ), INFO, WORK, RWORK )
728: ELSE
729: RCOND_TMP = 0.0D+0
730: END IF
731: *
732: * Cap the error at 1.0.
733: *
734: IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I
735: $ .AND. ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .GT. 1.0D+0 )
736: $ ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0
737: *
738: * Threshold the error (see LAWN).
739: *
740: IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN
741: ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0
742: ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 0.0D+0
743: IF ( PARAMS( LA_LINRX_CWISE_I ) .EQ. 1.0D+0
744: $ .AND. INFO.LT.N + J ) INFO = N + J
745: ELSE IF ( ERR_BNDS_COMP( J, LA_LINRX_ERR_I )
746: $ .LT. ERR_LBND ) THEN
747: ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = ERR_LBND
748: ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0
749: END IF
750: *
751: * Save the condition number.
752: *
753: IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN
754: ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = RCOND_TMP
755: END IF
756:
757: END DO
758: END IF
759: *
760: RETURN
761: *
762: * End of ZGBRFSX
763: *
764: END
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