Annotation of rpl/lapack/lapack/dla_gbrfsx_extended.f, revision 1.5
1.5 ! bertrand 1: *> \brief \b DLA_GBRFSX_EXTENDED
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DLA_GBRFSX_EXTENDED + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dla_gbrfsx_extended.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dla_gbrfsx_extended.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dla_gbrfsx_extended.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
! 22: * NRHS, AB, LDAB, AFB, LDAFB, IPIV,
! 23: * COLEQU, C, B, LDB, Y, LDY,
! 24: * BERR_OUT, N_NORMS, ERR_BNDS_NORM,
! 25: * ERR_BNDS_COMP, RES, AYB, DY,
! 26: * Y_TAIL, RCOND, ITHRESH, RTHRESH,
! 27: * DZ_UB, IGNORE_CWISE, INFO )
! 28: *
! 29: * .. Scalar Arguments ..
! 30: * INTEGER INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
! 31: * $ PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
! 32: * LOGICAL COLEQU, IGNORE_CWISE
! 33: * DOUBLE PRECISION RTHRESH, DZ_UB
! 34: * ..
! 35: * .. Array Arguments ..
! 36: * INTEGER IPIV( * )
! 37: * DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
! 38: * $ Y( LDY, * ), RES(*), DY(*), Y_TAIL(*)
! 39: * DOUBLE PRECISION C( * ), AYB(*), RCOND, BERR_OUT(*),
! 40: * $ ERR_BNDS_NORM( NRHS, * ),
! 41: * $ ERR_BNDS_COMP( NRHS, * )
! 42: * ..
! 43: *
! 44: *
! 45: *> \par Purpose:
! 46: * =============
! 47: *>
! 48: *> \verbatim
! 49: *>
! 50: *>
! 51: *> DLA_GBRFSX_EXTENDED improves the computed solution to a system of
! 52: *> linear equations by performing extra-precise iterative refinement
! 53: *> and provides error bounds and backward error estimates for the solution.
! 54: *> This subroutine is called by DGBRFSX to perform iterative refinement.
! 55: *> In addition to normwise error bound, the code provides maximum
! 56: *> componentwise error bound if possible. See comments for ERR_BNDS_NORM
! 57: *> and ERR_BNDS_COMP for details of the error bounds. Note that this
! 58: *> subroutine is only resonsible for setting the second fields of
! 59: *> ERR_BNDS_NORM and ERR_BNDS_COMP.
! 60: *> \endverbatim
! 61: *
! 62: * Arguments:
! 63: * ==========
! 64: *
! 65: *> \param[in] PREC_TYPE
! 66: *> \verbatim
! 67: *> PREC_TYPE is INTEGER
! 68: *> Specifies the intermediate precision to be used in refinement.
! 69: *> The value is defined by ILAPREC(P) where P is a CHARACTER and
! 70: *> P = 'S': Single
! 71: *> = 'D': Double
! 72: *> = 'I': Indigenous
! 73: *> = 'X', 'E': Extra
! 74: *> \endverbatim
! 75: *>
! 76: *> \param[in] TRANS_TYPE
! 77: *> \verbatim
! 78: *> TRANS_TYPE is INTEGER
! 79: *> Specifies the transposition operation on A.
! 80: *> The value is defined by ILATRANS(T) where T is a CHARACTER and
! 81: *> T = 'N': No transpose
! 82: *> = 'T': Transpose
! 83: *> = 'C': Conjugate transpose
! 84: *> \endverbatim
! 85: *>
! 86: *> \param[in] N
! 87: *> \verbatim
! 88: *> N is INTEGER
! 89: *> The number of linear equations, i.e., the order of the
! 90: *> matrix A. N >= 0.
! 91: *> \endverbatim
! 92: *>
! 93: *> \param[in] KL
! 94: *> \verbatim
! 95: *> KL is INTEGER
! 96: *> The number of subdiagonals within the band of A. KL >= 0.
! 97: *> \endverbatim
! 98: *>
! 99: *> \param[in] KU
! 100: *> \verbatim
! 101: *> KU is INTEGER
! 102: *> The number of superdiagonals within the band of A. KU >= 0
! 103: *> \endverbatim
! 104: *>
! 105: *> \param[in] NRHS
! 106: *> \verbatim
! 107: *> NRHS is INTEGER
! 108: *> The number of right-hand-sides, i.e., the number of columns of the
! 109: *> matrix B.
! 110: *> \endverbatim
! 111: *>
! 112: *> \param[in] AB
! 113: *> \verbatim
! 114: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
! 115: *> On entry, the N-by-N matrix AB.
! 116: *> \endverbatim
! 117: *>
! 118: *> \param[in] LDAB
! 119: *> \verbatim
! 120: *> LDAB is INTEGER
! 121: *> The leading dimension of the array AB. LDBA >= max(1,N).
! 122: *> \endverbatim
! 123: *>
! 124: *> \param[in] AFB
! 125: *> \verbatim
! 126: *> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
! 127: *> The factors L and U from the factorization
! 128: *> A = P*L*U as computed by DGBTRF.
! 129: *> \endverbatim
! 130: *>
! 131: *> \param[in] LDAFB
! 132: *> \verbatim
! 133: *> LDAFB is INTEGER
! 134: *> The leading dimension of the array AF. LDAFB >= max(1,N).
! 135: *> \endverbatim
! 136: *>
! 137: *> \param[in] IPIV
! 138: *> \verbatim
! 139: *> IPIV is INTEGER array, dimension (N)
! 140: *> The pivot indices from the factorization A = P*L*U
! 141: *> as computed by DGBTRF; row i of the matrix was interchanged
! 142: *> with row IPIV(i).
! 143: *> \endverbatim
! 144: *>
! 145: *> \param[in] COLEQU
! 146: *> \verbatim
! 147: *> COLEQU is LOGICAL
! 148: *> If .TRUE. then column equilibration was done to A before calling
! 149: *> this routine. This is needed to compute the solution and error
! 150: *> bounds correctly.
! 151: *> \endverbatim
! 152: *>
! 153: *> \param[in] C
! 154: *> \verbatim
! 155: *> C is DOUBLE PRECISION array, dimension (N)
! 156: *> The column scale factors for A. If COLEQU = .FALSE., C
! 157: *> is not accessed. If C is input, each element of C should be a power
! 158: *> of the radix to ensure a reliable solution and error estimates.
! 159: *> Scaling by powers of the radix does not cause rounding errors unless
! 160: *> the result underflows or overflows. Rounding errors during scaling
! 161: *> lead to refining with a matrix that is not equivalent to the
! 162: *> input matrix, producing error estimates that may not be
! 163: *> reliable.
! 164: *> \endverbatim
! 165: *>
! 166: *> \param[in] B
! 167: *> \verbatim
! 168: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
! 169: *> The right-hand-side matrix B.
! 170: *> \endverbatim
! 171: *>
! 172: *> \param[in] LDB
! 173: *> \verbatim
! 174: *> LDB is INTEGER
! 175: *> The leading dimension of the array B. LDB >= max(1,N).
! 176: *> \endverbatim
! 177: *>
! 178: *> \param[in,out] Y
! 179: *> \verbatim
! 180: *> Y is DOUBLE PRECISION array, dimension
! 181: *> (LDY,NRHS)
! 182: *> On entry, the solution matrix X, as computed by DGBTRS.
! 183: *> On exit, the improved solution matrix Y.
! 184: *> \endverbatim
! 185: *>
! 186: *> \param[in] LDY
! 187: *> \verbatim
! 188: *> LDY is INTEGER
! 189: *> The leading dimension of the array Y. LDY >= max(1,N).
! 190: *> \endverbatim
! 191: *>
! 192: *> \param[out] BERR_OUT
! 193: *> \verbatim
! 194: *> BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
! 195: *> On exit, BERR_OUT(j) contains the componentwise relative backward
! 196: *> error for right-hand-side j from the formula
! 197: *> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
! 198: *> where abs(Z) is the componentwise absolute value of the matrix
! 199: *> or vector Z. This is computed by DLA_LIN_BERR.
! 200: *> \endverbatim
! 201: *>
! 202: *> \param[in] N_NORMS
! 203: *> \verbatim
! 204: *> N_NORMS is INTEGER
! 205: *> Determines which error bounds to return (see ERR_BNDS_NORM
! 206: *> and ERR_BNDS_COMP).
! 207: *> If N_NORMS >= 1 return normwise error bounds.
! 208: *> If N_NORMS >= 2 return componentwise error bounds.
! 209: *> \endverbatim
! 210: *>
! 211: *> \param[in,out] ERR_BNDS_NORM
! 212: *> \verbatim
! 213: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension
! 214: *> (NRHS, N_ERR_BNDS)
! 215: *> For each right-hand side, this array contains information about
! 216: *> various error bounds and condition numbers corresponding to the
! 217: *> normwise relative error, which is defined as follows:
! 218: *>
! 219: *> Normwise relative error in the ith solution vector:
! 220: *> max_j (abs(XTRUE(j,i) - X(j,i)))
! 221: *> ------------------------------
! 222: *> max_j abs(X(j,i))
! 223: *>
! 224: *> The array is indexed by the type of error information as described
! 225: *> below. There currently are up to three pieces of information
! 226: *> returned.
! 227: *>
! 228: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
! 229: *> right-hand side.
! 230: *>
! 231: *> The second index in ERR_BNDS_NORM(:,err) contains the following
! 232: *> three fields:
! 233: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 234: *> reciprocal condition number is less than the threshold
! 235: *> sqrt(n) * slamch('Epsilon').
! 236: *>
! 237: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 238: *> almost certainly within a factor of 10 of the true error
! 239: *> so long as the next entry is greater than the threshold
! 240: *> sqrt(n) * slamch('Epsilon'). This error bound should only
! 241: *> be trusted if the previous boolean is true.
! 242: *>
! 243: *> err = 3 Reciprocal condition number: Estimated normwise
! 244: *> reciprocal condition number. Compared with the threshold
! 245: *> sqrt(n) * slamch('Epsilon') to determine if the error
! 246: *> estimate is "guaranteed". These reciprocal condition
! 247: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 248: *> appropriately scaled matrix Z.
! 249: *> Let Z = S*A, where S scales each row by a power of the
! 250: *> radix so all absolute row sums of Z are approximately 1.
! 251: *>
! 252: *> This subroutine is only responsible for setting the second field
! 253: *> above.
! 254: *> See Lapack Working Note 165 for further details and extra
! 255: *> cautions.
! 256: *> \endverbatim
! 257: *>
! 258: *> \param[in,out] ERR_BNDS_COMP
! 259: *> \verbatim
! 260: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension
! 261: *> (NRHS, N_ERR_BNDS)
! 262: *> For each right-hand side, this array contains information about
! 263: *> various error bounds and condition numbers corresponding to the
! 264: *> componentwise relative error, which is defined as follows:
! 265: *>
! 266: *> Componentwise relative error in the ith solution vector:
! 267: *> abs(XTRUE(j,i) - X(j,i))
! 268: *> max_j ----------------------
! 269: *> abs(X(j,i))
! 270: *>
! 271: *> The array is indexed by the right-hand side i (on which the
! 272: *> componentwise relative error depends), and the type of error
! 273: *> information as described below. There currently are up to three
! 274: *> pieces of information returned for each right-hand side. If
! 275: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
! 276: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
! 277: *> the first (:,N_ERR_BNDS) entries are returned.
! 278: *>
! 279: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
! 280: *> right-hand side.
! 281: *>
! 282: *> The second index in ERR_BNDS_COMP(:,err) contains the following
! 283: *> three fields:
! 284: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 285: *> reciprocal condition number is less than the threshold
! 286: *> sqrt(n) * slamch('Epsilon').
! 287: *>
! 288: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 289: *> almost certainly within a factor of 10 of the true error
! 290: *> so long as the next entry is greater than the threshold
! 291: *> sqrt(n) * slamch('Epsilon'). This error bound should only
! 292: *> be trusted if the previous boolean is true.
! 293: *>
! 294: *> err = 3 Reciprocal condition number: Estimated componentwise
! 295: *> reciprocal condition number. Compared with the threshold
! 296: *> sqrt(n) * slamch('Epsilon') to determine if the error
! 297: *> estimate is "guaranteed". These reciprocal condition
! 298: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 299: *> appropriately scaled matrix Z.
! 300: *> Let Z = S*(A*diag(x)), where x is the solution for the
! 301: *> current right-hand side and S scales each row of
! 302: *> A*diag(x) by a power of the radix so all absolute row
! 303: *> sums of Z are approximately 1.
! 304: *>
! 305: *> This subroutine is only responsible for setting the second field
! 306: *> above.
! 307: *> See Lapack Working Note 165 for further details and extra
! 308: *> cautions.
! 309: *> \endverbatim
! 310: *>
! 311: *> \param[in] RES
! 312: *> \verbatim
! 313: *> RES is DOUBLE PRECISION array, dimension (N)
! 314: *> Workspace to hold the intermediate residual.
! 315: *> \endverbatim
! 316: *>
! 317: *> \param[in] AYB
! 318: *> \verbatim
! 319: *> AYB is DOUBLE PRECISION array, dimension (N)
! 320: *> Workspace. This can be the same workspace passed for Y_TAIL.
! 321: *> \endverbatim
! 322: *>
! 323: *> \param[in] DY
! 324: *> \verbatim
! 325: *> DY is DOUBLE PRECISION array, dimension (N)
! 326: *> Workspace to hold the intermediate solution.
! 327: *> \endverbatim
! 328: *>
! 329: *> \param[in] Y_TAIL
! 330: *> \verbatim
! 331: *> Y_TAIL is DOUBLE PRECISION array, dimension (N)
! 332: *> Workspace to hold the trailing bits of the intermediate solution.
! 333: *> \endverbatim
! 334: *>
! 335: *> \param[in] RCOND
! 336: *> \verbatim
! 337: *> RCOND is DOUBLE PRECISION
! 338: *> Reciprocal scaled condition number. This is an estimate of the
! 339: *> reciprocal Skeel condition number of the matrix A after
! 340: *> equilibration (if done). If this is less than the machine
! 341: *> precision (in particular, if it is zero), the matrix is singular
! 342: *> to working precision. Note that the error may still be small even
! 343: *> if this number is very small and the matrix appears ill-
! 344: *> conditioned.
! 345: *> \endverbatim
! 346: *>
! 347: *> \param[in] ITHRESH
! 348: *> \verbatim
! 349: *> ITHRESH is INTEGER
! 350: *> The maximum number of residual computations allowed for
! 351: *> refinement. The default is 10. For 'aggressive' set to 100 to
! 352: *> permit convergence using approximate factorizations or
! 353: *> factorizations other than LU. If the factorization uses a
! 354: *> technique other than Gaussian elimination, the guarantees in
! 355: *> ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
! 356: *> \endverbatim
! 357: *>
! 358: *> \param[in] RTHRESH
! 359: *> \verbatim
! 360: *> RTHRESH is DOUBLE PRECISION
! 361: *> Determines when to stop refinement if the error estimate stops
! 362: *> decreasing. Refinement will stop when the next solution no longer
! 363: *> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
! 364: *> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
! 365: *> default value is 0.5. For 'aggressive' set to 0.9 to permit
! 366: *> convergence on extremely ill-conditioned matrices. See LAWN 165
! 367: *> for more details.
! 368: *> \endverbatim
! 369: *>
! 370: *> \param[in] DZ_UB
! 371: *> \verbatim
! 372: *> DZ_UB is DOUBLE PRECISION
! 373: *> Determines when to start considering componentwise convergence.
! 374: *> Componentwise convergence is only considered after each component
! 375: *> of the solution Y is stable, which we definte as the relative
! 376: *> change in each component being less than DZ_UB. The default value
! 377: *> is 0.25, requiring the first bit to be stable. See LAWN 165 for
! 378: *> more details.
! 379: *> \endverbatim
! 380: *>
! 381: *> \param[in] IGNORE_CWISE
! 382: *> \verbatim
! 383: *> IGNORE_CWISE is LOGICAL
! 384: *> If .TRUE. then ignore componentwise convergence. Default value
! 385: *> is .FALSE..
! 386: *> \endverbatim
! 387: *>
! 388: *> \param[out] INFO
! 389: *> \verbatim
! 390: *> INFO is INTEGER
! 391: *> = 0: Successful exit.
! 392: *> < 0: if INFO = -i, the ith argument to DGBTRS had an illegal
! 393: *> value
! 394: *> \endverbatim
! 395: *
! 396: * Authors:
! 397: * ========
! 398: *
! 399: *> \author Univ. of Tennessee
! 400: *> \author Univ. of California Berkeley
! 401: *> \author Univ. of Colorado Denver
! 402: *> \author NAG Ltd.
! 403: *
! 404: *> \date November 2011
! 405: *
! 406: *> \ingroup doubleGBcomputational
! 407: *
! 408: * =====================================================================
1.1 bertrand 409: SUBROUTINE DLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, KU,
410: $ NRHS, AB, LDAB, AFB, LDAFB, IPIV,
411: $ COLEQU, C, B, LDB, Y, LDY,
412: $ BERR_OUT, N_NORMS, ERR_BNDS_NORM,
413: $ ERR_BNDS_COMP, RES, AYB, DY,
414: $ Y_TAIL, RCOND, ITHRESH, RTHRESH,
415: $ DZ_UB, IGNORE_CWISE, INFO )
416: *
1.5 ! bertrand 417: * -- LAPACK computational routine (version 3.4.0) --
! 418: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 419: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 420: * November 2011
1.1 bertrand 421: *
422: * .. Scalar Arguments ..
423: INTEGER INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
424: $ PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
425: LOGICAL COLEQU, IGNORE_CWISE
426: DOUBLE PRECISION RTHRESH, DZ_UB
427: * ..
428: * .. Array Arguments ..
429: INTEGER IPIV( * )
430: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
431: $ Y( LDY, * ), RES(*), DY(*), Y_TAIL(*)
432: DOUBLE PRECISION C( * ), AYB(*), RCOND, BERR_OUT(*),
433: $ ERR_BNDS_NORM( NRHS, * ),
434: $ ERR_BNDS_COMP( NRHS, * )
435: * ..
436: *
437: * =====================================================================
438: *
439: * .. Local Scalars ..
440: CHARACTER TRANS
441: INTEGER CNT, I, J, M, X_STATE, Z_STATE, Y_PREC_STATE
442: DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
443: $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
444: $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
445: $ EPS, HUGEVAL, INCR_THRESH
446: LOGICAL INCR_PREC
447: * ..
448: * .. Parameters ..
449: INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
450: $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
451: $ EXTRA_Y
452: PARAMETER ( UNSTABLE_STATE = 0, WORKING_STATE = 1,
453: $ CONV_STATE = 2, NOPROG_STATE = 3 )
454: PARAMETER ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1,
455: $ EXTRA_Y = 2 )
456: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
457: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
458: INTEGER CMP_ERR_I, PIV_GROWTH_I
459: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
460: $ BERR_I = 3 )
461: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
462: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
463: $ PIV_GROWTH_I = 9 )
464: INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
465: $ LA_LINRX_CWISE_I
466: PARAMETER ( LA_LINRX_ITREF_I = 1,
467: $ LA_LINRX_ITHRESH_I = 2 )
468: PARAMETER ( LA_LINRX_CWISE_I = 3 )
469: INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
470: $ LA_LINRX_RCOND_I
471: PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 )
472: PARAMETER ( LA_LINRX_RCOND_I = 3 )
473: * ..
474: * .. External Subroutines ..
475: EXTERNAL DAXPY, DCOPY, DGBTRS, DGBMV, BLAS_DGBMV_X,
476: $ BLAS_DGBMV2_X, DLA_GBAMV, DLA_WWADDW, DLAMCH,
477: $ CHLA_TRANSTYPE, DLA_LIN_BERR
478: DOUBLE PRECISION DLAMCH
479: CHARACTER CHLA_TRANSTYPE
480: * ..
481: * .. Intrinsic Functions ..
482: INTRINSIC ABS, MAX, MIN
483: * ..
484: * .. Executable Statements ..
485: *
486: IF (INFO.NE.0) RETURN
487: TRANS = CHLA_TRANSTYPE(TRANS_TYPE)
488: EPS = DLAMCH( 'Epsilon' )
489: HUGEVAL = DLAMCH( 'Overflow' )
490: * Force HUGEVAL to Inf
491: HUGEVAL = HUGEVAL * HUGEVAL
492: * Using HUGEVAL may lead to spurious underflows.
493: INCR_THRESH = DBLE( N ) * EPS
494: M = KL+KU+1
495:
496: DO J = 1, NRHS
497: Y_PREC_STATE = EXTRA_RESIDUAL
498: IF ( Y_PREC_STATE .EQ. EXTRA_Y ) THEN
499: DO I = 1, N
500: Y_TAIL( I ) = 0.0D+0
501: END DO
502: END IF
503:
504: DXRAT = 0.0D+0
505: DXRATMAX = 0.0D+0
506: DZRAT = 0.0D+0
507: DZRATMAX = 0.0D+0
508: FINAL_DX_X = HUGEVAL
509: FINAL_DZ_Z = HUGEVAL
510: PREVNORMDX = HUGEVAL
511: PREV_DZ_Z = HUGEVAL
512: DZ_Z = HUGEVAL
513: DX_X = HUGEVAL
514:
515: X_STATE = WORKING_STATE
516: Z_STATE = UNSTABLE_STATE
517: INCR_PREC = .FALSE.
518:
519: DO CNT = 1, ITHRESH
520: *
521: * Compute residual RES = B_s - op(A_s) * Y,
522: * op(A) = A, A**T, or A**H depending on TRANS (and type).
523: *
524: CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
525: IF ( Y_PREC_STATE .EQ. BASE_RESIDUAL ) THEN
526: CALL DGBMV( TRANS, M, N, KL, KU, -1.0D+0, AB, LDAB,
527: $ Y( 1, J ), 1, 1.0D+0, RES, 1 )
528: ELSE IF ( Y_PREC_STATE .EQ. EXTRA_RESIDUAL ) THEN
529: CALL BLAS_DGBMV_X( TRANS_TYPE, N, N, KL, KU,
530: $ -1.0D+0, AB, LDAB, Y( 1, J ), 1, 1.0D+0, RES, 1,
531: $ PREC_TYPE )
532: ELSE
533: CALL BLAS_DGBMV2_X( TRANS_TYPE, N, N, KL, KU, -1.0D+0,
534: $ AB, LDAB, Y( 1, J ), Y_TAIL, 1, 1.0D+0, RES, 1,
535: $ PREC_TYPE )
536: END IF
537:
538: ! XXX: RES is no longer needed.
539: CALL DCOPY( N, RES, 1, DY, 1 )
540: CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, DY, N,
541: $ INFO )
542: *
543: * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
544: *
545: NORMX = 0.0D+0
546: NORMY = 0.0D+0
547: NORMDX = 0.0D+0
548: DZ_Z = 0.0D+0
549: YMIN = HUGEVAL
550:
551: DO I = 1, N
552: YK = ABS( Y( I, J ) )
553: DYK = ABS( DY( I ) )
554:
555: IF ( YK .NE. 0.0D+0 ) THEN
556: DZ_Z = MAX( DZ_Z, DYK / YK )
557: ELSE IF ( DYK .NE. 0.0D+0 ) THEN
558: DZ_Z = HUGEVAL
559: END IF
560:
561: YMIN = MIN( YMIN, YK )
562:
563: NORMY = MAX( NORMY, YK )
564:
565: IF ( COLEQU ) THEN
566: NORMX = MAX( NORMX, YK * C( I ) )
567: NORMDX = MAX( NORMDX, DYK * C( I ) )
568: ELSE
569: NORMX = NORMY
570: NORMDX = MAX( NORMDX, DYK )
571: END IF
572: END DO
573:
574: IF ( NORMX .NE. 0.0D+0 ) THEN
575: DX_X = NORMDX / NORMX
576: ELSE IF ( NORMDX .EQ. 0.0D+0 ) THEN
577: DX_X = 0.0D+0
578: ELSE
579: DX_X = HUGEVAL
580: END IF
581:
582: DXRAT = NORMDX / PREVNORMDX
583: DZRAT = DZ_Z / PREV_DZ_Z
584: *
585: * Check termination criteria.
586: *
587: IF ( .NOT.IGNORE_CWISE
588: $ .AND. YMIN*RCOND .LT. INCR_THRESH*NORMY
589: $ .AND. Y_PREC_STATE .LT. EXTRA_Y )
590: $ INCR_PREC = .TRUE.
591:
592: IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH )
593: $ X_STATE = WORKING_STATE
594: IF ( X_STATE .EQ. WORKING_STATE ) THEN
595: IF ( DX_X .LE. EPS ) THEN
596: X_STATE = CONV_STATE
597: ELSE IF ( DXRAT .GT. RTHRESH ) THEN
598: IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
599: INCR_PREC = .TRUE.
600: ELSE
601: X_STATE = NOPROG_STATE
602: END IF
603: ELSE
604: IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT
605: END IF
606: IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X
607: END IF
608:
609: IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB )
610: $ Z_STATE = WORKING_STATE
611: IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH )
612: $ Z_STATE = WORKING_STATE
613: IF ( Z_STATE .EQ. WORKING_STATE ) THEN
614: IF ( DZ_Z .LE. EPS ) THEN
615: Z_STATE = CONV_STATE
616: ELSE IF ( DZ_Z .GT. DZ_UB ) THEN
617: Z_STATE = UNSTABLE_STATE
618: DZRATMAX = 0.0D+0
619: FINAL_DZ_Z = HUGEVAL
620: ELSE IF ( DZRAT .GT. RTHRESH ) THEN
621: IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN
622: INCR_PREC = .TRUE.
623: ELSE
624: Z_STATE = NOPROG_STATE
625: END IF
626: ELSE
627: IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT
628: END IF
629: IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
630: END IF
631: *
632: * Exit if both normwise and componentwise stopped working,
633: * but if componentwise is unstable, let it go at least two
634: * iterations.
635: *
636: IF ( X_STATE.NE.WORKING_STATE ) THEN
637: IF ( IGNORE_CWISE ) GOTO 666
638: IF ( Z_STATE.EQ.NOPROG_STATE .OR. Z_STATE.EQ.CONV_STATE )
639: $ GOTO 666
640: IF ( Z_STATE.EQ.UNSTABLE_STATE .AND. CNT.GT.1 ) GOTO 666
641: END IF
642:
643: IF ( INCR_PREC ) THEN
644: INCR_PREC = .FALSE.
645: Y_PREC_STATE = Y_PREC_STATE + 1
646: DO I = 1, N
647: Y_TAIL( I ) = 0.0D+0
648: END DO
649: END IF
650:
651: PREVNORMDX = NORMDX
652: PREV_DZ_Z = DZ_Z
653: *
654: * Update soluton.
655: *
656: IF (Y_PREC_STATE .LT. EXTRA_Y) THEN
657: CALL DAXPY( N, 1.0D+0, DY, 1, Y(1,J), 1 )
658: ELSE
659: CALL DLA_WWADDW( N, Y(1,J), Y_TAIL, DY )
660: END IF
661:
662: END DO
663: * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
664: 666 CONTINUE
665: *
666: * Set final_* when cnt hits ithresh.
667: *
668: IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X
669: IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z
670: *
671: * Compute error bounds.
672: *
673: IF ( N_NORMS .GE. 1 ) THEN
674: ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) =
675: $ FINAL_DX_X / (1 - DXRATMAX)
676: END IF
677: IF (N_NORMS .GE. 2) THEN
678: ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) =
679: $ FINAL_DZ_Z / (1 - DZRATMAX)
680: END IF
681: *
682: * Compute componentwise relative backward error from formula
683: * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
684: * where abs(Z) is the componentwise absolute value of the matrix
685: * or vector Z.
686: *
687: * Compute residual RES = B_s - op(A_s) * Y,
688: * op(A) = A, A**T, or A**H depending on TRANS (and type).
689: *
690: CALL DCOPY( N, B( 1, J ), 1, RES, 1 )
691: CALL DGBMV(TRANS, N, N, KL, KU, -1.0D+0, AB, LDAB, Y(1,J),
692: $ 1, 1.0D+0, RES, 1 )
693:
694: DO I = 1, N
695: AYB( I ) = ABS( B( I, J ) )
696: END DO
697: *
698: * Compute abs(op(A_s))*abs(Y) + abs(B_s).
699: *
700: CALL DLA_GBAMV( TRANS_TYPE, N, N, KL, KU, 1.0D+0,
701: $ AB, LDAB, Y(1, J), 1, 1.0D+0, AYB, 1 )
702:
703: CALL DLA_LIN_BERR( N, N, 1, RES, AYB, BERR_OUT( J ) )
704: *
705: * End of loop for each RHS
706: *
707: END DO
708: *
709: RETURN
710: END
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