Annotation of rpl/lapack/lapack/dgbsvxx.f, revision 1.5
1.5 ! bertrand 1: *> \brief <b> DGBSVXX computes the solution to system of linear equations A * X = B for GB 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 DGBSVXX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgbsvxx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgbsvxx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgbsvxx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
! 22: * LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
! 23: * RCOND, RPVGRW, BERR, N_ERR_BNDS,
! 24: * ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
! 25: * WORK, IWORK, INFO )
! 26: *
! 27: * .. Scalar Arguments ..
! 28: * CHARACTER EQUED, FACT, TRANS
! 29: * INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
! 30: * $ N_ERR_BNDS, KL, KU
! 31: * DOUBLE PRECISION RCOND, RPVGRW
! 32: * ..
! 33: * .. Array Arguments ..
! 34: * INTEGER IPIV( * ), IWORK( * )
! 35: * DOUBLE PRECISION 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, * )
! 40: * ..
! 41: *
! 42: *
! 43: *> \par Purpose:
! 44: * =============
! 45: *>
! 46: *> \verbatim
! 47: *>
! 48: *> DGBSVXX uses the LU factorization to compute the solution to a
! 49: *> double precision system of linear equations A * X = B, where A is an
! 50: *> N-by-N matrix and X and B are N-by-NRHS matrices.
! 51: *>
! 52: *> If requested, both normwise and maximum componentwise error bounds
! 53: *> are returned. DGBSVXX 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: *> DGBSVXX 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: *> DGBSVXX 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 DGBSVXX 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: *> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
! 79: *> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
! 80: *> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
! 81: *>
! 82: *> Whether or not the system will be equilibrated depends on the
! 83: *> scaling of the matrix A, but if equilibration is used, A is
! 84: *> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
! 85: *> or diag(C)*B (if TRANS = 'T' or 'C').
! 86: *>
! 87: *> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
! 88: *> the matrix A (after equilibration if FACT = 'E') as
! 89: *>
! 90: *> A = P * L * U,
! 91: *>
! 92: *> where P is a permutation matrix, L is a unit lower triangular
! 93: *> matrix, and U is upper triangular.
! 94: *>
! 95: *> 3. If some U(i,i)=0, so that U is exactly singular, then the
! 96: *> routine returns with INFO = i. Otherwise, the factored form of A
! 97: *> is used to estimate the condition number of the matrix A (see
! 98: *> argument RCOND). If the reciprocal of the condition number is less
! 99: *> than machine precision, the routine still goes on to solve for X
! 100: *> and compute error bounds as described below.
! 101: *>
! 102: *> 4. The system of equations is solved for X using the factored form
! 103: *> of A.
! 104: *>
! 105: *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
! 106: *> the routine will use iterative refinement to try to get a small
! 107: *> error and error bounds. Refinement calculates the residual to at
! 108: *> least twice the working precision.
! 109: *>
! 110: *> 6. If equilibration was used, the matrix X is premultiplied by
! 111: *> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
! 112: *> that it solves the original system before equilibration.
! 113: *> \endverbatim
! 114: *
! 115: * Arguments:
! 116: * ==========
! 117: *
! 118: *> \verbatim
! 119: *> Some optional parameters are bundled in the PARAMS array. These
! 120: *> settings determine how refinement is performed, but often the
! 121: *> defaults are acceptable. If the defaults are acceptable, users
! 122: *> can pass NPARAMS = 0 which prevents the source code from accessing
! 123: *> the PARAMS argument.
! 124: *> \endverbatim
! 125: *>
! 126: *> \param[in] FACT
! 127: *> \verbatim
! 128: *> FACT is CHARACTER*1
! 129: *> Specifies whether or not the factored form of the matrix A is
! 130: *> supplied on entry, and if not, whether the matrix A should be
! 131: *> equilibrated before it is factored.
! 132: *> = 'F': On entry, AF and IPIV contain the factored form of A.
! 133: *> If EQUED is not 'N', the matrix A has been
! 134: *> equilibrated with scaling factors given by R and C.
! 135: *> A, AF, and IPIV are not modified.
! 136: *> = 'N': The matrix A will be copied to AF and factored.
! 137: *> = 'E': The matrix A will be equilibrated if necessary, then
! 138: *> copied to AF and factored.
! 139: *> \endverbatim
! 140: *>
! 141: *> \param[in] TRANS
! 142: *> \verbatim
! 143: *> TRANS is CHARACTER*1
! 144: *> Specifies the form of the system of equations:
! 145: *> = 'N': A * X = B (No transpose)
! 146: *> = 'T': A**T * X = B (Transpose)
! 147: *> = 'C': A**H * X = B (Conjugate Transpose = Transpose)
! 148: *> \endverbatim
! 149: *>
! 150: *> \param[in] N
! 151: *> \verbatim
! 152: *> N is INTEGER
! 153: *> The number of linear equations, i.e., the order of the
! 154: *> matrix A. N >= 0.
! 155: *> \endverbatim
! 156: *>
! 157: *> \param[in] KL
! 158: *> \verbatim
! 159: *> KL is INTEGER
! 160: *> The number of subdiagonals within the band of A. KL >= 0.
! 161: *> \endverbatim
! 162: *>
! 163: *> \param[in] KU
! 164: *> \verbatim
! 165: *> KU is INTEGER
! 166: *> The number of superdiagonals within the band of A. KU >= 0.
! 167: *> \endverbatim
! 168: *>
! 169: *> \param[in] NRHS
! 170: *> \verbatim
! 171: *> NRHS is INTEGER
! 172: *> The number of right hand sides, i.e., the number of columns
! 173: *> of the matrices B and X. NRHS >= 0.
! 174: *> \endverbatim
! 175: *>
! 176: *> \param[in,out] AB
! 177: *> \verbatim
! 178: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
! 179: *> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
! 180: *> The j-th column of A is stored in the j-th column of the
! 181: *> array AB as follows:
! 182: *> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
! 183: *>
! 184: *> If FACT = 'F' and EQUED is not 'N', then AB must have been
! 185: *> equilibrated by the scaling factors in R and/or C. AB is not
! 186: *> modified if FACT = 'F' or 'N', or if FACT = 'E' and
! 187: *> EQUED = 'N' on exit.
! 188: *>
! 189: *> On exit, if EQUED .ne. 'N', A is scaled as follows:
! 190: *> EQUED = 'R': A := diag(R) * A
! 191: *> EQUED = 'C': A := A * diag(C)
! 192: *> EQUED = 'B': A := diag(R) * A * diag(C).
! 193: *> \endverbatim
! 194: *>
! 195: *> \param[in] LDAB
! 196: *> \verbatim
! 197: *> LDAB is INTEGER
! 198: *> The leading dimension of the array AB. LDAB >= KL+KU+1.
! 199: *> \endverbatim
! 200: *>
! 201: *> \param[in,out] AFB
! 202: *> \verbatim
! 203: *> AFB is or output) DOUBLE PRECISION array, dimension (LDAFB,N)
! 204: *> If FACT = 'F', then AFB is an input argument and on entry
! 205: *> contains details of the LU factorization of the band matrix
! 206: *> A, as computed by DGBTRF. U is stored as an upper triangular
! 207: *> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
! 208: *> and the multipliers used during the factorization are stored
! 209: *> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
! 210: *> the factored form of the equilibrated matrix A.
! 211: *>
! 212: *> If FACT = 'N', then AF is an output argument and on exit
! 213: *> returns the factors L and U from the factorization A = P*L*U
! 214: *> of the original matrix A.
! 215: *>
! 216: *> If FACT = 'E', then AF is an output argument and on exit
! 217: *> returns the factors L and U from the factorization A = P*L*U
! 218: *> of the equilibrated matrix A (see the description of A for
! 219: *> the form of the equilibrated matrix).
! 220: *> \endverbatim
! 221: *>
! 222: *> \param[in] LDAFB
! 223: *> \verbatim
! 224: *> LDAFB is INTEGER
! 225: *> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
! 226: *> \endverbatim
! 227: *>
! 228: *> \param[in,out] IPIV
! 229: *> \verbatim
! 230: *> IPIV is or output) INTEGER array, dimension (N)
! 231: *> If FACT = 'F', then IPIV is an input argument and on entry
! 232: *> contains the pivot indices from the factorization A = P*L*U
! 233: *> as computed by DGETRF; row i of the matrix was interchanged
! 234: *> with row IPIV(i).
! 235: *>
! 236: *> If FACT = 'N', then IPIV is an output argument and on exit
! 237: *> contains the pivot indices from the factorization A = P*L*U
! 238: *> of the original matrix A.
! 239: *>
! 240: *> If FACT = 'E', then IPIV is an output argument and on exit
! 241: *> contains the pivot indices from the factorization A = P*L*U
! 242: *> of the equilibrated matrix A.
! 243: *> \endverbatim
! 244: *>
! 245: *> \param[in,out] EQUED
! 246: *> \verbatim
! 247: *> EQUED is or output) CHARACTER*1
! 248: *> Specifies the form of equilibration that was done.
! 249: *> = 'N': No equilibration (always true if FACT = 'N').
! 250: *> = 'R': Row equilibration, i.e., A has been premultiplied by
! 251: *> diag(R).
! 252: *> = 'C': Column equilibration, i.e., A has been postmultiplied
! 253: *> by diag(C).
! 254: *> = 'B': Both row and column equilibration, i.e., A has been
! 255: *> replaced by diag(R) * A * diag(C).
! 256: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
! 257: *> output argument.
! 258: *> \endverbatim
! 259: *>
! 260: *> \param[in,out] R
! 261: *> \verbatim
! 262: *> R is or output) DOUBLE PRECISION array, dimension (N)
! 263: *> The row scale factors for A. If EQUED = 'R' or 'B', A is
! 264: *> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
! 265: *> is not accessed. R is an input argument if FACT = 'F';
! 266: *> otherwise, R is an output argument. If FACT = 'F' and
! 267: *> EQUED = 'R' or 'B', each element of R must be positive.
! 268: *> If R is output, each element of R is a power of the radix.
! 269: *> If R is input, each element of R should be a power of the radix
! 270: *> to ensure a reliable solution and error estimates. Scaling by
! 271: *> powers of the radix does not cause rounding errors unless the
! 272: *> result underflows or overflows. Rounding errors during scaling
! 273: *> lead to refining with a matrix that is not equivalent to the
! 274: *> input matrix, producing error estimates that may not be
! 275: *> reliable.
! 276: *> \endverbatim
! 277: *>
! 278: *> \param[in,out] C
! 279: *> \verbatim
! 280: *> C is or output) DOUBLE PRECISION array, dimension (N)
! 281: *> The column scale factors for A. If EQUED = 'C' or 'B', A is
! 282: *> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
! 283: *> is not accessed. C is an input argument if FACT = 'F';
! 284: *> otherwise, C is an output argument. If FACT = 'F' and
! 285: *> EQUED = 'C' or 'B', each element of C must be positive.
! 286: *> If C is output, each element of C is a power of the radix.
! 287: *> If C is input, each element of C should be a power of the radix
! 288: *> to ensure a reliable solution and error estimates. Scaling by
! 289: *> powers of the radix does not cause rounding errors unless the
! 290: *> result underflows or overflows. Rounding errors during scaling
! 291: *> lead to refining with a matrix that is not equivalent to the
! 292: *> input matrix, producing error estimates that may not be
! 293: *> reliable.
! 294: *> \endverbatim
! 295: *>
! 296: *> \param[in,out] B
! 297: *> \verbatim
! 298: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
! 299: *> On entry, the N-by-NRHS right hand side matrix B.
! 300: *> On exit,
! 301: *> if EQUED = 'N', B is not modified;
! 302: *> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
! 303: *> diag(R)*B;
! 304: *> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
! 305: *> overwritten by diag(C)*B.
! 306: *> \endverbatim
! 307: *>
! 308: *> \param[in] LDB
! 309: *> \verbatim
! 310: *> LDB is INTEGER
! 311: *> The leading dimension of the array B. LDB >= max(1,N).
! 312: *> \endverbatim
! 313: *>
! 314: *> \param[out] X
! 315: *> \verbatim
! 316: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
! 317: *> If INFO = 0, the N-by-NRHS solution matrix X to the original
! 318: *> system of equations. Note that A and B are modified on exit
! 319: *> if EQUED .ne. 'N', and the solution to the equilibrated system is
! 320: *> inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
! 321: *> inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
! 322: *> \endverbatim
! 323: *>
! 324: *> \param[in] LDX
! 325: *> \verbatim
! 326: *> LDX is INTEGER
! 327: *> The leading dimension of the array X. LDX >= max(1,N).
! 328: *> \endverbatim
! 329: *>
! 330: *> \param[out] RCOND
! 331: *> \verbatim
! 332: *> RCOND is DOUBLE PRECISION
! 333: *> Reciprocal scaled condition number. This is an estimate of the
! 334: *> reciprocal Skeel condition number of the matrix A after
! 335: *> equilibration (if done). If this is less than the machine
! 336: *> precision (in particular, if it is zero), the matrix is singular
! 337: *> to working precision. Note that the error may still be small even
! 338: *> if this number is very small and the matrix appears ill-
! 339: *> conditioned.
! 340: *> \endverbatim
! 341: *>
! 342: *> \param[out] RPVGRW
! 343: *> \verbatim
! 344: *> RPVGRW is DOUBLE PRECISION
! 345: *> Reciprocal pivot growth. On exit, this contains the reciprocal
! 346: *> pivot growth factor norm(A)/norm(U). The "max absolute element"
! 347: *> norm is used. If this is much less than 1, then the stability of
! 348: *> the LU factorization of the (equilibrated) matrix A could be poor.
! 349: *> This also means that the solution X, estimated condition numbers,
! 350: *> and error bounds could be unreliable. If factorization fails with
! 351: *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
! 352: *> for the leading INFO columns of A. In DGESVX, this quantity is
! 353: *> returned in WORK(1).
! 354: *> \endverbatim
! 355: *>
! 356: *> \param[out] BERR
! 357: *> \verbatim
! 358: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 359: *> Componentwise relative backward error. This is the
! 360: *> componentwise relative backward error of each solution vector X(j)
! 361: *> (i.e., the smallest relative change in any element of A or B that
! 362: *> makes X(j) an exact solution).
! 363: *> \endverbatim
! 364: *>
! 365: *> \param[in] N_ERR_BNDS
! 366: *> \verbatim
! 367: *> N_ERR_BNDS is INTEGER
! 368: *> Number of error bounds to return for each right hand side
! 369: *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
! 370: *> ERR_BNDS_COMP below.
! 371: *> \endverbatim
! 372: *>
! 373: *> \param[out] ERR_BNDS_NORM
! 374: *> \verbatim
! 375: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
! 376: *> For each right-hand side, this array contains information about
! 377: *> various error bounds and condition numbers corresponding to the
! 378: *> normwise relative error, which is defined as follows:
! 379: *>
! 380: *> Normwise relative error in the ith solution vector:
! 381: *> max_j (abs(XTRUE(j,i) - X(j,i)))
! 382: *> ------------------------------
! 383: *> max_j abs(X(j,i))
! 384: *>
! 385: *> The array is indexed by the type of error information as described
! 386: *> below. There currently are up to three pieces of information
! 387: *> returned.
! 388: *>
! 389: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
! 390: *> right-hand side.
! 391: *>
! 392: *> The second index in ERR_BNDS_NORM(:,err) contains the following
! 393: *> three fields:
! 394: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 395: *> reciprocal condition number is less than the threshold
! 396: *> sqrt(n) * dlamch('Epsilon').
! 397: *>
! 398: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 399: *> almost certainly within a factor of 10 of the true error
! 400: *> so long as the next entry is greater than the threshold
! 401: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
! 402: *> be trusted if the previous boolean is true.
! 403: *>
! 404: *> err = 3 Reciprocal condition number: Estimated normwise
! 405: *> reciprocal condition number. Compared with the threshold
! 406: *> sqrt(n) * dlamch('Epsilon') to determine if the error
! 407: *> estimate is "guaranteed". These reciprocal condition
! 408: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 409: *> appropriately scaled matrix Z.
! 410: *> Let Z = S*A, where S scales each row by a power of the
! 411: *> radix so all absolute row sums of Z are approximately 1.
! 412: *>
! 413: *> See Lapack Working Note 165 for further details and extra
! 414: *> cautions.
! 415: *> \endverbatim
! 416: *>
! 417: *> \param[out] ERR_BNDS_COMP
! 418: *> \verbatim
! 419: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
! 420: *> For each right-hand side, this array contains information about
! 421: *> various error bounds and condition numbers corresponding to the
! 422: *> componentwise relative error, which is defined as follows:
! 423: *>
! 424: *> Componentwise relative error in the ith solution vector:
! 425: *> abs(XTRUE(j,i) - X(j,i))
! 426: *> max_j ----------------------
! 427: *> abs(X(j,i))
! 428: *>
! 429: *> The array is indexed by the right-hand side i (on which the
! 430: *> componentwise relative error depends), and the type of error
! 431: *> information as described below. There currently are up to three
! 432: *> pieces of information returned for each right-hand side. If
! 433: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
! 434: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
! 435: *> the first (:,N_ERR_BNDS) entries are returned.
! 436: *>
! 437: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
! 438: *> right-hand side.
! 439: *>
! 440: *> The second index in ERR_BNDS_COMP(:,err) contains the following
! 441: *> three fields:
! 442: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 443: *> reciprocal condition number is less than the threshold
! 444: *> sqrt(n) * dlamch('Epsilon').
! 445: *>
! 446: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 447: *> almost certainly within a factor of 10 of the true error
! 448: *> so long as the next entry is greater than the threshold
! 449: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
! 450: *> be trusted if the previous boolean is true.
! 451: *>
! 452: *> err = 3 Reciprocal condition number: Estimated componentwise
! 453: *> reciprocal condition number. Compared with the threshold
! 454: *> sqrt(n) * dlamch('Epsilon') to determine if the error
! 455: *> estimate is "guaranteed". These reciprocal condition
! 456: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 457: *> appropriately scaled matrix Z.
! 458: *> Let Z = S*(A*diag(x)), where x is the solution for the
! 459: *> current right-hand side and S scales each row of
! 460: *> A*diag(x) by a power of the radix so all absolute row
! 461: *> sums of Z are approximately 1.
! 462: *>
! 463: *> See Lapack Working Note 165 for further details and extra
! 464: *> cautions.
! 465: *> \endverbatim
! 466: *>
! 467: *> \param[in] NPARAMS
! 468: *> \verbatim
! 469: *> NPARAMS is INTEGER
! 470: *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
! 471: *> PARAMS array is never referenced and default values are used.
! 472: *> \endverbatim
! 473: *>
! 474: *> \param[in,out] PARAMS
! 475: *> \verbatim
! 476: *> PARAMS is / output) DOUBLE PRECISION array, dimension (NPARAMS)
! 477: *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
! 478: *> that entry will be filled with default value used for that
! 479: *> parameter. Only positions up to NPARAMS are accessed; defaults
! 480: *> are used for higher-numbered parameters.
! 481: *>
! 482: *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
! 483: *> refinement or not.
! 484: *> Default: 1.0D+0
! 485: *> = 0.0 : No refinement is performed, and no error bounds are
! 486: *> computed.
! 487: *> = 1.0 : Use the extra-precise refinement algorithm.
! 488: *> (other values are reserved for future use)
! 489: *>
! 490: *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
! 491: *> computations allowed for refinement.
! 492: *> Default: 10
! 493: *> Aggressive: Set to 100 to permit convergence using approximate
! 494: *> factorizations or factorizations other than LU. If
! 495: *> the factorization uses a technique other than
! 496: *> Gaussian elimination, the guarantees in
! 497: *> err_bnds_norm and err_bnds_comp may no longer be
! 498: *> trustworthy.
! 499: *>
! 500: *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
! 501: *> will attempt to find a solution with small componentwise
! 502: *> relative error in the double-precision algorithm. Positive
! 503: *> is true, 0.0 is false.
! 504: *> Default: 1.0 (attempt componentwise convergence)
! 505: *> \endverbatim
! 506: *>
! 507: *> \param[out] WORK
! 508: *> \verbatim
! 509: *> WORK is DOUBLE PRECISION array, dimension (4*N)
! 510: *> \endverbatim
! 511: *>
! 512: *> \param[out] IWORK
! 513: *> \verbatim
! 514: *> IWORK is INTEGER array, dimension (N)
! 515: *> \endverbatim
! 516: *>
! 517: *> \param[out] INFO
! 518: *> \verbatim
! 519: *> INFO is INTEGER
! 520: *> = 0: Successful exit. The solution to every right-hand side is
! 521: *> guaranteed.
! 522: *> < 0: If INFO = -i, the i-th argument had an illegal value
! 523: *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
! 524: *> has been completed, but the factor U is exactly singular, so
! 525: *> the solution and error bounds could not be computed. RCOND = 0
! 526: *> is returned.
! 527: *> = N+J: The solution corresponding to the Jth right-hand side is
! 528: *> not guaranteed. The solutions corresponding to other right-
! 529: *> hand sides K with K > J may not be guaranteed as well, but
! 530: *> only the first such right-hand side is reported. If a small
! 531: *> componentwise error is not requested (PARAMS(3) = 0.0) then
! 532: *> the Jth right-hand side is the first with a normwise error
! 533: *> bound that is not guaranteed (the smallest J such
! 534: *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
! 535: *> the Jth right-hand side is the first with either a normwise or
! 536: *> componentwise error bound that is not guaranteed (the smallest
! 537: *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
! 538: *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
! 539: *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
! 540: *> about all of the right-hand sides check ERR_BNDS_NORM or
! 541: *> ERR_BNDS_COMP.
! 542: *> \endverbatim
! 543: *
! 544: * Authors:
! 545: * ========
! 546: *
! 547: *> \author Univ. of Tennessee
! 548: *> \author Univ. of California Berkeley
! 549: *> \author Univ. of Colorado Denver
! 550: *> \author NAG Ltd.
! 551: *
! 552: *> \date November 2011
! 553: *
! 554: *> \ingroup doubleGBsolve
! 555: *
! 556: * =====================================================================
1.1 bertrand 557: SUBROUTINE DGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
558: $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
559: $ RCOND, RPVGRW, BERR, N_ERR_BNDS,
560: $ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
561: $ WORK, IWORK, INFO )
562: *
1.5 ! bertrand 563: * -- LAPACK driver routine (version 3.4.0) --
! 564: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 565: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 566: * November 2011
1.1 bertrand 567: *
568: * .. Scalar Arguments ..
569: CHARACTER EQUED, FACT, TRANS
570: INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
571: $ N_ERR_BNDS, KL, KU
572: DOUBLE PRECISION RCOND, RPVGRW
573: * ..
574: * .. Array Arguments ..
575: INTEGER IPIV( * ), IWORK( * )
576: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
577: $ X( LDX , * ),WORK( * )
578: DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
579: $ ERR_BNDS_NORM( NRHS, * ),
580: $ ERR_BNDS_COMP( NRHS, * )
581: * ..
582: *
1.5 ! bertrand 583: * ==================================================================
1.1 bertrand 584: *
585: * .. Parameters ..
586: DOUBLE PRECISION ZERO, ONE
587: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
588: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
589: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
590: INTEGER CMP_ERR_I, PIV_GROWTH_I
591: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
592: $ BERR_I = 3 )
593: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
594: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
595: $ PIV_GROWTH_I = 9 )
596: * ..
597: * .. Local Scalars ..
598: LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
599: INTEGER INFEQU, I, J
600: DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
601: $ ROWCND, SMLNUM
602: * ..
603: * .. External Functions ..
604: EXTERNAL LSAME, DLAMCH, DLA_GBRPVGRW
605: LOGICAL LSAME
606: DOUBLE PRECISION DLAMCH, DLA_GBRPVGRW
607: * ..
608: * .. External Subroutines ..
609: EXTERNAL DGBEQUB, DGBTRF, DGBTRS, DLACPY, DLAQGB,
610: $ XERBLA, DLASCL2, DGBRFSX
611: * ..
612: * .. Intrinsic Functions ..
613: INTRINSIC MAX, MIN
614: * ..
615: * .. Executable Statements ..
616: *
617: INFO = 0
618: NOFACT = LSAME( FACT, 'N' )
619: EQUIL = LSAME( FACT, 'E' )
620: NOTRAN = LSAME( TRANS, 'N' )
621: SMLNUM = DLAMCH( 'Safe minimum' )
622: BIGNUM = ONE / SMLNUM
623: IF( NOFACT .OR. EQUIL ) THEN
624: EQUED = 'N'
625: ROWEQU = .FALSE.
626: COLEQU = .FALSE.
627: ELSE
628: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
629: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
630: END IF
631: *
632: * Default is failure. If an input parameter is wrong or
633: * factorization fails, make everything look horrible. Only the
634: * pivot growth is set here, the rest is initialized in DGBRFSX.
635: *
636: RPVGRW = ZERO
637: *
638: * Test the input parameters. PARAMS is not tested until DGBRFSX.
639: *
640: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
641: $ LSAME( FACT, 'F' ) ) THEN
642: INFO = -1
643: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
644: $ LSAME( TRANS, 'C' ) ) THEN
645: INFO = -2
646: ELSE IF( N.LT.0 ) THEN
647: INFO = -3
648: ELSE IF( KL.LT.0 ) THEN
649: INFO = -4
650: ELSE IF( KU.LT.0 ) THEN
651: INFO = -5
652: ELSE IF( NRHS.LT.0 ) THEN
653: INFO = -6
654: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
655: INFO = -8
656: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
657: INFO = -10
658: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
659: $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
660: INFO = -12
661: ELSE
662: IF( ROWEQU ) THEN
663: RCMIN = BIGNUM
664: RCMAX = ZERO
665: DO 10 J = 1, N
666: RCMIN = MIN( RCMIN, R( J ) )
667: RCMAX = MAX( RCMAX, R( J ) )
668: 10 CONTINUE
669: IF( RCMIN.LE.ZERO ) THEN
670: INFO = -13
671: ELSE IF( N.GT.0 ) THEN
672: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
673: ELSE
674: ROWCND = ONE
675: END IF
676: END IF
677: IF( COLEQU .AND. INFO.EQ.0 ) THEN
678: RCMIN = BIGNUM
679: RCMAX = ZERO
680: DO 20 J = 1, N
681: RCMIN = MIN( RCMIN, C( J ) )
682: RCMAX = MAX( RCMAX, C( J ) )
683: 20 CONTINUE
684: IF( RCMIN.LE.ZERO ) THEN
685: INFO = -14
686: ELSE IF( N.GT.0 ) THEN
687: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
688: ELSE
689: COLCND = ONE
690: END IF
691: END IF
692: IF( INFO.EQ.0 ) THEN
693: IF( LDB.LT.MAX( 1, N ) ) THEN
694: INFO = -15
695: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
696: INFO = -16
697: END IF
698: END IF
699: END IF
700: *
701: IF( INFO.NE.0 ) THEN
702: CALL XERBLA( 'DGBSVXX', -INFO )
703: RETURN
704: END IF
705: *
706: IF( EQUIL ) THEN
707: *
708: * Compute row and column scalings to equilibrate the matrix A.
709: *
710: CALL DGBEQUB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
711: $ AMAX, INFEQU )
712: IF( INFEQU.EQ.0 ) THEN
713: *
714: * Equilibrate the matrix.
715: *
716: CALL DLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
717: $ AMAX, EQUED )
718: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
719: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
720: END IF
721: *
722: * If the scaling factors are not applied, set them to 1.0.
723: *
724: IF ( .NOT.ROWEQU ) THEN
725: DO J = 1, N
726: R( J ) = 1.0D+0
727: END DO
728: END IF
729: IF ( .NOT.COLEQU ) THEN
730: DO J = 1, N
731: C( J ) = 1.0D+0
732: END DO
733: END IF
734: END IF
735: *
736: * Scale the right hand side.
737: *
738: IF( NOTRAN ) THEN
739: IF( ROWEQU ) CALL DLASCL2(N, NRHS, R, B, LDB)
740: ELSE
741: IF( COLEQU ) CALL DLASCL2(N, NRHS, C, B, LDB)
742: END IF
743: *
744: IF( NOFACT .OR. EQUIL ) THEN
745: *
746: * Compute the LU factorization of A.
747: *
748: DO 40, J = 1, N
749: DO 30, I = KL+1, 2*KL+KU+1
750: AFB( I, J ) = AB( I-KL, J )
751: 30 CONTINUE
752: 40 CONTINUE
753: CALL DGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
754: *
755: * Return if INFO is non-zero.
756: *
757: IF( INFO.GT.0 ) THEN
758: *
759: * Pivot in column INFO is exactly 0
760: * Compute the reciprocal pivot growth factor of the
761: * leading rank-deficient INFO columns of A.
762: *
763: RPVGRW = DLA_GBRPVGRW( N, KL, KU, INFO, AB, LDAB, AFB,
764: $ LDAFB )
765: RETURN
766: END IF
767: END IF
768: *
769: * Compute the reciprocal pivot growth factor RPVGRW.
770: *
771: RPVGRW = DLA_GBRPVGRW( N, KL, KU, N, AB, LDAB, AFB, LDAFB )
772: *
773: * Compute the solution matrix X.
774: *
775: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
776: CALL DGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
777: $ INFO )
778: *
779: * Use iterative refinement to improve the computed solution and
780: * compute error bounds and backward error estimates for it.
781: *
782: CALL DGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
783: $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR,
784: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
785: $ WORK, IWORK, INFO )
786: *
787: * Scale solutions.
788: *
789: IF ( COLEQU .AND. NOTRAN ) THEN
790: CALL DLASCL2 ( N, NRHS, C, X, LDX )
791: ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
792: CALL DLASCL2 ( N, NRHS, R, X, LDX )
793: END IF
794: *
795: RETURN
796: *
797: * End of DGBSVXX
798: *
799: END
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