Annotation of rpl/lapack/lapack/zgesvxx.f, revision 1.5
1.5 ! bertrand 1: *> \brief <b> ZGESVXX computes the solution to system of linear equations A * X = B for GE 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 ZGESVXX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgesvxx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgesvxx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgesvxx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
! 22: * EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
! 23: * BERR, N_ERR_BNDS, ERR_BNDS_NORM,
! 24: * ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
! 25: * INFO )
! 26: *
! 27: * .. Scalar Arguments ..
! 28: * CHARACTER EQUED, FACT, TRANS
! 29: * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
! 30: * $ N_ERR_BNDS
! 31: * DOUBLE PRECISION RCOND, RPVGRW
! 32: * ..
! 33: * .. Array Arguments ..
! 34: * INTEGER IPIV( * )
! 35: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), 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: *> ZGESVXX uses the LU factorization to compute the solution to a
! 49: *> complex*16 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. ZGESVXX 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: *> ZGESVXX 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: *> ZGESVXX 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 ZGESVXX 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)
! 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] NRHS
! 158: *> \verbatim
! 159: *> NRHS is INTEGER
! 160: *> The number of right hand sides, i.e., the number of columns
! 161: *> of the matrices B and X. NRHS >= 0.
! 162: *> \endverbatim
! 163: *>
! 164: *> \param[in,out] A
! 165: *> \verbatim
! 166: *> A is COMPLEX*16 array, dimension (LDA,N)
! 167: *> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
! 168: *> not 'N', then A must have been equilibrated by the scaling
! 169: *> factors in R and/or C. A is not modified if FACT = 'F' or
! 170: *> 'N', or if FACT = 'E' and EQUED = 'N' on exit.
! 171: *>
! 172: *> On exit, if EQUED .ne. 'N', A is scaled as follows:
! 173: *> EQUED = 'R': A := diag(R) * A
! 174: *> EQUED = 'C': A := A * diag(C)
! 175: *> EQUED = 'B': A := diag(R) * A * diag(C).
! 176: *> \endverbatim
! 177: *>
! 178: *> \param[in] LDA
! 179: *> \verbatim
! 180: *> LDA is INTEGER
! 181: *> The leading dimension of the array A. LDA >= max(1,N).
! 182: *> \endverbatim
! 183: *>
! 184: *> \param[in,out] AF
! 185: *> \verbatim
! 186: *> AF is or output) COMPLEX*16 array, dimension (LDAF,N)
! 187: *> If FACT = 'F', then AF is an input argument and on entry
! 188: *> contains the factors L and U from the factorization
! 189: *> A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then
! 190: *> AF is the factored form of the equilibrated matrix A.
! 191: *>
! 192: *> If FACT = 'N', then AF is an output argument and on exit
! 193: *> returns the factors L and U from the factorization A = P*L*U
! 194: *> of the original matrix A.
! 195: *>
! 196: *> If FACT = 'E', then AF is an output argument and on exit
! 197: *> returns the factors L and U from the factorization A = P*L*U
! 198: *> of the equilibrated matrix A (see the description of A for
! 199: *> the form of the equilibrated matrix).
! 200: *> \endverbatim
! 201: *>
! 202: *> \param[in] LDAF
! 203: *> \verbatim
! 204: *> LDAF is INTEGER
! 205: *> The leading dimension of the array AF. LDAF >= max(1,N).
! 206: *> \endverbatim
! 207: *>
! 208: *> \param[in,out] IPIV
! 209: *> \verbatim
! 210: *> IPIV is or output) INTEGER array, dimension (N)
! 211: *> If FACT = 'F', then IPIV is an input argument and on entry
! 212: *> contains the pivot indices from the factorization A = P*L*U
! 213: *> as computed by ZGETRF; row i of the matrix was interchanged
! 214: *> with row IPIV(i).
! 215: *>
! 216: *> If FACT = 'N', then IPIV is an output argument and on exit
! 217: *> contains the pivot indices from the factorization A = P*L*U
! 218: *> of the original matrix A.
! 219: *>
! 220: *> If FACT = 'E', then IPIV is an output argument and on exit
! 221: *> contains the pivot indices from the factorization A = P*L*U
! 222: *> of the equilibrated matrix A.
! 223: *> \endverbatim
! 224: *>
! 225: *> \param[in,out] EQUED
! 226: *> \verbatim
! 227: *> EQUED is or output) CHARACTER*1
! 228: *> Specifies the form of equilibration that was done.
! 229: *> = 'N': No equilibration (always true if FACT = 'N').
! 230: *> = 'R': Row equilibration, i.e., A has been premultiplied by
! 231: *> diag(R).
! 232: *> = 'C': Column equilibration, i.e., A has been postmultiplied
! 233: *> by diag(C).
! 234: *> = 'B': Both row and column equilibration, i.e., A has been
! 235: *> replaced by diag(R) * A * diag(C).
! 236: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
! 237: *> output argument.
! 238: *> \endverbatim
! 239: *>
! 240: *> \param[in,out] R
! 241: *> \verbatim
! 242: *> R is or output) DOUBLE PRECISION array, dimension (N)
! 243: *> The row scale factors for A. If EQUED = 'R' or 'B', A is
! 244: *> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
! 245: *> is not accessed. R is an input argument if FACT = 'F';
! 246: *> otherwise, R is an output argument. If FACT = 'F' and
! 247: *> EQUED = 'R' or 'B', each element of R must be positive.
! 248: *> If R is output, each element of R is a power of the radix.
! 249: *> If R is input, each element of R should be a power of the radix
! 250: *> to ensure a reliable solution and error estimates. Scaling by
! 251: *> powers of the radix does not cause rounding errors unless the
! 252: *> result underflows or overflows. Rounding errors during scaling
! 253: *> lead to refining with a matrix that is not equivalent to the
! 254: *> input matrix, producing error estimates that may not be
! 255: *> reliable.
! 256: *> \endverbatim
! 257: *>
! 258: *> \param[in,out] C
! 259: *> \verbatim
! 260: *> C is or output) DOUBLE PRECISION array, dimension (N)
! 261: *> The column scale factors for A. If EQUED = 'C' or 'B', A is
! 262: *> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
! 263: *> is not accessed. C is an input argument if FACT = 'F';
! 264: *> otherwise, C is an output argument. If FACT = 'F' and
! 265: *> EQUED = 'C' or 'B', each element of C must be positive.
! 266: *> If C is output, each element of C is a power of the radix.
! 267: *> If C is input, each element of C should be a power of the radix
! 268: *> to ensure a reliable solution and error estimates. Scaling by
! 269: *> powers of the radix does not cause rounding errors unless the
! 270: *> result underflows or overflows. Rounding errors during scaling
! 271: *> lead to refining with a matrix that is not equivalent to the
! 272: *> input matrix, producing error estimates that may not be
! 273: *> reliable.
! 274: *> \endverbatim
! 275: *>
! 276: *> \param[in,out] B
! 277: *> \verbatim
! 278: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
! 279: *> On entry, the N-by-NRHS right hand side matrix B.
! 280: *> On exit,
! 281: *> if EQUED = 'N', B is not modified;
! 282: *> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
! 283: *> diag(R)*B;
! 284: *> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
! 285: *> overwritten by diag(C)*B.
! 286: *> \endverbatim
! 287: *>
! 288: *> \param[in] LDB
! 289: *> \verbatim
! 290: *> LDB is INTEGER
! 291: *> The leading dimension of the array B. LDB >= max(1,N).
! 292: *> \endverbatim
! 293: *>
! 294: *> \param[out] X
! 295: *> \verbatim
! 296: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
! 297: *> If INFO = 0, the N-by-NRHS solution matrix X to the original
! 298: *> system of equations. Note that A and B are modified on exit
! 299: *> if EQUED .ne. 'N', and the solution to the equilibrated system is
! 300: *> inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
! 301: *> inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
! 302: *> \endverbatim
! 303: *>
! 304: *> \param[in] LDX
! 305: *> \verbatim
! 306: *> LDX is INTEGER
! 307: *> The leading dimension of the array X. LDX >= max(1,N).
! 308: *> \endverbatim
! 309: *>
! 310: *> \param[out] RCOND
! 311: *> \verbatim
! 312: *> RCOND is DOUBLE PRECISION
! 313: *> Reciprocal scaled condition number. This is an estimate of the
! 314: *> reciprocal Skeel condition number of the matrix A after
! 315: *> equilibration (if done). If this is less than the machine
! 316: *> precision (in particular, if it is zero), the matrix is singular
! 317: *> to working precision. Note that the error may still be small even
! 318: *> if this number is very small and the matrix appears ill-
! 319: *> conditioned.
! 320: *> \endverbatim
! 321: *>
! 322: *> \param[out] RPVGRW
! 323: *> \verbatim
! 324: *> RPVGRW is DOUBLE PRECISION
! 325: *> Reciprocal pivot growth. On exit, this contains the reciprocal
! 326: *> pivot growth factor norm(A)/norm(U). The "max absolute element"
! 327: *> norm is used. If this is much less than 1, then the stability of
! 328: *> the LU factorization of the (equilibrated) matrix A could be poor.
! 329: *> This also means that the solution X, estimated condition numbers,
! 330: *> and error bounds could be unreliable. If factorization fails with
! 331: *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
! 332: *> for the leading INFO columns of A. In ZGESVX, this quantity is
! 333: *> returned in WORK(1).
! 334: *> \endverbatim
! 335: *>
! 336: *> \param[out] BERR
! 337: *> \verbatim
! 338: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 339: *> Componentwise relative backward error. This is the
! 340: *> componentwise relative backward error of each solution vector X(j)
! 341: *> (i.e., the smallest relative change in any element of A or B that
! 342: *> makes X(j) an exact solution).
! 343: *> \endverbatim
! 344: *>
! 345: *> \param[in] N_ERR_BNDS
! 346: *> \verbatim
! 347: *> N_ERR_BNDS is INTEGER
! 348: *> Number of error bounds to return for each right hand side
! 349: *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
! 350: *> ERR_BNDS_COMP below.
! 351: *> \endverbatim
! 352: *>
! 353: *> \param[out] ERR_BNDS_NORM
! 354: *> \verbatim
! 355: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
! 356: *> For each right-hand side, this array contains information about
! 357: *> various error bounds and condition numbers corresponding to the
! 358: *> normwise relative error, which is defined as follows:
! 359: *>
! 360: *> Normwise relative error in the ith solution vector:
! 361: *> max_j (abs(XTRUE(j,i) - X(j,i)))
! 362: *> ------------------------------
! 363: *> max_j abs(X(j,i))
! 364: *>
! 365: *> The array is indexed by the type of error information as described
! 366: *> below. There currently are up to three pieces of information
! 367: *> returned.
! 368: *>
! 369: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
! 370: *> right-hand side.
! 371: *>
! 372: *> The second index in ERR_BNDS_NORM(:,err) contains the following
! 373: *> three fields:
! 374: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 375: *> reciprocal condition number is less than the threshold
! 376: *> sqrt(n) * dlamch('Epsilon').
! 377: *>
! 378: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 379: *> almost certainly within a factor of 10 of the true error
! 380: *> so long as the next entry is greater than the threshold
! 381: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
! 382: *> be trusted if the previous boolean is true.
! 383: *>
! 384: *> err = 3 Reciprocal condition number: Estimated normwise
! 385: *> reciprocal condition number. Compared with the threshold
! 386: *> sqrt(n) * dlamch('Epsilon') to determine if the error
! 387: *> estimate is "guaranteed". These reciprocal condition
! 388: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 389: *> appropriately scaled matrix Z.
! 390: *> Let Z = S*A, where S scales each row by a power of the
! 391: *> radix so all absolute row sums of Z are approximately 1.
! 392: *>
! 393: *> See Lapack Working Note 165 for further details and extra
! 394: *> cautions.
! 395: *> \endverbatim
! 396: *>
! 397: *> \param[out] ERR_BNDS_COMP
! 398: *> \verbatim
! 399: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
! 400: *> For each right-hand side, this array contains information about
! 401: *> various error bounds and condition numbers corresponding to the
! 402: *> componentwise relative error, which is defined as follows:
! 403: *>
! 404: *> Componentwise relative error in the ith solution vector:
! 405: *> abs(XTRUE(j,i) - X(j,i))
! 406: *> max_j ----------------------
! 407: *> abs(X(j,i))
! 408: *>
! 409: *> The array is indexed by the right-hand side i (on which the
! 410: *> componentwise relative error depends), and the type of error
! 411: *> information as described below. There currently are up to three
! 412: *> pieces of information returned for each right-hand side. If
! 413: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
! 414: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
! 415: *> the first (:,N_ERR_BNDS) entries are returned.
! 416: *>
! 417: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
! 418: *> right-hand side.
! 419: *>
! 420: *> The second index in ERR_BNDS_COMP(:,err) contains the following
! 421: *> three fields:
! 422: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 423: *> reciprocal condition number is less than the threshold
! 424: *> sqrt(n) * dlamch('Epsilon').
! 425: *>
! 426: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 427: *> almost certainly within a factor of 10 of the true error
! 428: *> so long as the next entry is greater than the threshold
! 429: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
! 430: *> be trusted if the previous boolean is true.
! 431: *>
! 432: *> err = 3 Reciprocal condition number: Estimated componentwise
! 433: *> reciprocal condition number. Compared with the threshold
! 434: *> sqrt(n) * dlamch('Epsilon') to determine if the error
! 435: *> estimate is "guaranteed". These reciprocal condition
! 436: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 437: *> appropriately scaled matrix Z.
! 438: *> Let Z = S*(A*diag(x)), where x is the solution for the
! 439: *> current right-hand side and S scales each row of
! 440: *> A*diag(x) by a power of the radix so all absolute row
! 441: *> sums of Z are approximately 1.
! 442: *>
! 443: *> See Lapack Working Note 165 for further details and extra
! 444: *> cautions.
! 445: *> \endverbatim
! 446: *>
! 447: *> \param[in] NPARAMS
! 448: *> \verbatim
! 449: *> NPARAMS is INTEGER
! 450: *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
! 451: *> PARAMS array is never referenced and default values are used.
! 452: *> \endverbatim
! 453: *>
! 454: *> \param[in,out] PARAMS
! 455: *> \verbatim
! 456: *> PARAMS is / output) DOUBLE PRECISION array, dimension NPARAMS
! 457: *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
! 458: *> that entry will be filled with default value used for that
! 459: *> parameter. Only positions up to NPARAMS are accessed; defaults
! 460: *> are used for higher-numbered parameters.
! 461: *>
! 462: *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
! 463: *> refinement or not.
! 464: *> Default: 1.0D+0
! 465: *> = 0.0 : No refinement is performed, and no error bounds are
! 466: *> computed.
! 467: *> = 1.0 : Use the extra-precise refinement algorithm.
! 468: *> (other values are reserved for future use)
! 469: *>
! 470: *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
! 471: *> computations allowed for refinement.
! 472: *> Default: 10
! 473: *> Aggressive: Set to 100 to permit convergence using approximate
! 474: *> factorizations or factorizations other than LU. If
! 475: *> the factorization uses a technique other than
! 476: *> Gaussian elimination, the guarantees in
! 477: *> err_bnds_norm and err_bnds_comp may no longer be
! 478: *> trustworthy.
! 479: *>
! 480: *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
! 481: *> will attempt to find a solution with small componentwise
! 482: *> relative error in the double-precision algorithm. Positive
! 483: *> is true, 0.0 is false.
! 484: *> Default: 1.0 (attempt componentwise convergence)
! 485: *> \endverbatim
! 486: *>
! 487: *> \param[out] WORK
! 488: *> \verbatim
! 489: *> WORK is COMPLEX*16 array, dimension (2*N)
! 490: *> \endverbatim
! 491: *>
! 492: *> \param[out] RWORK
! 493: *> \verbatim
! 494: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
! 495: *> \endverbatim
! 496: *>
! 497: *> \param[out] INFO
! 498: *> \verbatim
! 499: *> INFO is INTEGER
! 500: *> = 0: Successful exit. The solution to every right-hand side is
! 501: *> guaranteed.
! 502: *> < 0: If INFO = -i, the i-th argument had an illegal value
! 503: *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
! 504: *> has been completed, but the factor U is exactly singular, so
! 505: *> the solution and error bounds could not be computed. RCOND = 0
! 506: *> is returned.
! 507: *> = N+J: The solution corresponding to the Jth right-hand side is
! 508: *> not guaranteed. The solutions corresponding to other right-
! 509: *> hand sides K with K > J may not be guaranteed as well, but
! 510: *> only the first such right-hand side is reported. If a small
! 511: *> componentwise error is not requested (PARAMS(3) = 0.0) then
! 512: *> the Jth right-hand side is the first with a normwise error
! 513: *> bound that is not guaranteed (the smallest J such
! 514: *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
! 515: *> the Jth right-hand side is the first with either a normwise or
! 516: *> componentwise error bound that is not guaranteed (the smallest
! 517: *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
! 518: *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
! 519: *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
! 520: *> about all of the right-hand sides check ERR_BNDS_NORM or
! 521: *> ERR_BNDS_COMP.
! 522: *> \endverbatim
! 523: *
! 524: * Authors:
! 525: * ========
! 526: *
! 527: *> \author Univ. of Tennessee
! 528: *> \author Univ. of California Berkeley
! 529: *> \author Univ. of Colorado Denver
! 530: *> \author NAG Ltd.
! 531: *
! 532: *> \date November 2011
! 533: *
! 534: *> \ingroup complex16GEsolve
! 535: *
! 536: * =====================================================================
1.1 bertrand 537: SUBROUTINE ZGESVXX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
538: $ EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
539: $ BERR, N_ERR_BNDS, ERR_BNDS_NORM,
540: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
541: $ INFO )
542: *
1.5 ! bertrand 543: * -- LAPACK driver routine (version 3.4.0) --
! 544: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 545: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 546: * November 2011
1.1 bertrand 547: *
548: * .. Scalar Arguments ..
549: CHARACTER EQUED, FACT, TRANS
550: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
551: $ N_ERR_BNDS
552: DOUBLE PRECISION RCOND, RPVGRW
553: * ..
554: * .. Array Arguments ..
555: INTEGER IPIV( * )
556: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
557: $ X( LDX , * ),WORK( * )
558: DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ),
559: $ ERR_BNDS_NORM( NRHS, * ),
560: $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
561: * ..
562: *
1.5 ! bertrand 563: * ==================================================================
1.1 bertrand 564: *
565: * .. Parameters ..
566: DOUBLE PRECISION ZERO, ONE
567: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
568: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
569: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
570: INTEGER CMP_ERR_I, PIV_GROWTH_I
571: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
572: $ BERR_I = 3 )
573: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
574: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
575: $ PIV_GROWTH_I = 9 )
576: * ..
577: * .. Local Scalars ..
578: LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
579: INTEGER INFEQU, J
580: DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN,
581: $ ROWCND, SMLNUM
582: * ..
583: * .. External Functions ..
584: EXTERNAL LSAME, DLAMCH, ZLA_RPVGRW
585: LOGICAL LSAME
586: DOUBLE PRECISION DLAMCH, ZLA_RPVGRW
587: * ..
588: * .. External Subroutines ..
589: EXTERNAL ZGEEQUB, ZGETRF, ZGETRS, ZLACPY, ZLAQGE,
590: $ XERBLA, ZLASCL2, ZGERFSX
591: * ..
592: * .. Intrinsic Functions ..
593: INTRINSIC MAX, MIN
594: * ..
595: * .. Executable Statements ..
596: *
597: INFO = 0
598: NOFACT = LSAME( FACT, 'N' )
599: EQUIL = LSAME( FACT, 'E' )
600: NOTRAN = LSAME( TRANS, 'N' )
601: SMLNUM = DLAMCH( 'Safe minimum' )
602: BIGNUM = ONE / SMLNUM
603: IF( NOFACT .OR. EQUIL ) THEN
604: EQUED = 'N'
605: ROWEQU = .FALSE.
606: COLEQU = .FALSE.
607: ELSE
608: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
609: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
610: END IF
611: *
612: * Default is failure. If an input parameter is wrong or
613: * factorization fails, make everything look horrible. Only the
614: * pivot growth is set here, the rest is initialized in ZGERFSX.
615: *
616: RPVGRW = ZERO
617: *
618: * Test the input parameters. PARAMS is not tested until ZGERFSX.
619: *
620: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
621: $ LSAME( FACT, 'F' ) ) THEN
622: INFO = -1
623: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
624: $ LSAME( TRANS, 'C' ) ) THEN
625: INFO = -2
626: ELSE IF( N.LT.0 ) THEN
627: INFO = -3
628: ELSE IF( NRHS.LT.0 ) THEN
629: INFO = -4
630: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
631: INFO = -6
632: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
633: INFO = -8
634: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
635: $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
636: INFO = -10
637: ELSE
638: IF( ROWEQU ) THEN
639: RCMIN = BIGNUM
640: RCMAX = ZERO
641: DO 10 J = 1, N
642: RCMIN = MIN( RCMIN, R( J ) )
643: RCMAX = MAX( RCMAX, R( J ) )
644: 10 CONTINUE
645: IF( RCMIN.LE.ZERO ) THEN
646: INFO = -11
647: ELSE IF( N.GT.0 ) THEN
648: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
649: ELSE
650: ROWCND = ONE
651: END IF
652: END IF
653: IF( COLEQU .AND. INFO.EQ.0 ) THEN
654: RCMIN = BIGNUM
655: RCMAX = ZERO
656: DO 20 J = 1, N
657: RCMIN = MIN( RCMIN, C( J ) )
658: RCMAX = MAX( RCMAX, C( J ) )
659: 20 CONTINUE
660: IF( RCMIN.LE.ZERO ) THEN
661: INFO = -12
662: ELSE IF( N.GT.0 ) THEN
663: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
664: ELSE
665: COLCND = ONE
666: END IF
667: END IF
668: IF( INFO.EQ.0 ) THEN
669: IF( LDB.LT.MAX( 1, N ) ) THEN
670: INFO = -14
671: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
672: INFO = -16
673: END IF
674: END IF
675: END IF
676: *
677: IF( INFO.NE.0 ) THEN
678: CALL XERBLA( 'ZGESVXX', -INFO )
679: RETURN
680: END IF
681: *
682: IF( EQUIL ) THEN
683: *
684: * Compute row and column scalings to equilibrate the matrix A.
685: *
686: CALL ZGEEQUB( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
687: $ INFEQU )
688: IF( INFEQU.EQ.0 ) THEN
689: *
690: * Equilibrate the matrix.
691: *
692: CALL ZLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
693: $ EQUED )
694: ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
695: COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
696: END IF
697: *
698: * If the scaling factors are not applied, set them to 1.0.
699: *
700: IF ( .NOT.ROWEQU ) THEN
701: DO J = 1, N
702: R( J ) = 1.0D+0
703: END DO
704: END IF
705: IF ( .NOT.COLEQU ) THEN
706: DO J = 1, N
707: C( J ) = 1.0D+0
708: END DO
709: END IF
710: END IF
711: *
712: * Scale the right-hand side.
713: *
714: IF( NOTRAN ) THEN
715: IF( ROWEQU ) CALL ZLASCL2( N, NRHS, R, B, LDB )
716: ELSE
717: IF( COLEQU ) CALL ZLASCL2( N, NRHS, C, B, LDB )
718: END IF
719: *
720: IF( NOFACT .OR. EQUIL ) THEN
721: *
722: * Compute the LU factorization of A.
723: *
724: CALL ZLACPY( 'Full', N, N, A, LDA, AF, LDAF )
725: CALL ZGETRF( N, N, AF, LDAF, IPIV, INFO )
726: *
727: * Return if INFO is non-zero.
728: *
729: IF( INFO.GT.0 ) THEN
730: *
731: * Pivot in column INFO is exactly 0
732: * Compute the reciprocal pivot growth factor of the
733: * leading rank-deficient INFO columns of A.
734: *
735: RPVGRW = ZLA_RPVGRW( N, INFO, A, LDA, AF, LDAF )
736: RETURN
737: END IF
738: END IF
739: *
740: * Compute the reciprocal pivot growth factor RPVGRW.
741: *
742: RPVGRW = ZLA_RPVGRW( N, N, A, LDA, AF, LDAF )
743: *
744: * Compute the solution matrix X.
745: *
746: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
747: CALL ZGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
748: *
749: * Use iterative refinement to improve the computed solution and
750: * compute error bounds and backward error estimates for it.
751: *
752: CALL ZGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF,
753: $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR,
754: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
755: $ WORK, RWORK, INFO )
756: *
757: * Scale solutions.
758: *
759: IF ( COLEQU .AND. NOTRAN ) THEN
760: CALL ZLASCL2 ( N, NRHS, C, X, LDX )
761: ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN
762: CALL ZLASCL2 ( N, NRHS, R, X, LDX )
763: END IF
764: *
765: RETURN
766: *
767: * End of ZGESVXX
768: *
769: END
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