Annotation of rpl/lapack/lapack/zposvxx.f, revision 1.5
1.5 ! bertrand 1: *> \brief <b> ZPOSVXX computes the solution to system of linear equations A * X = B for PO 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 ZPOSVXX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zposvxx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zposvxx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zposvxx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
! 22: * S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
! 23: * N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
! 24: * NPARAMS, PARAMS, WORK, RWORK, INFO )
! 25: *
! 26: * .. Scalar Arguments ..
! 27: * CHARACTER EQUED, FACT, UPLO
! 28: * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
! 29: * $ N_ERR_BNDS
! 30: * DOUBLE PRECISION RCOND, RPVGRW
! 31: * ..
! 32: * .. Array Arguments ..
! 33: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
! 34: * $ WORK( * ), X( LDX, * )
! 35: * DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
! 36: * $ ERR_BNDS_NORM( NRHS, * ),
! 37: * $ ERR_BNDS_COMP( NRHS, * )
! 38: * ..
! 39: *
! 40: *
! 41: *> \par Purpose:
! 42: * =============
! 43: *>
! 44: *> \verbatim
! 45: *>
! 46: *> ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
! 47: *> to compute the solution to a complex*16 system of linear equations
! 48: *> A * X = B, where A is an N-by-N symmetric positive definite matrix
! 49: *> and X and B are N-by-NRHS matrices.
! 50: *>
! 51: *> If requested, both normwise and maximum componentwise error bounds
! 52: *> are returned. ZPOSVXX will return a solution with a tiny
! 53: *> guaranteed error (O(eps) where eps is the working machine
! 54: *> precision) unless the matrix is very ill-conditioned, in which
! 55: *> case a warning is returned. Relevant condition numbers also are
! 56: *> calculated and returned.
! 57: *>
! 58: *> ZPOSVXX accepts user-provided factorizations and equilibration
! 59: *> factors; see the definitions of the FACT and EQUED options.
! 60: *> Solving with refinement and using a factorization from a previous
! 61: *> ZPOSVXX call will also produce a solution with either O(eps)
! 62: *> errors or warnings, but we cannot make that claim for general
! 63: *> user-provided factorizations and equilibration factors if they
! 64: *> differ from what ZPOSVXX would itself produce.
! 65: *> \endverbatim
! 66: *
! 67: *> \par Description:
! 68: * =================
! 69: *>
! 70: *> \verbatim
! 71: *>
! 72: *> The following steps are performed:
! 73: *>
! 74: *> 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
! 75: *> the system:
! 76: *>
! 77: *> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
! 78: *>
! 79: *> Whether or not the system will be equilibrated depends on the
! 80: *> scaling of the matrix A, but if equilibration is used, A is
! 81: *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
! 82: *>
! 83: *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
! 84: *> factor the matrix A (after equilibration if FACT = 'E') as
! 85: *> A = U**T* U, if UPLO = 'U', or
! 86: *> A = L * L**T, if UPLO = 'L',
! 87: *> where U is an upper triangular matrix and L is a lower triangular
! 88: *> matrix.
! 89: *>
! 90: *> 3. If the leading i-by-i principal minor is not positive definite,
! 91: *> then the routine returns with INFO = i. Otherwise, the factored
! 92: *> form of A is used to estimate the condition number of the matrix
! 93: *> A (see argument RCOND). If the reciprocal of the condition number
! 94: *> is less than machine precision, the routine still goes on to solve
! 95: *> for X and compute error bounds as described below.
! 96: *>
! 97: *> 4. The system of equations is solved for X using the factored form
! 98: *> of A.
! 99: *>
! 100: *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
! 101: *> the routine will use iterative refinement to try to get a small
! 102: *> error and error bounds. Refinement calculates the residual to at
! 103: *> least twice the working precision.
! 104: *>
! 105: *> 6. If equilibration was used, the matrix X is premultiplied by
! 106: *> diag(S) so that it solves the original system before
! 107: *> equilibration.
! 108: *> \endverbatim
! 109: *
! 110: * Arguments:
! 111: * ==========
! 112: *
! 113: *> \verbatim
! 114: *> Some optional parameters are bundled in the PARAMS array. These
! 115: *> settings determine how refinement is performed, but often the
! 116: *> defaults are acceptable. If the defaults are acceptable, users
! 117: *> can pass NPARAMS = 0 which prevents the source code from accessing
! 118: *> the PARAMS argument.
! 119: *> \endverbatim
! 120: *>
! 121: *> \param[in] FACT
! 122: *> \verbatim
! 123: *> FACT is CHARACTER*1
! 124: *> Specifies whether or not the factored form of the matrix A is
! 125: *> supplied on entry, and if not, whether the matrix A should be
! 126: *> equilibrated before it is factored.
! 127: *> = 'F': On entry, AF contains the factored form of A.
! 128: *> If EQUED is not 'N', the matrix A has been
! 129: *> equilibrated with scaling factors given by S.
! 130: *> A and AF are not modified.
! 131: *> = 'N': The matrix A will be copied to AF and factored.
! 132: *> = 'E': The matrix A will be equilibrated if necessary, then
! 133: *> copied to AF and factored.
! 134: *> \endverbatim
! 135: *>
! 136: *> \param[in] UPLO
! 137: *> \verbatim
! 138: *> UPLO is CHARACTER*1
! 139: *> = 'U': Upper triangle of A is stored;
! 140: *> = 'L': Lower triangle of A is stored.
! 141: *> \endverbatim
! 142: *>
! 143: *> \param[in] N
! 144: *> \verbatim
! 145: *> N is INTEGER
! 146: *> The number of linear equations, i.e., the order of the
! 147: *> matrix A. N >= 0.
! 148: *> \endverbatim
! 149: *>
! 150: *> \param[in] NRHS
! 151: *> \verbatim
! 152: *> NRHS is INTEGER
! 153: *> The number of right hand sides, i.e., the number of columns
! 154: *> of the matrices B and X. NRHS >= 0.
! 155: *> \endverbatim
! 156: *>
! 157: *> \param[in,out] A
! 158: *> \verbatim
! 159: *> A is COMPLEX*16 array, dimension (LDA,N)
! 160: *> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
! 161: *> 'Y', then A must contain the equilibrated matrix
! 162: *> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
! 163: *> triangular part of A contains the upper triangular part of the
! 164: *> matrix A, and the strictly lower triangular part of A is not
! 165: *> referenced. If UPLO = 'L', the leading N-by-N lower triangular
! 166: *> part of A contains the lower triangular part of the matrix A, and
! 167: *> the strictly upper triangular part of A is not referenced. A is
! 168: *> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
! 169: *> 'N' on exit.
! 170: *>
! 171: *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
! 172: *> diag(S)*A*diag(S).
! 173: *> \endverbatim
! 174: *>
! 175: *> \param[in] LDA
! 176: *> \verbatim
! 177: *> LDA is INTEGER
! 178: *> The leading dimension of the array A. LDA >= max(1,N).
! 179: *> \endverbatim
! 180: *>
! 181: *> \param[in,out] AF
! 182: *> \verbatim
! 183: *> AF is or output) COMPLEX*16 array, dimension (LDAF,N)
! 184: *> If FACT = 'F', then AF is an input argument and on entry
! 185: *> contains the triangular factor U or L from the Cholesky
! 186: *> factorization A = U**T*U or A = L*L**T, in the same storage
! 187: *> format as A. If EQUED .ne. 'N', then AF is the factored
! 188: *> form of the equilibrated matrix diag(S)*A*diag(S).
! 189: *>
! 190: *> If FACT = 'N', then AF is an output argument and on exit
! 191: *> returns the triangular factor U or L from the Cholesky
! 192: *> factorization A = U**T*U or A = L*L**T of the original
! 193: *> matrix A.
! 194: *>
! 195: *> If FACT = 'E', then AF is an output argument and on exit
! 196: *> returns the triangular factor U or L from the Cholesky
! 197: *> factorization A = U**T*U or A = L*L**T of the equilibrated
! 198: *> matrix A (see the description of A for the form of the
! 199: *> 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] EQUED
! 209: *> \verbatim
! 210: *> EQUED is or output) CHARACTER*1
! 211: *> Specifies the form of equilibration that was done.
! 212: *> = 'N': No equilibration (always true if FACT = 'N').
! 213: *> = 'Y': Both row and column equilibration, i.e., A has been
! 214: *> replaced by diag(S) * A * diag(S).
! 215: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
! 216: *> output argument.
! 217: *> \endverbatim
! 218: *>
! 219: *> \param[in,out] S
! 220: *> \verbatim
! 221: *> S is or output) DOUBLE PRECISION array, dimension (N)
! 222: *> The row scale factors for A. If EQUED = 'Y', A is multiplied on
! 223: *> the left and right by diag(S). S is an input argument if FACT =
! 224: *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
! 225: *> = 'Y', each element of S must be positive. If S is output, each
! 226: *> element of S is a power of the radix. If S is input, each element
! 227: *> of S should be a power of the radix to ensure a reliable solution
! 228: *> and error estimates. Scaling by powers of the radix does not cause
! 229: *> rounding errors unless the result underflows or overflows.
! 230: *> Rounding errors during scaling lead to refining with a matrix that
! 231: *> is not equivalent to the input matrix, producing error estimates
! 232: *> that may not be reliable.
! 233: *> \endverbatim
! 234: *>
! 235: *> \param[in,out] B
! 236: *> \verbatim
! 237: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
! 238: *> On entry, the N-by-NRHS right hand side matrix B.
! 239: *> On exit,
! 240: *> if EQUED = 'N', B is not modified;
! 241: *> if EQUED = 'Y', B is overwritten by diag(S)*B;
! 242: *> \endverbatim
! 243: *>
! 244: *> \param[in] LDB
! 245: *> \verbatim
! 246: *> LDB is INTEGER
! 247: *> The leading dimension of the array B. LDB >= max(1,N).
! 248: *> \endverbatim
! 249: *>
! 250: *> \param[out] X
! 251: *> \verbatim
! 252: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
! 253: *> If INFO = 0, the N-by-NRHS solution matrix X to the original
! 254: *> system of equations. Note that A and B are modified on exit if
! 255: *> EQUED .ne. 'N', and the solution to the equilibrated system is
! 256: *> inv(diag(S))*X.
! 257: *> \endverbatim
! 258: *>
! 259: *> \param[in] LDX
! 260: *> \verbatim
! 261: *> LDX is INTEGER
! 262: *> The leading dimension of the array X. LDX >= max(1,N).
! 263: *> \endverbatim
! 264: *>
! 265: *> \param[out] RCOND
! 266: *> \verbatim
! 267: *> RCOND is DOUBLE PRECISION
! 268: *> Reciprocal scaled condition number. This is an estimate of the
! 269: *> reciprocal Skeel condition number of the matrix A after
! 270: *> equilibration (if done). If this is less than the machine
! 271: *> precision (in particular, if it is zero), the matrix is singular
! 272: *> to working precision. Note that the error may still be small even
! 273: *> if this number is very small and the matrix appears ill-
! 274: *> conditioned.
! 275: *> \endverbatim
! 276: *>
! 277: *> \param[out] RPVGRW
! 278: *> \verbatim
! 279: *> RPVGRW is DOUBLE PRECISION
! 280: *> Reciprocal pivot growth. On exit, this contains the reciprocal
! 281: *> pivot growth factor norm(A)/norm(U). The "max absolute element"
! 282: *> norm is used. If this is much less than 1, then the stability of
! 283: *> the LU factorization of the (equilibrated) matrix A could be poor.
! 284: *> This also means that the solution X, estimated condition numbers,
! 285: *> and error bounds could be unreliable. If factorization fails with
! 286: *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
! 287: *> for the leading INFO columns of A.
! 288: *> \endverbatim
! 289: *>
! 290: *> \param[out] BERR
! 291: *> \verbatim
! 292: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 293: *> Componentwise relative backward error. This is the
! 294: *> componentwise relative backward error of each solution vector X(j)
! 295: *> (i.e., the smallest relative change in any element of A or B that
! 296: *> makes X(j) an exact solution).
! 297: *> \endverbatim
! 298: *>
! 299: *> \param[in] N_ERR_BNDS
! 300: *> \verbatim
! 301: *> N_ERR_BNDS is INTEGER
! 302: *> Number of error bounds to return for each right hand side
! 303: *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
! 304: *> ERR_BNDS_COMP below.
! 305: *> \endverbatim
! 306: *>
! 307: *> \param[out] ERR_BNDS_NORM
! 308: *> \verbatim
! 309: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
! 310: *> For each right-hand side, this array contains information about
! 311: *> various error bounds and condition numbers corresponding to the
! 312: *> normwise relative error, which is defined as follows:
! 313: *>
! 314: *> Normwise relative error in the ith solution vector:
! 315: *> max_j (abs(XTRUE(j,i) - X(j,i)))
! 316: *> ------------------------------
! 317: *> max_j abs(X(j,i))
! 318: *>
! 319: *> The array is indexed by the type of error information as described
! 320: *> below. There currently are up to three pieces of information
! 321: *> returned.
! 322: *>
! 323: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
! 324: *> right-hand side.
! 325: *>
! 326: *> The second index in ERR_BNDS_NORM(:,err) contains the following
! 327: *> three fields:
! 328: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 329: *> reciprocal condition number is less than the threshold
! 330: *> sqrt(n) * dlamch('Epsilon').
! 331: *>
! 332: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 333: *> almost certainly within a factor of 10 of the true error
! 334: *> so long as the next entry is greater than the threshold
! 335: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
! 336: *> be trusted if the previous boolean is true.
! 337: *>
! 338: *> err = 3 Reciprocal condition number: Estimated normwise
! 339: *> reciprocal condition number. Compared with the threshold
! 340: *> sqrt(n) * dlamch('Epsilon') to determine if the error
! 341: *> estimate is "guaranteed". These reciprocal condition
! 342: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 343: *> appropriately scaled matrix Z.
! 344: *> Let Z = S*A, where S scales each row by a power of the
! 345: *> radix so all absolute row sums of Z are approximately 1.
! 346: *>
! 347: *> See Lapack Working Note 165 for further details and extra
! 348: *> cautions.
! 349: *> \endverbatim
! 350: *>
! 351: *> \param[out] ERR_BNDS_COMP
! 352: *> \verbatim
! 353: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
! 354: *> For each right-hand side, this array contains information about
! 355: *> various error bounds and condition numbers corresponding to the
! 356: *> componentwise relative error, which is defined as follows:
! 357: *>
! 358: *> Componentwise relative error in the ith solution vector:
! 359: *> abs(XTRUE(j,i) - X(j,i))
! 360: *> max_j ----------------------
! 361: *> abs(X(j,i))
! 362: *>
! 363: *> The array is indexed by the right-hand side i (on which the
! 364: *> componentwise relative error depends), and the type of error
! 365: *> information as described below. There currently are up to three
! 366: *> pieces of information returned for each right-hand side. If
! 367: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
! 368: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
! 369: *> the first (:,N_ERR_BNDS) entries are returned.
! 370: *>
! 371: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
! 372: *> right-hand side.
! 373: *>
! 374: *> The second index in ERR_BNDS_COMP(:,err) contains the following
! 375: *> three fields:
! 376: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 377: *> reciprocal condition number is less than the threshold
! 378: *> sqrt(n) * dlamch('Epsilon').
! 379: *>
! 380: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 381: *> almost certainly within a factor of 10 of the true error
! 382: *> so long as the next entry is greater than the threshold
! 383: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
! 384: *> be trusted if the previous boolean is true.
! 385: *>
! 386: *> err = 3 Reciprocal condition number: Estimated componentwise
! 387: *> reciprocal condition number. Compared with the threshold
! 388: *> sqrt(n) * dlamch('Epsilon') to determine if the error
! 389: *> estimate is "guaranteed". These reciprocal condition
! 390: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 391: *> appropriately scaled matrix Z.
! 392: *> Let Z = S*(A*diag(x)), where x is the solution for the
! 393: *> current right-hand side and S scales each row of
! 394: *> A*diag(x) by a power of the radix so all absolute row
! 395: *> sums of Z are approximately 1.
! 396: *>
! 397: *> See Lapack Working Note 165 for further details and extra
! 398: *> cautions.
! 399: *> \endverbatim
! 400: *>
! 401: *> \param[in] NPARAMS
! 402: *> \verbatim
! 403: *> NPARAMS is INTEGER
! 404: *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
! 405: *> PARAMS array is never referenced and default values are used.
! 406: *> \endverbatim
! 407: *>
! 408: *> \param[in,out] PARAMS
! 409: *> \verbatim
! 410: *> PARAMS is / output) DOUBLE PRECISION array, dimension NPARAMS
! 411: *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
! 412: *> that entry will be filled with default value used for that
! 413: *> parameter. Only positions up to NPARAMS are accessed; defaults
! 414: *> are used for higher-numbered parameters.
! 415: *>
! 416: *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
! 417: *> refinement or not.
! 418: *> Default: 1.0D+0
! 419: *> = 0.0 : No refinement is performed, and no error bounds are
! 420: *> computed.
! 421: *> = 1.0 : Use the extra-precise refinement algorithm.
! 422: *> (other values are reserved for future use)
! 423: *>
! 424: *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
! 425: *> computations allowed for refinement.
! 426: *> Default: 10
! 427: *> Aggressive: Set to 100 to permit convergence using approximate
! 428: *> factorizations or factorizations other than LU. If
! 429: *> the factorization uses a technique other than
! 430: *> Gaussian elimination, the guarantees in
! 431: *> err_bnds_norm and err_bnds_comp may no longer be
! 432: *> trustworthy.
! 433: *>
! 434: *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
! 435: *> will attempt to find a solution with small componentwise
! 436: *> relative error in the double-precision algorithm. Positive
! 437: *> is true, 0.0 is false.
! 438: *> Default: 1.0 (attempt componentwise convergence)
! 439: *> \endverbatim
! 440: *>
! 441: *> \param[out] WORK
! 442: *> \verbatim
! 443: *> WORK is COMPLEX*16 array, dimension (2*N)
! 444: *> \endverbatim
! 445: *>
! 446: *> \param[out] RWORK
! 447: *> \verbatim
! 448: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
! 449: *> \endverbatim
! 450: *>
! 451: *> \param[out] INFO
! 452: *> \verbatim
! 453: *> INFO is INTEGER
! 454: *> = 0: Successful exit. The solution to every right-hand side is
! 455: *> guaranteed.
! 456: *> < 0: If INFO = -i, the i-th argument had an illegal value
! 457: *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
! 458: *> has been completed, but the factor U is exactly singular, so
! 459: *> the solution and error bounds could not be computed. RCOND = 0
! 460: *> is returned.
! 461: *> = N+J: The solution corresponding to the Jth right-hand side is
! 462: *> not guaranteed. The solutions corresponding to other right-
! 463: *> hand sides K with K > J may not be guaranteed as well, but
! 464: *> only the first such right-hand side is reported. If a small
! 465: *> componentwise error is not requested (PARAMS(3) = 0.0) then
! 466: *> the Jth right-hand side is the first with a normwise error
! 467: *> bound that is not guaranteed (the smallest J such
! 468: *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
! 469: *> the Jth right-hand side is the first with either a normwise or
! 470: *> componentwise error bound that is not guaranteed (the smallest
! 471: *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
! 472: *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
! 473: *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
! 474: *> about all of the right-hand sides check ERR_BNDS_NORM or
! 475: *> ERR_BNDS_COMP.
! 476: *> \endverbatim
! 477: *
! 478: * Authors:
! 479: * ========
! 480: *
! 481: *> \author Univ. of Tennessee
! 482: *> \author Univ. of California Berkeley
! 483: *> \author Univ. of Colorado Denver
! 484: *> \author NAG Ltd.
! 485: *
! 486: *> \date November 2011
! 487: *
! 488: *> \ingroup complex16POsolve
! 489: *
! 490: * =====================================================================
1.1 bertrand 491: SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
492: $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
493: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
494: $ NPARAMS, PARAMS, WORK, RWORK, INFO )
495: *
1.5 ! bertrand 496: * -- LAPACK driver routine (version 3.4.0) --
! 497: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 498: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 499: * November 2011
1.1 bertrand 500: *
501: * .. Scalar Arguments ..
502: CHARACTER EQUED, FACT, UPLO
503: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
504: $ N_ERR_BNDS
505: DOUBLE PRECISION RCOND, RPVGRW
506: * ..
507: * .. Array Arguments ..
508: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
509: $ WORK( * ), X( LDX, * )
510: DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
511: $ ERR_BNDS_NORM( NRHS, * ),
512: $ ERR_BNDS_COMP( NRHS, * )
513: * ..
514: *
1.5 ! bertrand 515: * ==================================================================
1.1 bertrand 516: *
517: * .. Parameters ..
518: DOUBLE PRECISION ZERO, ONE
519: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
520: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
521: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
522: INTEGER CMP_ERR_I, PIV_GROWTH_I
523: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
524: $ BERR_I = 3 )
525: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
526: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
527: $ PIV_GROWTH_I = 9 )
528: * ..
529: * .. Local Scalars ..
530: LOGICAL EQUIL, NOFACT, RCEQU
531: INTEGER INFEQU, J
532: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
533: * ..
534: * .. External Functions ..
535: EXTERNAL LSAME, DLAMCH, ZLA_PORPVGRW
536: LOGICAL LSAME
537: DOUBLE PRECISION DLAMCH, ZLA_PORPVGRW
538: * ..
539: * .. External Subroutines ..
540: EXTERNAL ZPOCON, ZPOEQUB, ZPOTRF, ZPOTRS, ZLACPY,
541: $ ZLAQHE, XERBLA, ZLASCL2, ZPORFSX
542: * ..
543: * .. Intrinsic Functions ..
544: INTRINSIC MAX, MIN
545: * ..
546: * .. Executable Statements ..
547: *
548: INFO = 0
549: NOFACT = LSAME( FACT, 'N' )
550: EQUIL = LSAME( FACT, 'E' )
551: SMLNUM = DLAMCH( 'Safe minimum' )
552: BIGNUM = ONE / SMLNUM
553: IF( NOFACT .OR. EQUIL ) THEN
554: EQUED = 'N'
555: RCEQU = .FALSE.
556: ELSE
557: RCEQU = LSAME( EQUED, 'Y' )
558: ENDIF
559: *
560: * Default is failure. If an input parameter is wrong or
561: * factorization fails, make everything look horrible. Only the
562: * pivot growth is set here, the rest is initialized in ZPORFSX.
563: *
564: RPVGRW = ZERO
565: *
566: * Test the input parameters. PARAMS is not tested until ZPORFSX.
567: *
568: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
569: $ LSAME( FACT, 'F' ) ) THEN
570: INFO = -1
571: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
572: $ .NOT.LSAME( UPLO, 'L' ) ) THEN
573: INFO = -2
574: ELSE IF( N.LT.0 ) THEN
575: INFO = -3
576: ELSE IF( NRHS.LT.0 ) THEN
577: INFO = -4
578: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
579: INFO = -6
580: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
581: INFO = -8
582: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
583: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
584: INFO = -9
585: ELSE
586: IF ( RCEQU ) THEN
587: SMIN = BIGNUM
588: SMAX = ZERO
589: DO 10 J = 1, N
590: SMIN = MIN( SMIN, S( J ) )
591: SMAX = MAX( SMAX, S( J ) )
592: 10 CONTINUE
593: IF( SMIN.LE.ZERO ) THEN
594: INFO = -10
595: ELSE IF( N.GT.0 ) THEN
596: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
597: ELSE
598: SCOND = ONE
599: END IF
600: END IF
601: IF( INFO.EQ.0 ) THEN
602: IF( LDB.LT.MAX( 1, N ) ) THEN
603: INFO = -12
604: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
605: INFO = -14
606: END IF
607: END IF
608: END IF
609: *
610: IF( INFO.NE.0 ) THEN
611: CALL XERBLA( 'ZPOSVXX', -INFO )
612: RETURN
613: END IF
614: *
615: IF( EQUIL ) THEN
616: *
617: * Compute row and column scalings to equilibrate the matrix A.
618: *
619: CALL ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU )
620: IF( INFEQU.EQ.0 ) THEN
621: *
622: * Equilibrate the matrix.
623: *
624: CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
625: RCEQU = LSAME( EQUED, 'Y' )
626: END IF
627: END IF
628: *
629: * Scale the right-hand side.
630: *
631: IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB )
632: *
633: IF( NOFACT .OR. EQUIL ) THEN
634: *
635: * Compute the Cholesky factorization of A.
636: *
637: CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
638: CALL ZPOTRF( UPLO, N, AF, LDAF, INFO )
639: *
640: * Return if INFO is non-zero.
641: *
642: IF( INFO.GT.0 ) THEN
643: *
644: * Pivot in column INFO is exactly 0
645: * Compute the reciprocal pivot growth factor of the
646: * leading rank-deficient INFO columns of A.
647: *
648: RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK )
649: RETURN
650: END IF
651: END IF
652: *
653: * Compute the reciprocal pivot growth factor RPVGRW.
654: *
655: RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK )
656: *
657: * Compute the solution matrix X.
658: *
659: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
660: CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
661: *
662: * Use iterative refinement to improve the computed solution and
663: * compute error bounds and backward error estimates for it.
664: *
665: CALL ZPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF,
666: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
667: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
668:
669: *
670: * Scale solutions.
671: *
672: IF ( RCEQU ) THEN
673: CALL ZLASCL2( N, NRHS, S, X, LDX )
674: END IF
675: *
676: RETURN
677: *
678: * End of ZPOSVXX
679: *
680: END
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