Annotation of rpl/lapack/lapack/dposvxx.f, revision 1.5
1.5 ! bertrand 1: *> \brief <b> DPOSVXX 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 DPOSVXX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dposvxx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dposvxx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dposvxx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DPOSVXX( 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, IWORK, 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: * INTEGER IWORK( * )
! 34: * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
! 35: * $ X( LDX, * ), WORK( * )
! 36: * DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
! 37: * $ ERR_BNDS_NORM( NRHS, * ),
! 38: * $ ERR_BNDS_COMP( NRHS, * )
! 39: * ..
! 40: *
! 41: *
! 42: *> \par Purpose:
! 43: * =============
! 44: *>
! 45: *> \verbatim
! 46: *>
! 47: *> DPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
! 48: *> to compute the solution to a double precision system of linear equations
! 49: *> A * X = B, where A is an N-by-N symmetric positive definite matrix
! 50: *> and X and B are N-by-NRHS matrices.
! 51: *>
! 52: *> If requested, both normwise and maximum componentwise error bounds
! 53: *> are returned. DPOSVXX 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: *> DPOSVXX 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: *> DPOSVXX 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 DPOSVXX 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: *> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
! 79: *>
! 80: *> Whether or not the system will be equilibrated depends on the
! 81: *> scaling of the matrix A, but if equilibration is used, A is
! 82: *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
! 83: *>
! 84: *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
! 85: *> factor the matrix A (after equilibration if FACT = 'E') as
! 86: *> A = U**T* U, if UPLO = 'U', or
! 87: *> A = L * L**T, if UPLO = 'L',
! 88: *> where U is an upper triangular matrix and L is a lower triangular
! 89: *> matrix.
! 90: *>
! 91: *> 3. If the leading i-by-i principal minor is not positive definite,
! 92: *> then the routine returns with INFO = i. Otherwise, the factored
! 93: *> form of A is used to estimate the condition number of the matrix
! 94: *> A (see argument RCOND). If the reciprocal of the condition number
! 95: *> is less than machine precision, the routine still goes on to solve
! 96: *> for X and compute error bounds as described below.
! 97: *>
! 98: *> 4. The system of equations is solved for X using the factored form
! 99: *> of A.
! 100: *>
! 101: *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
! 102: *> the routine will use iterative refinement to try to get a small
! 103: *> error and error bounds. Refinement calculates the residual to at
! 104: *> least twice the working precision.
! 105: *>
! 106: *> 6. If equilibration was used, the matrix X is premultiplied by
! 107: *> diag(S) so that it solves the original system before
! 108: *> equilibration.
! 109: *> \endverbatim
! 110: *
! 111: * Arguments:
! 112: * ==========
! 113: *
! 114: *> \verbatim
! 115: *> Some optional parameters are bundled in the PARAMS array. These
! 116: *> settings determine how refinement is performed, but often the
! 117: *> defaults are acceptable. If the defaults are acceptable, users
! 118: *> can pass NPARAMS = 0 which prevents the source code from accessing
! 119: *> the PARAMS argument.
! 120: *> \endverbatim
! 121: *>
! 122: *> \param[in] FACT
! 123: *> \verbatim
! 124: *> FACT is CHARACTER*1
! 125: *> Specifies whether or not the factored form of the matrix A is
! 126: *> supplied on entry, and if not, whether the matrix A should be
! 127: *> equilibrated before it is factored.
! 128: *> = 'F': On entry, AF contains the factored form of A.
! 129: *> If EQUED is not 'N', the matrix A has been
! 130: *> equilibrated with scaling factors given by S.
! 131: *> A and AF are not modified.
! 132: *> = 'N': The matrix A will be copied to AF and factored.
! 133: *> = 'E': The matrix A will be equilibrated if necessary, then
! 134: *> copied to AF and factored.
! 135: *> \endverbatim
! 136: *>
! 137: *> \param[in] UPLO
! 138: *> \verbatim
! 139: *> UPLO is CHARACTER*1
! 140: *> = 'U': Upper triangle of A is stored;
! 141: *> = 'L': Lower triangle of A is stored.
! 142: *> \endverbatim
! 143: *>
! 144: *> \param[in] N
! 145: *> \verbatim
! 146: *> N is INTEGER
! 147: *> The number of linear equations, i.e., the order of the
! 148: *> matrix A. N >= 0.
! 149: *> \endverbatim
! 150: *>
! 151: *> \param[in] NRHS
! 152: *> \verbatim
! 153: *> NRHS is INTEGER
! 154: *> The number of right hand sides, i.e., the number of columns
! 155: *> of the matrices B and X. NRHS >= 0.
! 156: *> \endverbatim
! 157: *>
! 158: *> \param[in,out] A
! 159: *> \verbatim
! 160: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 161: *> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
! 162: *> 'Y', then A must contain the equilibrated matrix
! 163: *> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
! 164: *> triangular part of A contains the upper triangular part of the
! 165: *> matrix A, and the strictly lower triangular part of A is not
! 166: *> referenced. If UPLO = 'L', the leading N-by-N lower triangular
! 167: *> part of A contains the lower triangular part of the matrix A, and
! 168: *> the strictly upper triangular part of A is not referenced. A is
! 169: *> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
! 170: *> 'N' on exit.
! 171: *>
! 172: *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
! 173: *> diag(S)*A*diag(S).
! 174: *> \endverbatim
! 175: *>
! 176: *> \param[in] LDA
! 177: *> \verbatim
! 178: *> LDA is INTEGER
! 179: *> The leading dimension of the array A. LDA >= max(1,N).
! 180: *> \endverbatim
! 181: *>
! 182: *> \param[in,out] AF
! 183: *> \verbatim
! 184: *> AF is or output) DOUBLE PRECISION array, dimension (LDAF,N)
! 185: *> If FACT = 'F', then AF is an input argument and on entry
! 186: *> contains the triangular factor U or L from the Cholesky
! 187: *> factorization A = U**T*U or A = L*L**T, in the same storage
! 188: *> format as A. If EQUED .ne. 'N', then AF is the factored
! 189: *> form of the equilibrated matrix diag(S)*A*diag(S).
! 190: *>
! 191: *> If FACT = 'N', then AF is an output argument and on exit
! 192: *> returns the triangular factor U or L from the Cholesky
! 193: *> factorization A = U**T*U or A = L*L**T of the original
! 194: *> matrix A.
! 195: *>
! 196: *> If FACT = 'E', then AF is an output argument and on exit
! 197: *> returns the triangular factor U or L from the Cholesky
! 198: *> factorization A = U**T*U or A = L*L**T of the equilibrated
! 199: *> matrix A (see the description of A for the form of the
! 200: *> equilibrated matrix).
! 201: *> \endverbatim
! 202: *>
! 203: *> \param[in] LDAF
! 204: *> \verbatim
! 205: *> LDAF is INTEGER
! 206: *> The leading dimension of the array AF. LDAF >= max(1,N).
! 207: *> \endverbatim
! 208: *>
! 209: *> \param[in,out] EQUED
! 210: *> \verbatim
! 211: *> EQUED is or output) CHARACTER*1
! 212: *> Specifies the form of equilibration that was done.
! 213: *> = 'N': No equilibration (always true if FACT = 'N').
! 214: *> = 'Y': Both row and column equilibration, i.e., A has been
! 215: *> replaced by diag(S) * A * diag(S).
! 216: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
! 217: *> output argument.
! 218: *> \endverbatim
! 219: *>
! 220: *> \param[in,out] S
! 221: *> \verbatim
! 222: *> S is or output) DOUBLE PRECISION array, dimension (N)
! 223: *> The row scale factors for A. If EQUED = 'Y', A is multiplied on
! 224: *> the left and right by diag(S). S is an input argument if FACT =
! 225: *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
! 226: *> = 'Y', each element of S must be positive. If S is output, each
! 227: *> element of S is a power of the radix. If S is input, each element
! 228: *> of S should be a power of the radix to ensure a reliable solution
! 229: *> and error estimates. Scaling by powers of the radix does not cause
! 230: *> rounding errors unless the result underflows or overflows.
! 231: *> Rounding errors during scaling lead to refining with a matrix that
! 232: *> is not equivalent to the input matrix, producing error estimates
! 233: *> that may not be reliable.
! 234: *> \endverbatim
! 235: *>
! 236: *> \param[in,out] B
! 237: *> \verbatim
! 238: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
! 239: *> On entry, the N-by-NRHS right hand side matrix B.
! 240: *> On exit,
! 241: *> if EQUED = 'N', B is not modified;
! 242: *> if EQUED = 'Y', B is overwritten by diag(S)*B;
! 243: *> \endverbatim
! 244: *>
! 245: *> \param[in] LDB
! 246: *> \verbatim
! 247: *> LDB is INTEGER
! 248: *> The leading dimension of the array B. LDB >= max(1,N).
! 249: *> \endverbatim
! 250: *>
! 251: *> \param[out] X
! 252: *> \verbatim
! 253: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
! 254: *> If INFO = 0, the N-by-NRHS solution matrix X to the original
! 255: *> system of equations. Note that A and B are modified on exit if
! 256: *> EQUED .ne. 'N', and the solution to the equilibrated system is
! 257: *> inv(diag(S))*X.
! 258: *> \endverbatim
! 259: *>
! 260: *> \param[in] LDX
! 261: *> \verbatim
! 262: *> LDX is INTEGER
! 263: *> The leading dimension of the array X. LDX >= max(1,N).
! 264: *> \endverbatim
! 265: *>
! 266: *> \param[out] RCOND
! 267: *> \verbatim
! 268: *> RCOND is DOUBLE PRECISION
! 269: *> Reciprocal scaled condition number. This is an estimate of the
! 270: *> reciprocal Skeel condition number of the matrix A after
! 271: *> equilibration (if done). If this is less than the machine
! 272: *> precision (in particular, if it is zero), the matrix is singular
! 273: *> to working precision. Note that the error may still be small even
! 274: *> if this number is very small and the matrix appears ill-
! 275: *> conditioned.
! 276: *> \endverbatim
! 277: *>
! 278: *> \param[out] RPVGRW
! 279: *> \verbatim
! 280: *> RPVGRW is DOUBLE PRECISION
! 281: *> Reciprocal pivot growth. On exit, this contains the reciprocal
! 282: *> pivot growth factor norm(A)/norm(U). The "max absolute element"
! 283: *> norm is used. If this is much less than 1, then the stability of
! 284: *> the LU factorization of the (equilibrated) matrix A could be poor.
! 285: *> This also means that the solution X, estimated condition numbers,
! 286: *> and error bounds could be unreliable. If factorization fails with
! 287: *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
! 288: *> for the leading INFO columns of A.
! 289: *> \endverbatim
! 290: *>
! 291: *> \param[out] BERR
! 292: *> \verbatim
! 293: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 294: *> Componentwise relative backward error. This is the
! 295: *> componentwise relative backward error of each solution vector X(j)
! 296: *> (i.e., the smallest relative change in any element of A or B that
! 297: *> makes X(j) an exact solution).
! 298: *> \endverbatim
! 299: *>
! 300: *> \param[in] N_ERR_BNDS
! 301: *> \verbatim
! 302: *> N_ERR_BNDS is INTEGER
! 303: *> Number of error bounds to return for each right hand side
! 304: *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
! 305: *> ERR_BNDS_COMP below.
! 306: *> \endverbatim
! 307: *>
! 308: *> \param[out] ERR_BNDS_NORM
! 309: *> \verbatim
! 310: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
! 311: *> For each right-hand side, this array contains information about
! 312: *> various error bounds and condition numbers corresponding to the
! 313: *> normwise relative error, which is defined as follows:
! 314: *>
! 315: *> Normwise relative error in the ith solution vector:
! 316: *> max_j (abs(XTRUE(j,i) - X(j,i)))
! 317: *> ------------------------------
! 318: *> max_j abs(X(j,i))
! 319: *>
! 320: *> The array is indexed by the type of error information as described
! 321: *> below. There currently are up to three pieces of information
! 322: *> returned.
! 323: *>
! 324: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
! 325: *> right-hand side.
! 326: *>
! 327: *> The second index in ERR_BNDS_NORM(:,err) contains the following
! 328: *> three fields:
! 329: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 330: *> reciprocal condition number is less than the threshold
! 331: *> sqrt(n) * dlamch('Epsilon').
! 332: *>
! 333: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 334: *> almost certainly within a factor of 10 of the true error
! 335: *> so long as the next entry is greater than the threshold
! 336: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
! 337: *> be trusted if the previous boolean is true.
! 338: *>
! 339: *> err = 3 Reciprocal condition number: Estimated normwise
! 340: *> reciprocal condition number. Compared with the threshold
! 341: *> sqrt(n) * dlamch('Epsilon') to determine if the error
! 342: *> estimate is "guaranteed". These reciprocal condition
! 343: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 344: *> appropriately scaled matrix Z.
! 345: *> Let Z = S*A, where S scales each row by a power of the
! 346: *> radix so all absolute row sums of Z are approximately 1.
! 347: *>
! 348: *> See Lapack Working Note 165 for further details and extra
! 349: *> cautions.
! 350: *> \endverbatim
! 351: *>
! 352: *> \param[out] ERR_BNDS_COMP
! 353: *> \verbatim
! 354: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
! 355: *> For each right-hand side, this array contains information about
! 356: *> various error bounds and condition numbers corresponding to the
! 357: *> componentwise relative error, which is defined as follows:
! 358: *>
! 359: *> Componentwise relative error in the ith solution vector:
! 360: *> abs(XTRUE(j,i) - X(j,i))
! 361: *> max_j ----------------------
! 362: *> abs(X(j,i))
! 363: *>
! 364: *> The array is indexed by the right-hand side i (on which the
! 365: *> componentwise relative error depends), and the type of error
! 366: *> information as described below. There currently are up to three
! 367: *> pieces of information returned for each right-hand side. If
! 368: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
! 369: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
! 370: *> the first (:,N_ERR_BNDS) entries are returned.
! 371: *>
! 372: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
! 373: *> right-hand side.
! 374: *>
! 375: *> The second index in ERR_BNDS_COMP(:,err) contains the following
! 376: *> three fields:
! 377: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
! 378: *> reciprocal condition number is less than the threshold
! 379: *> sqrt(n) * dlamch('Epsilon').
! 380: *>
! 381: *> err = 2 "Guaranteed" error bound: The estimated forward error,
! 382: *> almost certainly within a factor of 10 of the true error
! 383: *> so long as the next entry is greater than the threshold
! 384: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
! 385: *> be trusted if the previous boolean is true.
! 386: *>
! 387: *> err = 3 Reciprocal condition number: Estimated componentwise
! 388: *> reciprocal condition number. Compared with the threshold
! 389: *> sqrt(n) * dlamch('Epsilon') to determine if the error
! 390: *> estimate is "guaranteed". These reciprocal condition
! 391: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
! 392: *> appropriately scaled matrix Z.
! 393: *> Let Z = S*(A*diag(x)), where x is the solution for the
! 394: *> current right-hand side and S scales each row of
! 395: *> A*diag(x) by a power of the radix so all absolute row
! 396: *> sums of Z are approximately 1.
! 397: *>
! 398: *> See Lapack Working Note 165 for further details and extra
! 399: *> cautions.
! 400: *> \endverbatim
! 401: *>
! 402: *> \param[in] NPARAMS
! 403: *> \verbatim
! 404: *> NPARAMS is INTEGER
! 405: *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
! 406: *> PARAMS array is never referenced and default values are used.
! 407: *> \endverbatim
! 408: *>
! 409: *> \param[in,out] PARAMS
! 410: *> \verbatim
! 411: *> PARAMS is / output) DOUBLE PRECISION array, dimension NPARAMS
! 412: *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
! 413: *> that entry will be filled with default value used for that
! 414: *> parameter. Only positions up to NPARAMS are accessed; defaults
! 415: *> are used for higher-numbered parameters.
! 416: *>
! 417: *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
! 418: *> refinement or not.
! 419: *> Default: 1.0D+0
! 420: *> = 0.0 : No refinement is performed, and no error bounds are
! 421: *> computed.
! 422: *> = 1.0 : Use the extra-precise refinement algorithm.
! 423: *> (other values are reserved for future use)
! 424: *>
! 425: *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
! 426: *> computations allowed for refinement.
! 427: *> Default: 10
! 428: *> Aggressive: Set to 100 to permit convergence using approximate
! 429: *> factorizations or factorizations other than LU. If
! 430: *> the factorization uses a technique other than
! 431: *> Gaussian elimination, the guarantees in
! 432: *> err_bnds_norm and err_bnds_comp may no longer be
! 433: *> trustworthy.
! 434: *>
! 435: *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
! 436: *> will attempt to find a solution with small componentwise
! 437: *> relative error in the double-precision algorithm. Positive
! 438: *> is true, 0.0 is false.
! 439: *> Default: 1.0 (attempt componentwise convergence)
! 440: *> \endverbatim
! 441: *>
! 442: *> \param[out] WORK
! 443: *> \verbatim
! 444: *> WORK is DOUBLE PRECISION array, dimension (4*N)
! 445: *> \endverbatim
! 446: *>
! 447: *> \param[out] IWORK
! 448: *> \verbatim
! 449: *> IWORK is INTEGER array, dimension (N)
! 450: *> \endverbatim
! 451: *>
! 452: *> \param[out] INFO
! 453: *> \verbatim
! 454: *> INFO is INTEGER
! 455: *> = 0: Successful exit. The solution to every right-hand side is
! 456: *> guaranteed.
! 457: *> < 0: If INFO = -i, the i-th argument had an illegal value
! 458: *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
! 459: *> has been completed, but the factor U is exactly singular, so
! 460: *> the solution and error bounds could not be computed. RCOND = 0
! 461: *> is returned.
! 462: *> = N+J: The solution corresponding to the Jth right-hand side is
! 463: *> not guaranteed. The solutions corresponding to other right-
! 464: *> hand sides K with K > J may not be guaranteed as well, but
! 465: *> only the first such right-hand side is reported. If a small
! 466: *> componentwise error is not requested (PARAMS(3) = 0.0) then
! 467: *> the Jth right-hand side is the first with a normwise error
! 468: *> bound that is not guaranteed (the smallest J such
! 469: *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
! 470: *> the Jth right-hand side is the first with either a normwise or
! 471: *> componentwise error bound that is not guaranteed (the smallest
! 472: *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
! 473: *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
! 474: *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
! 475: *> about all of the right-hand sides check ERR_BNDS_NORM or
! 476: *> ERR_BNDS_COMP.
! 477: *> \endverbatim
! 478: *
! 479: * Authors:
! 480: * ========
! 481: *
! 482: *> \author Univ. of Tennessee
! 483: *> \author Univ. of California Berkeley
! 484: *> \author Univ. of Colorado Denver
! 485: *> \author NAG Ltd.
! 486: *
! 487: *> \date November 2011
! 488: *
! 489: *> \ingroup doublePOsolve
! 490: *
! 491: * =====================================================================
1.1 bertrand 492: SUBROUTINE DPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
493: $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
494: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
495: $ NPARAMS, PARAMS, WORK, IWORK, INFO )
496: *
1.5 ! bertrand 497: * -- LAPACK driver routine (version 3.4.0) --
! 498: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 499: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 500: * November 2011
1.1 bertrand 501: *
502: * .. Scalar Arguments ..
503: CHARACTER EQUED, FACT, UPLO
504: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
505: $ N_ERR_BNDS
506: DOUBLE PRECISION RCOND, RPVGRW
507: * ..
508: * .. Array Arguments ..
509: INTEGER IWORK( * )
510: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
511: $ X( LDX, * ), WORK( * )
512: DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
513: $ ERR_BNDS_NORM( NRHS, * ),
514: $ ERR_BNDS_COMP( NRHS, * )
515: * ..
516: *
1.5 ! bertrand 517: * ==================================================================
1.1 bertrand 518: *
519: * .. Parameters ..
520: DOUBLE PRECISION ZERO, ONE
521: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
522: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
523: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
524: INTEGER CMP_ERR_I, PIV_GROWTH_I
525: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
526: $ BERR_I = 3 )
527: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
528: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
529: $ PIV_GROWTH_I = 9 )
530: * ..
531: * .. Local Scalars ..
532: LOGICAL EQUIL, NOFACT, RCEQU
533: INTEGER INFEQU, J
534: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX,
535: $ SCOND, SMLNUM
536: * ..
537: * .. External Functions ..
538: EXTERNAL LSAME, DLAMCH, DLA_PORPVGRW
539: LOGICAL LSAME
540: DOUBLE PRECISION DLAMCH, DLA_PORPVGRW
541: * ..
542: * .. External Subroutines ..
543: EXTERNAL DPOEQUB, DPOTRF, DPOTRS, DLACPY, DLAQSY,
544: $ XERBLA, DLASCL2, DPORFSX
545: * ..
546: * .. Intrinsic Functions ..
547: INTRINSIC MAX, MIN
548: * ..
549: * .. Executable Statements ..
550: *
551: INFO = 0
552: NOFACT = LSAME( FACT, 'N' )
553: EQUIL = LSAME( FACT, 'E' )
554: SMLNUM = DLAMCH( 'Safe minimum' )
555: BIGNUM = ONE / SMLNUM
556: IF( NOFACT .OR. EQUIL ) THEN
557: EQUED = 'N'
558: RCEQU = .FALSE.
559: ELSE
560: RCEQU = LSAME( EQUED, 'Y' )
561: ENDIF
562: *
563: * Default is failure. If an input parameter is wrong or
564: * factorization fails, make everything look horrible. Only the
565: * pivot growth is set here, the rest is initialized in DPORFSX.
566: *
567: RPVGRW = ZERO
568: *
569: * Test the input parameters. PARAMS is not tested until DPORFSX.
570: *
571: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
572: $ LSAME( FACT, 'F' ) ) THEN
573: INFO = -1
574: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
575: $ .NOT.LSAME( UPLO, 'L' ) ) THEN
576: INFO = -2
577: ELSE IF( N.LT.0 ) THEN
578: INFO = -3
579: ELSE IF( NRHS.LT.0 ) THEN
580: INFO = -4
581: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
582: INFO = -6
583: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
584: INFO = -8
585: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
586: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
587: INFO = -9
588: ELSE
589: IF ( RCEQU ) THEN
590: SMIN = BIGNUM
591: SMAX = ZERO
592: DO 10 J = 1, N
593: SMIN = MIN( SMIN, S( J ) )
594: SMAX = MAX( SMAX, S( J ) )
595: 10 CONTINUE
596: IF( SMIN.LE.ZERO ) THEN
597: INFO = -10
598: ELSE IF( N.GT.0 ) THEN
599: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
600: ELSE
601: SCOND = ONE
602: END IF
603: END IF
604: IF( INFO.EQ.0 ) THEN
605: IF( LDB.LT.MAX( 1, N ) ) THEN
606: INFO = -12
607: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
608: INFO = -14
609: END IF
610: END IF
611: END IF
612: *
613: IF( INFO.NE.0 ) THEN
614: CALL XERBLA( 'DPOSVXX', -INFO )
615: RETURN
616: END IF
617: *
618: IF( EQUIL ) THEN
619: *
620: * Compute row and column scalings to equilibrate the matrix A.
621: *
622: CALL DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU )
623: IF( INFEQU.EQ.0 ) THEN
624: *
625: * Equilibrate the matrix.
626: *
627: CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
628: RCEQU = LSAME( EQUED, 'Y' )
629: END IF
630: END IF
631: *
632: * Scale the right-hand side.
633: *
634: IF( RCEQU ) CALL DLASCL2( N, NRHS, S, B, LDB )
635: *
636: IF( NOFACT .OR. EQUIL ) THEN
637: *
638: * Compute the Cholesky factorization of A.
639: *
640: CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
641: CALL DPOTRF( UPLO, N, AF, LDAF, INFO )
642: *
643: * Return if INFO is non-zero.
644: *
645: IF( INFO.NE.0 ) THEN
646: *
647: * Pivot in column INFO is exactly 0
648: * Compute the reciprocal pivot growth factor of the
649: * leading rank-deficient INFO columns of A.
650: *
651: RPVGRW = DLA_PORPVGRW( UPLO, INFO, A, LDA, AF, LDAF, WORK )
652: RETURN
653: ENDIF
654: END IF
655: *
656: * Compute the reciprocal growth factor RPVGRW.
657: *
658: RPVGRW = DLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, WORK )
659: *
660: * Compute the solution matrix X.
661: *
662: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
663: CALL DPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
664: *
665: * Use iterative refinement to improve the computed solution and
666: * compute error bounds and backward error estimates for it.
667: *
668: CALL DPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF,
669: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
670: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO )
671:
672: *
673: * Scale solutions.
674: *
675: IF ( RCEQU ) THEN
676: CALL DLASCL2 ( N, NRHS, S, X, LDX )
677: END IF
678: *
679: RETURN
680: *
681: * End of DPOSVXX
682: *
683: END
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