Annotation of rpl/lapack/lapack/zppsvx.f, revision 1.9
1.9 ! bertrand 1: *> \brief <b> ZPPSVX computes the solution to system of linear equations A * X = B for OTHER 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 ZPPSVX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zppsvx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zppsvx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zppsvx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
! 22: * X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER EQUED, FACT, UPLO
! 26: * INTEGER INFO, LDB, LDX, N, NRHS
! 27: * DOUBLE PRECISION RCOND
! 28: * ..
! 29: * .. Array Arguments ..
! 30: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
! 31: * COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
! 32: * $ X( LDX, * )
! 33: * ..
! 34: *
! 35: *
! 36: *> \par Purpose:
! 37: * =============
! 38: *>
! 39: *> \verbatim
! 40: *>
! 41: *> ZPPSVX uses the Cholesky factorization A = U**H * U or A = L * L**H to
! 42: *> compute the solution to a complex system of linear equations
! 43: *> A * X = B,
! 44: *> where A is an N-by-N Hermitian positive definite matrix stored in
! 45: *> packed format and X and B are N-by-NRHS matrices.
! 46: *>
! 47: *> Error bounds on the solution and a condition estimate are also
! 48: *> provided.
! 49: *> \endverbatim
! 50: *
! 51: *> \par Description:
! 52: * =================
! 53: *>
! 54: *> \verbatim
! 55: *>
! 56: *> The following steps are performed:
! 57: *>
! 58: *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
! 59: *> the system:
! 60: *> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
! 61: *> Whether or not the system will be equilibrated depends on the
! 62: *> scaling of the matrix A, but if equilibration is used, A is
! 63: *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
! 64: *>
! 65: *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
! 66: *> factor the matrix A (after equilibration if FACT = 'E') as
! 67: *> A = U**H * U , if UPLO = 'U', or
! 68: *> A = L * L**H, if UPLO = 'L',
! 69: *> where U is an upper triangular matrix, L is a lower triangular
! 70: *> matrix, and **H indicates conjugate transpose.
! 71: *>
! 72: *> 3. If the leading i-by-i principal minor is not positive definite,
! 73: *> then the routine returns with INFO = i. Otherwise, the factored
! 74: *> form of A is used to estimate the condition number of the matrix
! 75: *> A. If the reciprocal of the condition number is less than machine
! 76: *> precision, INFO = N+1 is returned as a warning, but the routine
! 77: *> still goes on to solve for X and compute error bounds as
! 78: *> described below.
! 79: *>
! 80: *> 4. The system of equations is solved for X using the factored form
! 81: *> of A.
! 82: *>
! 83: *> 5. Iterative refinement is applied to improve the computed solution
! 84: *> matrix and calculate error bounds and backward error estimates
! 85: *> for it.
! 86: *>
! 87: *> 6. If equilibration was used, the matrix X is premultiplied by
! 88: *> diag(S) so that it solves the original system before
! 89: *> equilibration.
! 90: *> \endverbatim
! 91: *
! 92: * Arguments:
! 93: * ==========
! 94: *
! 95: *> \param[in] FACT
! 96: *> \verbatim
! 97: *> FACT is CHARACTER*1
! 98: *> Specifies whether or not the factored form of the matrix A is
! 99: *> supplied on entry, and if not, whether the matrix A should be
! 100: *> equilibrated before it is factored.
! 101: *> = 'F': On entry, AFP contains the factored form of A.
! 102: *> If EQUED = 'Y', the matrix A has been equilibrated
! 103: *> with scaling factors given by S. AP and AFP will not
! 104: *> be modified.
! 105: *> = 'N': The matrix A will be copied to AFP and factored.
! 106: *> = 'E': The matrix A will be equilibrated if necessary, then
! 107: *> copied to AFP and factored.
! 108: *> \endverbatim
! 109: *>
! 110: *> \param[in] UPLO
! 111: *> \verbatim
! 112: *> UPLO is CHARACTER*1
! 113: *> = 'U': Upper triangle of A is stored;
! 114: *> = 'L': Lower triangle of A is stored.
! 115: *> \endverbatim
! 116: *>
! 117: *> \param[in] N
! 118: *> \verbatim
! 119: *> N is INTEGER
! 120: *> The number of linear equations, i.e., the order of the
! 121: *> matrix A. N >= 0.
! 122: *> \endverbatim
! 123: *>
! 124: *> \param[in] NRHS
! 125: *> \verbatim
! 126: *> NRHS is INTEGER
! 127: *> The number of right hand sides, i.e., the number of columns
! 128: *> of the matrices B and X. NRHS >= 0.
! 129: *> \endverbatim
! 130: *>
! 131: *> \param[in,out] AP
! 132: *> \verbatim
! 133: *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
! 134: *> On entry, the upper or lower triangle of the Hermitian matrix
! 135: *> A, packed columnwise in a linear array, except if FACT = 'F'
! 136: *> and EQUED = 'Y', then A must contain the equilibrated matrix
! 137: *> diag(S)*A*diag(S). The j-th column of A is stored in the
! 138: *> array AP as follows:
! 139: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 140: *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
! 141: *> See below for further details. A is not modified if
! 142: *> FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
! 143: *>
! 144: *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
! 145: *> diag(S)*A*diag(S).
! 146: *> \endverbatim
! 147: *>
! 148: *> \param[in,out] AFP
! 149: *> \verbatim
! 150: *> AFP is or output) COMPLEX*16 array, dimension (N*(N+1)/2)
! 151: *> If FACT = 'F', then AFP is an input argument and on entry
! 152: *> contains the triangular factor U or L from the Cholesky
! 153: *> factorization A = U**H*U or A = L*L**H, in the same storage
! 154: *> format as A. If EQUED .ne. 'N', then AFP is the factored
! 155: *> form of the equilibrated matrix A.
! 156: *>
! 157: *> If FACT = 'N', then AFP is an output argument and on exit
! 158: *> returns the triangular factor U or L from the Cholesky
! 159: *> factorization A = U**H * U or A = L * L**H of the original
! 160: *> matrix A.
! 161: *>
! 162: *> If FACT = 'E', then AFP is an output argument and on exit
! 163: *> returns the triangular factor U or L from the Cholesky
! 164: *> factorization A = U**H * U or A = L * L**H of the equilibrated
! 165: *> matrix A (see the description of AP for the form of the
! 166: *> equilibrated matrix).
! 167: *> \endverbatim
! 168: *>
! 169: *> \param[in,out] EQUED
! 170: *> \verbatim
! 171: *> EQUED is or output) CHARACTER*1
! 172: *> Specifies the form of equilibration that was done.
! 173: *> = 'N': No equilibration (always true if FACT = 'N').
! 174: *> = 'Y': Equilibration was done, i.e., A has been replaced by
! 175: *> diag(S) * A * diag(S).
! 176: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
! 177: *> output argument.
! 178: *> \endverbatim
! 179: *>
! 180: *> \param[in,out] S
! 181: *> \verbatim
! 182: *> S is or output) DOUBLE PRECISION array, dimension (N)
! 183: *> The scale factors for A; not accessed if EQUED = 'N'. S is
! 184: *> an input argument if FACT = 'F'; otherwise, S is an output
! 185: *> argument. If FACT = 'F' and EQUED = 'Y', each element of S
! 186: *> must be positive.
! 187: *> \endverbatim
! 188: *>
! 189: *> \param[in,out] B
! 190: *> \verbatim
! 191: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
! 192: *> On entry, the N-by-NRHS right hand side matrix B.
! 193: *> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
! 194: *> B is overwritten by diag(S) * B.
! 195: *> \endverbatim
! 196: *>
! 197: *> \param[in] LDB
! 198: *> \verbatim
! 199: *> LDB is INTEGER
! 200: *> The leading dimension of the array B. LDB >= max(1,N).
! 201: *> \endverbatim
! 202: *>
! 203: *> \param[out] X
! 204: *> \verbatim
! 205: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
! 206: *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
! 207: *> the original system of equations. Note that if EQUED = 'Y',
! 208: *> A and B are modified on exit, and the solution to the
! 209: *> equilibrated system is inv(diag(S))*X.
! 210: *> \endverbatim
! 211: *>
! 212: *> \param[in] LDX
! 213: *> \verbatim
! 214: *> LDX is INTEGER
! 215: *> The leading dimension of the array X. LDX >= max(1,N).
! 216: *> \endverbatim
! 217: *>
! 218: *> \param[out] RCOND
! 219: *> \verbatim
! 220: *> RCOND is DOUBLE PRECISION
! 221: *> The estimate of the reciprocal condition number of the matrix
! 222: *> A after equilibration (if done). If RCOND is less than the
! 223: *> machine precision (in particular, if RCOND = 0), the matrix
! 224: *> is singular to working precision. This condition is
! 225: *> indicated by a return code of INFO > 0.
! 226: *> \endverbatim
! 227: *>
! 228: *> \param[out] FERR
! 229: *> \verbatim
! 230: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
! 231: *> The estimated forward error bound for each solution vector
! 232: *> X(j) (the j-th column of the solution matrix X).
! 233: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
! 234: *> is an estimated upper bound for the magnitude of the largest
! 235: *> element in (X(j) - XTRUE) divided by the magnitude of the
! 236: *> largest element in X(j). The estimate is as reliable as
! 237: *> the estimate for RCOND, and is almost always a slight
! 238: *> overestimate of the true error.
! 239: *> \endverbatim
! 240: *>
! 241: *> \param[out] BERR
! 242: *> \verbatim
! 243: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 244: *> The componentwise relative backward error of each solution
! 245: *> vector X(j) (i.e., the smallest relative change in
! 246: *> any element of A or B that makes X(j) an exact solution).
! 247: *> \endverbatim
! 248: *>
! 249: *> \param[out] WORK
! 250: *> \verbatim
! 251: *> WORK is COMPLEX*16 array, dimension (2*N)
! 252: *> \endverbatim
! 253: *>
! 254: *> \param[out] RWORK
! 255: *> \verbatim
! 256: *> RWORK is DOUBLE PRECISION array, dimension (N)
! 257: *> \endverbatim
! 258: *>
! 259: *> \param[out] INFO
! 260: *> \verbatim
! 261: *> INFO is INTEGER
! 262: *> = 0: successful exit
! 263: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 264: *> > 0: if INFO = i, and i is
! 265: *> <= N: the leading minor of order i of A is
! 266: *> not positive definite, so the factorization
! 267: *> could not be completed, and the solution has not
! 268: *> been computed. RCOND = 0 is returned.
! 269: *> = N+1: U is nonsingular, but RCOND is less than machine
! 270: *> precision, meaning that the matrix is singular
! 271: *> to working precision. Nevertheless, the
! 272: *> solution and error bounds are computed because
! 273: *> there are a number of situations where the
! 274: *> computed solution can be more accurate than the
! 275: *> value of RCOND would suggest.
! 276: *> \endverbatim
! 277: *
! 278: * Authors:
! 279: * ========
! 280: *
! 281: *> \author Univ. of Tennessee
! 282: *> \author Univ. of California Berkeley
! 283: *> \author Univ. of Colorado Denver
! 284: *> \author NAG Ltd.
! 285: *
! 286: *> \date November 2011
! 287: *
! 288: *> \ingroup complex16OTHERsolve
! 289: *
! 290: *> \par Further Details:
! 291: * =====================
! 292: *>
! 293: *> \verbatim
! 294: *>
! 295: *> The packed storage scheme is illustrated by the following example
! 296: *> when N = 4, UPLO = 'U':
! 297: *>
! 298: *> Two-dimensional storage of the Hermitian matrix A:
! 299: *>
! 300: *> a11 a12 a13 a14
! 301: *> a22 a23 a24
! 302: *> a33 a34 (aij = conjg(aji))
! 303: *> a44
! 304: *>
! 305: *> Packed storage of the upper triangle of A:
! 306: *>
! 307: *> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
! 308: *> \endverbatim
! 309: *>
! 310: * =====================================================================
1.1 bertrand 311: SUBROUTINE ZPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
312: $ X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
313: *
1.9 ! bertrand 314: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 315: * -- LAPACK is a software package provided by Univ. of Tennessee, --
316: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 317: * November 2011
1.1 bertrand 318: *
319: * .. Scalar Arguments ..
320: CHARACTER EQUED, FACT, UPLO
321: INTEGER INFO, LDB, LDX, N, NRHS
322: DOUBLE PRECISION RCOND
323: * ..
324: * .. Array Arguments ..
325: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
326: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
327: $ X( LDX, * )
328: * ..
329: *
330: * =====================================================================
331: *
332: * .. Parameters ..
333: DOUBLE PRECISION ZERO, ONE
334: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
335: * ..
336: * .. Local Scalars ..
337: LOGICAL EQUIL, NOFACT, RCEQU
338: INTEGER I, INFEQU, J
339: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
340: * ..
341: * .. External Functions ..
342: LOGICAL LSAME
343: DOUBLE PRECISION DLAMCH, ZLANHP
344: EXTERNAL LSAME, DLAMCH, ZLANHP
345: * ..
346: * .. External Subroutines ..
347: EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAQHP, ZPPCON, ZPPEQU,
348: $ ZPPRFS, ZPPTRF, ZPPTRS
349: * ..
350: * .. Intrinsic Functions ..
351: INTRINSIC MAX, MIN
352: * ..
353: * .. Executable Statements ..
354: *
355: INFO = 0
356: NOFACT = LSAME( FACT, 'N' )
357: EQUIL = LSAME( FACT, 'E' )
358: IF( NOFACT .OR. EQUIL ) THEN
359: EQUED = 'N'
360: RCEQU = .FALSE.
361: ELSE
362: RCEQU = LSAME( EQUED, 'Y' )
363: SMLNUM = DLAMCH( 'Safe minimum' )
364: BIGNUM = ONE / SMLNUM
365: END IF
366: *
367: * Test the input parameters.
368: *
369: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
370: $ THEN
371: INFO = -1
372: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
373: $ THEN
374: INFO = -2
375: ELSE IF( N.LT.0 ) THEN
376: INFO = -3
377: ELSE IF( NRHS.LT.0 ) THEN
378: INFO = -4
379: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
380: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
381: INFO = -7
382: ELSE
383: IF( RCEQU ) THEN
384: SMIN = BIGNUM
385: SMAX = ZERO
386: DO 10 J = 1, N
387: SMIN = MIN( SMIN, S( J ) )
388: SMAX = MAX( SMAX, S( J ) )
389: 10 CONTINUE
390: IF( SMIN.LE.ZERO ) THEN
391: INFO = -8
392: ELSE IF( N.GT.0 ) THEN
393: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
394: ELSE
395: SCOND = ONE
396: END IF
397: END IF
398: IF( INFO.EQ.0 ) THEN
399: IF( LDB.LT.MAX( 1, N ) ) THEN
400: INFO = -10
401: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
402: INFO = -12
403: END IF
404: END IF
405: END IF
406: *
407: IF( INFO.NE.0 ) THEN
408: CALL XERBLA( 'ZPPSVX', -INFO )
409: RETURN
410: END IF
411: *
412: IF( EQUIL ) THEN
413: *
414: * Compute row and column scalings to equilibrate the matrix A.
415: *
416: CALL ZPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU )
417: IF( INFEQU.EQ.0 ) THEN
418: *
419: * Equilibrate the matrix.
420: *
421: CALL ZLAQHP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
422: RCEQU = LSAME( EQUED, 'Y' )
423: END IF
424: END IF
425: *
426: * Scale the right-hand side.
427: *
428: IF( RCEQU ) THEN
429: DO 30 J = 1, NRHS
430: DO 20 I = 1, N
431: B( I, J ) = S( I )*B( I, J )
432: 20 CONTINUE
433: 30 CONTINUE
434: END IF
435: *
436: IF( NOFACT .OR. EQUIL ) THEN
437: *
1.8 bertrand 438: * Compute the Cholesky factorization A = U**H * U or A = L * L**H.
1.1 bertrand 439: *
440: CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
441: CALL ZPPTRF( UPLO, N, AFP, INFO )
442: *
443: * Return if INFO is non-zero.
444: *
445: IF( INFO.GT.0 )THEN
446: RCOND = ZERO
447: RETURN
448: END IF
449: END IF
450: *
451: * Compute the norm of the matrix A.
452: *
453: ANORM = ZLANHP( 'I', UPLO, N, AP, RWORK )
454: *
455: * Compute the reciprocal of the condition number of A.
456: *
457: CALL ZPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, RWORK, INFO )
458: *
459: * Compute the solution matrix X.
460: *
461: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
462: CALL ZPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO )
463: *
464: * Use iterative refinement to improve the computed solution and
465: * compute error bounds and backward error estimates for it.
466: *
467: CALL ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR,
468: $ WORK, RWORK, INFO )
469: *
470: * Transform the solution matrix X to a solution of the original
471: * system.
472: *
473: IF( RCEQU ) THEN
474: DO 50 J = 1, NRHS
475: DO 40 I = 1, N
476: X( I, J ) = S( I )*X( I, J )
477: 40 CONTINUE
478: 50 CONTINUE
479: DO 60 J = 1, NRHS
480: FERR( J ) = FERR( J ) / SCOND
481: 60 CONTINUE
482: END IF
483: *
484: * Set INFO = N+1 if the matrix is singular to working precision.
485: *
486: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
487: $ INFO = N + 1
488: *
489: RETURN
490: *
491: * End of ZPPSVX
492: *
493: END
CVSweb interface <joel.bertrand@systella.fr>
![[BACK]](/cvsweb/icons/back.gif){kind=icon}
Return to zppsvx.f CVS log ![[TXT]](/cvsweb/icons/text.gif){kind=icon}
![[DIR]](/cvsweb/icons/dir.gif){kind=icon}
Up to [local] / rpl / lapack / lapack