Annotation of rpl/lapack/lapack/zpbsvx.f, revision 1.9
1.9 ! bertrand 1: *> \brief <b> ZPBSVX 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 ZPBSVX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpbsvx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpbsvx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpbsvx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
! 22: * EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
! 23: * WORK, RWORK, INFO )
! 24: *
! 25: * .. Scalar Arguments ..
! 26: * CHARACTER EQUED, FACT, UPLO
! 27: * INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
! 28: * DOUBLE PRECISION RCOND
! 29: * ..
! 30: * .. Array Arguments ..
! 31: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
! 32: * COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
! 33: * $ WORK( * ), X( LDX, * )
! 34: * ..
! 35: *
! 36: *
! 37: *> \par Purpose:
! 38: * =============
! 39: *>
! 40: *> \verbatim
! 41: *>
! 42: *> ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
! 43: *> compute the solution to a complex system of linear equations
! 44: *> A * X = B,
! 45: *> where A is an N-by-N Hermitian positive definite band matrix and X
! 46: *> and B are N-by-NRHS matrices.
! 47: *>
! 48: *> Error bounds on the solution and a condition estimate are also
! 49: *> provided.
! 50: *> \endverbatim
! 51: *
! 52: *> \par Description:
! 53: * =================
! 54: *>
! 55: *> \verbatim
! 56: *>
! 57: *> The following steps are performed:
! 58: *>
! 59: *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
! 60: *> the system:
! 61: *> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
! 62: *> Whether or not the system will be equilibrated depends on the
! 63: *> scaling of the matrix A, but if equilibration is used, A is
! 64: *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
! 65: *>
! 66: *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
! 67: *> factor the matrix A (after equilibration if FACT = 'E') as
! 68: *> A = U**H * U, if UPLO = 'U', or
! 69: *> A = L * L**H, if UPLO = 'L',
! 70: *> where U is an upper triangular band matrix, and L is a lower
! 71: *> triangular band matrix.
! 72: *>
! 73: *> 3. If the leading i-by-i principal minor is not positive definite,
! 74: *> then the routine returns with INFO = i. Otherwise, the factored
! 75: *> form of A is used to estimate the condition number of the matrix
! 76: *> A. If the reciprocal of the condition number is less than machine
! 77: *> precision, INFO = N+1 is returned as a warning, but the routine
! 78: *> still goes on to solve for X and compute error bounds as
! 79: *> described below.
! 80: *>
! 81: *> 4. The system of equations is solved for X using the factored form
! 82: *> of A.
! 83: *>
! 84: *> 5. Iterative refinement is applied to improve the computed solution
! 85: *> matrix and calculate error bounds and backward error estimates
! 86: *> for it.
! 87: *>
! 88: *> 6. If equilibration was used, the matrix X is premultiplied by
! 89: *> diag(S) so that it solves the original system before
! 90: *> equilibration.
! 91: *> \endverbatim
! 92: *
! 93: * Arguments:
! 94: * ==========
! 95: *
! 96: *> \param[in] FACT
! 97: *> \verbatim
! 98: *> FACT is CHARACTER*1
! 99: *> Specifies whether or not the factored form of the matrix A is
! 100: *> supplied on entry, and if not, whether the matrix A should be
! 101: *> equilibrated before it is factored.
! 102: *> = 'F': On entry, AFB contains the factored form of A.
! 103: *> If EQUED = 'Y', the matrix A has been equilibrated
! 104: *> with scaling factors given by S. AB and AFB will not
! 105: *> be modified.
! 106: *> = 'N': The matrix A will be copied to AFB and factored.
! 107: *> = 'E': The matrix A will be equilibrated if necessary, then
! 108: *> copied to AFB and factored.
! 109: *> \endverbatim
! 110: *>
! 111: *> \param[in] UPLO
! 112: *> \verbatim
! 113: *> UPLO is CHARACTER*1
! 114: *> = 'U': Upper triangle of A is stored;
! 115: *> = 'L': Lower triangle of A is stored.
! 116: *> \endverbatim
! 117: *>
! 118: *> \param[in] N
! 119: *> \verbatim
! 120: *> N is INTEGER
! 121: *> The number of linear equations, i.e., the order of the
! 122: *> matrix A. N >= 0.
! 123: *> \endverbatim
! 124: *>
! 125: *> \param[in] KD
! 126: *> \verbatim
! 127: *> KD is INTEGER
! 128: *> The number of superdiagonals of the matrix A if UPLO = 'U',
! 129: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
! 130: *> \endverbatim
! 131: *>
! 132: *> \param[in] NRHS
! 133: *> \verbatim
! 134: *> NRHS is INTEGER
! 135: *> The number of right-hand sides, i.e., the number of columns
! 136: *> of the matrices B and X. NRHS >= 0.
! 137: *> \endverbatim
! 138: *>
! 139: *> \param[in,out] AB
! 140: *> \verbatim
! 141: *> AB is COMPLEX*16 array, dimension (LDAB,N)
! 142: *> On entry, the upper or lower triangle of the Hermitian band
! 143: *> matrix A, stored in the first KD+1 rows of the array, except
! 144: *> if FACT = 'F' and EQUED = 'Y', then A must contain the
! 145: *> equilibrated matrix diag(S)*A*diag(S). The j-th column of A
! 146: *> is stored in the j-th column of the array AB as follows:
! 147: *> if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
! 148: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
! 149: *> See below for further details.
! 150: *>
! 151: *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
! 152: *> diag(S)*A*diag(S).
! 153: *> \endverbatim
! 154: *>
! 155: *> \param[in] LDAB
! 156: *> \verbatim
! 157: *> LDAB is INTEGER
! 158: *> The leading dimension of the array A. LDAB >= KD+1.
! 159: *> \endverbatim
! 160: *>
! 161: *> \param[in,out] AFB
! 162: *> \verbatim
! 163: *> AFB is or output) COMPLEX*16 array, dimension (LDAFB,N)
! 164: *> If FACT = 'F', then AFB is an input argument and on entry
! 165: *> contains the triangular factor U or L from the Cholesky
! 166: *> factorization A = U**H *U or A = L*L**H of the band matrix
! 167: *> A, in the same storage format as A (see AB). If EQUED = 'Y',
! 168: *> then AFB is the factored form of the equilibrated matrix A.
! 169: *>
! 170: *> If FACT = 'N', then AFB is an output argument and on exit
! 171: *> returns the triangular factor U or L from the Cholesky
! 172: *> factorization A = U**H *U or A = L*L**H.
! 173: *>
! 174: *> If FACT = 'E', then AFB is an output argument and on exit
! 175: *> returns the triangular factor U or L from the Cholesky
! 176: *> factorization A = U**H *U or A = L*L**H of the equilibrated
! 177: *> matrix A (see the description of A for the form of the
! 178: *> equilibrated matrix).
! 179: *> \endverbatim
! 180: *>
! 181: *> \param[in] LDAFB
! 182: *> \verbatim
! 183: *> LDAFB is INTEGER
! 184: *> The leading dimension of the array AFB. LDAFB >= KD+1.
! 185: *> \endverbatim
! 186: *>
! 187: *> \param[in,out] EQUED
! 188: *> \verbatim
! 189: *> EQUED is or output) CHARACTER*1
! 190: *> Specifies the form of equilibration that was done.
! 191: *> = 'N': No equilibration (always true if FACT = 'N').
! 192: *> = 'Y': Equilibration was done, i.e., A has been replaced by
! 193: *> diag(S) * A * diag(S).
! 194: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
! 195: *> output argument.
! 196: *> \endverbatim
! 197: *>
! 198: *> \param[in,out] S
! 199: *> \verbatim
! 200: *> S is or output) DOUBLE PRECISION array, dimension (N)
! 201: *> The scale factors for A; not accessed if EQUED = 'N'. S is
! 202: *> an input argument if FACT = 'F'; otherwise, S is an output
! 203: *> argument. If FACT = 'F' and EQUED = 'Y', each element of S
! 204: *> must be positive.
! 205: *> \endverbatim
! 206: *>
! 207: *> \param[in,out] B
! 208: *> \verbatim
! 209: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
! 210: *> On entry, the N-by-NRHS right hand side matrix B.
! 211: *> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
! 212: *> B is overwritten by diag(S) * B.
! 213: *> \endverbatim
! 214: *>
! 215: *> \param[in] LDB
! 216: *> \verbatim
! 217: *> LDB is INTEGER
! 218: *> The leading dimension of the array B. LDB >= max(1,N).
! 219: *> \endverbatim
! 220: *>
! 221: *> \param[out] X
! 222: *> \verbatim
! 223: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
! 224: *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
! 225: *> the original system of equations. Note that if EQUED = 'Y',
! 226: *> A and B are modified on exit, and the solution to the
! 227: *> equilibrated system is inv(diag(S))*X.
! 228: *> \endverbatim
! 229: *>
! 230: *> \param[in] LDX
! 231: *> \verbatim
! 232: *> LDX is INTEGER
! 233: *> The leading dimension of the array X. LDX >= max(1,N).
! 234: *> \endverbatim
! 235: *>
! 236: *> \param[out] RCOND
! 237: *> \verbatim
! 238: *> RCOND is DOUBLE PRECISION
! 239: *> The estimate of the reciprocal condition number of the matrix
! 240: *> A after equilibration (if done). If RCOND is less than the
! 241: *> machine precision (in particular, if RCOND = 0), the matrix
! 242: *> is singular to working precision. This condition is
! 243: *> indicated by a return code of INFO > 0.
! 244: *> \endverbatim
! 245: *>
! 246: *> \param[out] FERR
! 247: *> \verbatim
! 248: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
! 249: *> The estimated forward error bound for each solution vector
! 250: *> X(j) (the j-th column of the solution matrix X).
! 251: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
! 252: *> is an estimated upper bound for the magnitude of the largest
! 253: *> element in (X(j) - XTRUE) divided by the magnitude of the
! 254: *> largest element in X(j). The estimate is as reliable as
! 255: *> the estimate for RCOND, and is almost always a slight
! 256: *> overestimate of the true error.
! 257: *> \endverbatim
! 258: *>
! 259: *> \param[out] BERR
! 260: *> \verbatim
! 261: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 262: *> The componentwise relative backward error of each solution
! 263: *> vector X(j) (i.e., the smallest relative change in
! 264: *> any element of A or B that makes X(j) an exact solution).
! 265: *> \endverbatim
! 266: *>
! 267: *> \param[out] WORK
! 268: *> \verbatim
! 269: *> WORK is COMPLEX*16 array, dimension (2*N)
! 270: *> \endverbatim
! 271: *>
! 272: *> \param[out] RWORK
! 273: *> \verbatim
! 274: *> RWORK is DOUBLE PRECISION array, dimension (N)
! 275: *> \endverbatim
! 276: *>
! 277: *> \param[out] INFO
! 278: *> \verbatim
! 279: *> INFO is INTEGER
! 280: *> = 0: successful exit
! 281: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 282: *> > 0: if INFO = i, and i is
! 283: *> <= N: the leading minor of order i of A is
! 284: *> not positive definite, so the factorization
! 285: *> could not be completed, and the solution has not
! 286: *> been computed. RCOND = 0 is returned.
! 287: *> = N+1: U is nonsingular, but RCOND is less than machine
! 288: *> precision, meaning that the matrix is singular
! 289: *> to working precision. Nevertheless, the
! 290: *> solution and error bounds are computed because
! 291: *> there are a number of situations where the
! 292: *> computed solution can be more accurate than the
! 293: *> value of RCOND would suggest.
! 294: *> \endverbatim
! 295: *
! 296: * Authors:
! 297: * ========
! 298: *
! 299: *> \author Univ. of Tennessee
! 300: *> \author Univ. of California Berkeley
! 301: *> \author Univ. of Colorado Denver
! 302: *> \author NAG Ltd.
! 303: *
! 304: *> \date November 2011
! 305: *
! 306: *> \ingroup complex16OTHERsolve
! 307: *
! 308: *> \par Further Details:
! 309: * =====================
! 310: *>
! 311: *> \verbatim
! 312: *>
! 313: *> The band storage scheme is illustrated by the following example, when
! 314: *> N = 6, KD = 2, and UPLO = 'U':
! 315: *>
! 316: *> Two-dimensional storage of the Hermitian matrix A:
! 317: *>
! 318: *> a11 a12 a13
! 319: *> a22 a23 a24
! 320: *> a33 a34 a35
! 321: *> a44 a45 a46
! 322: *> a55 a56
! 323: *> (aij=conjg(aji)) a66
! 324: *>
! 325: *> Band storage of the upper triangle of A:
! 326: *>
! 327: *> * * a13 a24 a35 a46
! 328: *> * a12 a23 a34 a45 a56
! 329: *> a11 a22 a33 a44 a55 a66
! 330: *>
! 331: *> Similarly, if UPLO = 'L' the format of A is as follows:
! 332: *>
! 333: *> a11 a22 a33 a44 a55 a66
! 334: *> a21 a32 a43 a54 a65 *
! 335: *> a31 a42 a53 a64 * *
! 336: *>
! 337: *> Array elements marked * are not used by the routine.
! 338: *> \endverbatim
! 339: *>
! 340: * =====================================================================
1.1 bertrand 341: SUBROUTINE ZPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
342: $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
343: $ WORK, RWORK, INFO )
344: *
1.9 ! bertrand 345: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 346: * -- LAPACK is a software package provided by Univ. of Tennessee, --
347: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 348: * November 2011
1.1 bertrand 349: *
350: * .. Scalar Arguments ..
351: CHARACTER EQUED, FACT, UPLO
352: INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
353: DOUBLE PRECISION RCOND
354: * ..
355: * .. Array Arguments ..
356: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
357: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
358: $ WORK( * ), X( LDX, * )
359: * ..
360: *
361: * =====================================================================
362: *
363: * .. Parameters ..
364: DOUBLE PRECISION ZERO, ONE
365: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
366: * ..
367: * .. Local Scalars ..
368: LOGICAL EQUIL, NOFACT, RCEQU, UPPER
369: INTEGER I, INFEQU, J, J1, J2
370: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
371: * ..
372: * .. External Functions ..
373: LOGICAL LSAME
374: DOUBLE PRECISION DLAMCH, ZLANHB
375: EXTERNAL LSAME, DLAMCH, ZLANHB
376: * ..
377: * .. External Subroutines ..
378: EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAQHB, ZPBCON, ZPBEQU,
379: $ ZPBRFS, ZPBTRF, ZPBTRS
380: * ..
381: * .. Intrinsic Functions ..
382: INTRINSIC MAX, MIN
383: * ..
384: * .. Executable Statements ..
385: *
386: INFO = 0
387: NOFACT = LSAME( FACT, 'N' )
388: EQUIL = LSAME( FACT, 'E' )
389: UPPER = LSAME( UPLO, 'U' )
390: IF( NOFACT .OR. EQUIL ) THEN
391: EQUED = 'N'
392: RCEQU = .FALSE.
393: ELSE
394: RCEQU = LSAME( EQUED, 'Y' )
395: SMLNUM = DLAMCH( 'Safe minimum' )
396: BIGNUM = ONE / SMLNUM
397: END IF
398: *
399: * Test the input parameters.
400: *
401: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
402: $ THEN
403: INFO = -1
404: ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
405: INFO = -2
406: ELSE IF( N.LT.0 ) THEN
407: INFO = -3
408: ELSE IF( KD.LT.0 ) THEN
409: INFO = -4
410: ELSE IF( NRHS.LT.0 ) THEN
411: INFO = -5
412: ELSE IF( LDAB.LT.KD+1 ) THEN
413: INFO = -7
414: ELSE IF( LDAFB.LT.KD+1 ) THEN
415: INFO = -9
416: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
417: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
418: INFO = -10
419: ELSE
420: IF( RCEQU ) THEN
421: SMIN = BIGNUM
422: SMAX = ZERO
423: DO 10 J = 1, N
424: SMIN = MIN( SMIN, S( J ) )
425: SMAX = MAX( SMAX, S( J ) )
426: 10 CONTINUE
427: IF( SMIN.LE.ZERO ) THEN
428: INFO = -11
429: ELSE IF( N.GT.0 ) THEN
430: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
431: ELSE
432: SCOND = ONE
433: END IF
434: END IF
435: IF( INFO.EQ.0 ) THEN
436: IF( LDB.LT.MAX( 1, N ) ) THEN
437: INFO = -13
438: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
439: INFO = -15
440: END IF
441: END IF
442: END IF
443: *
444: IF( INFO.NE.0 ) THEN
445: CALL XERBLA( 'ZPBSVX', -INFO )
446: RETURN
447: END IF
448: *
449: IF( EQUIL ) THEN
450: *
451: * Compute row and column scalings to equilibrate the matrix A.
452: *
453: CALL ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
454: IF( INFEQU.EQ.0 ) THEN
455: *
456: * Equilibrate the matrix.
457: *
458: CALL ZLAQHB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
459: RCEQU = LSAME( EQUED, 'Y' )
460: END IF
461: END IF
462: *
463: * Scale the right-hand side.
464: *
465: IF( RCEQU ) THEN
466: DO 30 J = 1, NRHS
467: DO 20 I = 1, N
468: B( I, J ) = S( I )*B( I, J )
469: 20 CONTINUE
470: 30 CONTINUE
471: END IF
472: *
473: IF( NOFACT .OR. EQUIL ) THEN
474: *
1.8 bertrand 475: * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
1.1 bertrand 476: *
477: IF( UPPER ) THEN
478: DO 40 J = 1, N
479: J1 = MAX( J-KD, 1 )
480: CALL ZCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
481: $ AFB( KD+1-J+J1, J ), 1 )
482: 40 CONTINUE
483: ELSE
484: DO 50 J = 1, N
485: J2 = MIN( J+KD, N )
486: CALL ZCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
487: 50 CONTINUE
488: END IF
489: *
490: CALL ZPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
491: *
492: * Return if INFO is non-zero.
493: *
494: IF( INFO.GT.0 )THEN
495: RCOND = ZERO
496: RETURN
497: END IF
498: END IF
499: *
500: * Compute the norm of the matrix A.
501: *
502: ANORM = ZLANHB( '1', UPLO, N, KD, AB, LDAB, RWORK )
503: *
504: * Compute the reciprocal of the condition number of A.
505: *
506: CALL ZPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, RWORK,
507: $ INFO )
508: *
509: * Compute the solution matrix X.
510: *
511: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
512: CALL ZPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
513: *
514: * Use iterative refinement to improve the computed solution and
515: * compute error bounds and backward error estimates for it.
516: *
517: CALL ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
518: $ LDX, FERR, BERR, WORK, RWORK, INFO )
519: *
520: * Transform the solution matrix X to a solution of the original
521: * system.
522: *
523: IF( RCEQU ) THEN
524: DO 70 J = 1, NRHS
525: DO 60 I = 1, N
526: X( I, J ) = S( I )*X( I, J )
527: 60 CONTINUE
528: 70 CONTINUE
529: DO 80 J = 1, NRHS
530: FERR( J ) = FERR( J ) / SCOND
531: 80 CONTINUE
532: END IF
533: *
534: * Set INFO = N+1 if the matrix is singular to working precision.
535: *
536: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
537: $ INFO = N + 1
538: *
539: RETURN
540: *
541: * End of ZPBSVX
542: *
543: END
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