Annotation of rpl/lapack/lapack/zhpsvx.f, revision 1.9
1.9 ! bertrand 1: *> \brief <b> ZHPSVX 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 ZHPSVX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhpsvx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhpsvx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhpsvx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZHPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
! 22: * LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER FACT, UPLO
! 26: * INTEGER INFO, LDB, LDX, N, NRHS
! 27: * DOUBLE PRECISION RCOND
! 28: * ..
! 29: * .. Array Arguments ..
! 30: * INTEGER IPIV( * )
! 31: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
! 32: * COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
! 33: * $ X( LDX, * )
! 34: * ..
! 35: *
! 36: *
! 37: *> \par Purpose:
! 38: * =============
! 39: *>
! 40: *> \verbatim
! 41: *>
! 42: *> ZHPSVX uses the diagonal pivoting factorization A = U*D*U**H or
! 43: *> A = L*D*L**H to compute the solution to a complex system of linear
! 44: *> equations A * X = B, where A is an N-by-N Hermitian matrix stored
! 45: *> in 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 = 'N', the diagonal pivoting method is used to factor A as
! 59: *> A = U * D * U**H, if UPLO = 'U', or
! 60: *> A = L * D * L**H, if UPLO = 'L',
! 61: *> where U (or L) is a product of permutation and unit upper (lower)
! 62: *> triangular matrices and D is Hermitian and block diagonal with
! 63: *> 1-by-1 and 2-by-2 diagonal blocks.
! 64: *>
! 65: *> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
! 66: *> returns with INFO = i. Otherwise, the factored form of A is used
! 67: *> to estimate the condition number of the matrix A. If the
! 68: *> reciprocal of the condition number is less than machine precision,
! 69: *> INFO = N+1 is returned as a warning, but the routine still goes on
! 70: *> to solve for X and compute error bounds as described below.
! 71: *>
! 72: *> 3. The system of equations is solved for X using the factored form
! 73: *> of A.
! 74: *>
! 75: *> 4. Iterative refinement is applied to improve the computed solution
! 76: *> matrix and calculate error bounds and backward error estimates
! 77: *> for it.
! 78: *> \endverbatim
! 79: *
! 80: * Arguments:
! 81: * ==========
! 82: *
! 83: *> \param[in] FACT
! 84: *> \verbatim
! 85: *> FACT is CHARACTER*1
! 86: *> Specifies whether or not the factored form of A has been
! 87: *> supplied on entry.
! 88: *> = 'F': On entry, AFP and IPIV contain the factored form of
! 89: *> A. AFP and IPIV will not be modified.
! 90: *> = 'N': The matrix A will be copied to AFP and factored.
! 91: *> \endverbatim
! 92: *>
! 93: *> \param[in] UPLO
! 94: *> \verbatim
! 95: *> UPLO is CHARACTER*1
! 96: *> = 'U': Upper triangle of A is stored;
! 97: *> = 'L': Lower triangle of A is stored.
! 98: *> \endverbatim
! 99: *>
! 100: *> \param[in] N
! 101: *> \verbatim
! 102: *> N is INTEGER
! 103: *> The number of linear equations, i.e., the order of the
! 104: *> matrix A. N >= 0.
! 105: *> \endverbatim
! 106: *>
! 107: *> \param[in] NRHS
! 108: *> \verbatim
! 109: *> NRHS is INTEGER
! 110: *> The number of right hand sides, i.e., the number of columns
! 111: *> of the matrices B and X. NRHS >= 0.
! 112: *> \endverbatim
! 113: *>
! 114: *> \param[in] AP
! 115: *> \verbatim
! 116: *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
! 117: *> The upper or lower triangle of the Hermitian matrix A, packed
! 118: *> columnwise in a linear array. The j-th column of A is stored
! 119: *> in the array AP as follows:
! 120: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 121: *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
! 122: *> See below for further details.
! 123: *> \endverbatim
! 124: *>
! 125: *> \param[in,out] AFP
! 126: *> \verbatim
! 127: *> AFP is or output) COMPLEX*16 array, dimension (N*(N+1)/2)
! 128: *> If FACT = 'F', then AFP is an input argument and on entry
! 129: *> contains the block diagonal matrix D and the multipliers used
! 130: *> to obtain the factor U or L from the factorization
! 131: *> A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, stored as
! 132: *> a packed triangular matrix in the same storage format as A.
! 133: *>
! 134: *> If FACT = 'N', then AFP is an output argument and on exit
! 135: *> contains the block diagonal matrix D and the multipliers used
! 136: *> to obtain the factor U or L from the factorization
! 137: *> A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, stored as
! 138: *> a packed triangular matrix in the same storage format as A.
! 139: *> \endverbatim
! 140: *>
! 141: *> \param[in,out] IPIV
! 142: *> \verbatim
! 143: *> IPIV is or output) INTEGER array, dimension (N)
! 144: *> If FACT = 'F', then IPIV is an input argument and on entry
! 145: *> contains details of the interchanges and the block structure
! 146: *> of D, as determined by ZHPTRF.
! 147: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
! 148: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
! 149: *> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
! 150: *> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
! 151: *> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
! 152: *> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
! 153: *> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
! 154: *>
! 155: *> If FACT = 'N', then IPIV is an output argument and on exit
! 156: *> contains details of the interchanges and the block structure
! 157: *> of D, as determined by ZHPTRF.
! 158: *> \endverbatim
! 159: *>
! 160: *> \param[in] B
! 161: *> \verbatim
! 162: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
! 163: *> The N-by-NRHS right hand side matrix B.
! 164: *> \endverbatim
! 165: *>
! 166: *> \param[in] LDB
! 167: *> \verbatim
! 168: *> LDB is INTEGER
! 169: *> The leading dimension of the array B. LDB >= max(1,N).
! 170: *> \endverbatim
! 171: *>
! 172: *> \param[out] X
! 173: *> \verbatim
! 174: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
! 175: *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
! 176: *> \endverbatim
! 177: *>
! 178: *> \param[in] LDX
! 179: *> \verbatim
! 180: *> LDX is INTEGER
! 181: *> The leading dimension of the array X. LDX >= max(1,N).
! 182: *> \endverbatim
! 183: *>
! 184: *> \param[out] RCOND
! 185: *> \verbatim
! 186: *> RCOND is DOUBLE PRECISION
! 187: *> The estimate of the reciprocal condition number of the matrix
! 188: *> A. If RCOND is less than the machine precision (in
! 189: *> particular, if RCOND = 0), the matrix is singular to working
! 190: *> precision. This condition is indicated by a return code of
! 191: *> INFO > 0.
! 192: *> \endverbatim
! 193: *>
! 194: *> \param[out] FERR
! 195: *> \verbatim
! 196: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
! 197: *> The estimated forward error bound for each solution vector
! 198: *> X(j) (the j-th column of the solution matrix X).
! 199: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
! 200: *> is an estimated upper bound for the magnitude of the largest
! 201: *> element in (X(j) - XTRUE) divided by the magnitude of the
! 202: *> largest element in X(j). The estimate is as reliable as
! 203: *> the estimate for RCOND, and is almost always a slight
! 204: *> overestimate of the true error.
! 205: *> \endverbatim
! 206: *>
! 207: *> \param[out] BERR
! 208: *> \verbatim
! 209: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 210: *> The componentwise relative backward error of each solution
! 211: *> vector X(j) (i.e., the smallest relative change in
! 212: *> any element of A or B that makes X(j) an exact solution).
! 213: *> \endverbatim
! 214: *>
! 215: *> \param[out] WORK
! 216: *> \verbatim
! 217: *> WORK is COMPLEX*16 array, dimension (2*N)
! 218: *> \endverbatim
! 219: *>
! 220: *> \param[out] RWORK
! 221: *> \verbatim
! 222: *> RWORK is DOUBLE PRECISION array, dimension (N)
! 223: *> \endverbatim
! 224: *>
! 225: *> \param[out] INFO
! 226: *> \verbatim
! 227: *> INFO is INTEGER
! 228: *> = 0: successful exit
! 229: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 230: *> > 0: if INFO = i, and i is
! 231: *> <= N: D(i,i) is exactly zero. The factorization
! 232: *> has been completed but the factor D is exactly
! 233: *> singular, so the solution and error bounds could
! 234: *> not be computed. RCOND = 0 is returned.
! 235: *> = N+1: D is nonsingular, but RCOND is less than machine
! 236: *> precision, meaning that the matrix is singular
! 237: *> to working precision. Nevertheless, the
! 238: *> solution and error bounds are computed because
! 239: *> there are a number of situations where the
! 240: *> computed solution can be more accurate than the
! 241: *> value of RCOND would suggest.
! 242: *> \endverbatim
! 243: *
! 244: * Authors:
! 245: * ========
! 246: *
! 247: *> \author Univ. of Tennessee
! 248: *> \author Univ. of California Berkeley
! 249: *> \author Univ. of Colorado Denver
! 250: *> \author NAG Ltd.
! 251: *
! 252: *> \date November 2011
! 253: *
! 254: *> \ingroup complex16OTHERsolve
! 255: *
! 256: *> \par Further Details:
! 257: * =====================
! 258: *>
! 259: *> \verbatim
! 260: *>
! 261: *> The packed storage scheme is illustrated by the following example
! 262: *> when N = 4, UPLO = 'U':
! 263: *>
! 264: *> Two-dimensional storage of the Hermitian matrix A:
! 265: *>
! 266: *> a11 a12 a13 a14
! 267: *> a22 a23 a24
! 268: *> a33 a34 (aij = conjg(aji))
! 269: *> a44
! 270: *>
! 271: *> Packed storage of the upper triangle of A:
! 272: *>
! 273: *> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
! 274: *> \endverbatim
! 275: *>
! 276: * =====================================================================
1.1 bertrand 277: SUBROUTINE ZHPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
278: $ LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
279: *
1.9 ! bertrand 280: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 281: * -- LAPACK is a software package provided by Univ. of Tennessee, --
282: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 283: * November 2011
1.1 bertrand 284: *
285: * .. Scalar Arguments ..
286: CHARACTER FACT, UPLO
287: INTEGER INFO, LDB, LDX, N, NRHS
288: DOUBLE PRECISION RCOND
289: * ..
290: * .. Array Arguments ..
291: INTEGER IPIV( * )
292: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
293: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
294: $ X( LDX, * )
295: * ..
296: *
297: * =====================================================================
298: *
299: * .. Parameters ..
300: DOUBLE PRECISION ZERO
301: PARAMETER ( ZERO = 0.0D+0 )
302: * ..
303: * .. Local Scalars ..
304: LOGICAL NOFACT
305: DOUBLE PRECISION ANORM
306: * ..
307: * .. External Functions ..
308: LOGICAL LSAME
309: DOUBLE PRECISION DLAMCH, ZLANHP
310: EXTERNAL LSAME, DLAMCH, ZLANHP
311: * ..
312: * .. External Subroutines ..
313: EXTERNAL XERBLA, ZCOPY, ZHPCON, ZHPRFS, ZHPTRF, ZHPTRS,
314: $ ZLACPY
315: * ..
316: * .. Intrinsic Functions ..
317: INTRINSIC MAX
318: * ..
319: * .. Executable Statements ..
320: *
321: * Test the input parameters.
322: *
323: INFO = 0
324: NOFACT = LSAME( FACT, 'N' )
325: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
326: INFO = -1
327: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
328: $ THEN
329: INFO = -2
330: ELSE IF( N.LT.0 ) THEN
331: INFO = -3
332: ELSE IF( NRHS.LT.0 ) THEN
333: INFO = -4
334: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
335: INFO = -9
336: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
337: INFO = -11
338: END IF
339: IF( INFO.NE.0 ) THEN
340: CALL XERBLA( 'ZHPSVX', -INFO )
341: RETURN
342: END IF
343: *
344: IF( NOFACT ) THEN
345: *
1.8 bertrand 346: * Compute the factorization A = U*D*U**H or A = L*D*L**H.
1.1 bertrand 347: *
348: CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
349: CALL ZHPTRF( UPLO, N, AFP, IPIV, INFO )
350: *
351: * Return if INFO is non-zero.
352: *
353: IF( INFO.GT.0 )THEN
354: RCOND = ZERO
355: RETURN
356: END IF
357: END IF
358: *
359: * Compute the norm of the matrix A.
360: *
361: ANORM = ZLANHP( 'I', UPLO, N, AP, RWORK )
362: *
363: * Compute the reciprocal of the condition number of A.
364: *
365: CALL ZHPCON( UPLO, N, AFP, IPIV, ANORM, RCOND, WORK, INFO )
366: *
367: * Compute the solution vectors X.
368: *
369: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
370: CALL ZHPTRS( UPLO, N, NRHS, AFP, IPIV, X, LDX, INFO )
371: *
372: * Use iterative refinement to improve the computed solutions and
373: * compute error bounds and backward error estimates for them.
374: *
375: CALL ZHPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR,
376: $ BERR, WORK, RWORK, INFO )
377: *
378: * Set INFO = N+1 if the matrix is singular to working precision.
379: *
380: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
381: $ INFO = N + 1
382: *
383: RETURN
384: *
385: * End of ZHPSVX
386: *
387: END
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