Annotation of rpl/lapack/lapack/zptsvx.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b ZPTSVX
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZPTSVX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zptsvx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zptsvx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zptsvx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
! 22: * RCOND, FERR, BERR, WORK, RWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER FACT
! 26: * INTEGER INFO, LDB, LDX, N, NRHS
! 27: * DOUBLE PRECISION RCOND
! 28: * ..
! 29: * .. Array Arguments ..
! 30: * DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
! 31: * $ RWORK( * )
! 32: * COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
! 33: * $ X( LDX, * )
! 34: * ..
! 35: *
! 36: *
! 37: *> \par Purpose:
! 38: * =============
! 39: *>
! 40: *> \verbatim
! 41: *>
! 42: *> ZPTSVX uses the factorization A = L*D*L**H to compute the solution
! 43: *> to a complex system of linear equations A*X = B, where A is an
! 44: *> N-by-N Hermitian positive definite tridiagonal matrix and X and B
! 45: *> 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 matrix A is factored as A = L*D*L**H, where L
! 59: *> is a unit lower bidiagonal matrix and D is diagonal. The
! 60: *> factorization can also be regarded as having the form
! 61: *> A = U**H*D*U.
! 62: *>
! 63: *> 2. If the leading i-by-i principal minor is not positive definite,
! 64: *> then the routine returns with INFO = i. Otherwise, the factored
! 65: *> form of A is used to estimate the condition number of the matrix
! 66: *> A. If the reciprocal of the condition number is less than machine
! 67: *> precision, INFO = N+1 is returned as a warning, but the routine
! 68: *> still goes on to solve for X and compute error bounds as
! 69: *> described below.
! 70: *>
! 71: *> 3. The system of equations is solved for X using the factored form
! 72: *> of A.
! 73: *>
! 74: *> 4. Iterative refinement is applied to improve the computed solution
! 75: *> matrix and calculate error bounds and backward error estimates
! 76: *> for it.
! 77: *> \endverbatim
! 78: *
! 79: * Arguments:
! 80: * ==========
! 81: *
! 82: *> \param[in] FACT
! 83: *> \verbatim
! 84: *> FACT is CHARACTER*1
! 85: *> Specifies whether or not the factored form of the matrix
! 86: *> A is supplied on entry.
! 87: *> = 'F': On entry, DF and EF contain the factored form of A.
! 88: *> D, E, DF, and EF will not be modified.
! 89: *> = 'N': The matrix A will be copied to DF and EF and
! 90: *> factored.
! 91: *> \endverbatim
! 92: *>
! 93: *> \param[in] N
! 94: *> \verbatim
! 95: *> N is INTEGER
! 96: *> The order of the matrix A. N >= 0.
! 97: *> \endverbatim
! 98: *>
! 99: *> \param[in] NRHS
! 100: *> \verbatim
! 101: *> NRHS is INTEGER
! 102: *> The number of right hand sides, i.e., the number of columns
! 103: *> of the matrices B and X. NRHS >= 0.
! 104: *> \endverbatim
! 105: *>
! 106: *> \param[in] D
! 107: *> \verbatim
! 108: *> D is DOUBLE PRECISION array, dimension (N)
! 109: *> The n diagonal elements of the tridiagonal matrix A.
! 110: *> \endverbatim
! 111: *>
! 112: *> \param[in] E
! 113: *> \verbatim
! 114: *> E is COMPLEX*16 array, dimension (N-1)
! 115: *> The (n-1) subdiagonal elements of the tridiagonal matrix A.
! 116: *> \endverbatim
! 117: *>
! 118: *> \param[in,out] DF
! 119: *> \verbatim
! 120: *> DF is or output) DOUBLE PRECISION array, dimension (N)
! 121: *> If FACT = 'F', then DF is an input argument and on entry
! 122: *> contains the n diagonal elements of the diagonal matrix D
! 123: *> from the L*D*L**H factorization of A.
! 124: *> If FACT = 'N', then DF is an output argument and on exit
! 125: *> contains the n diagonal elements of the diagonal matrix D
! 126: *> from the L*D*L**H factorization of A.
! 127: *> \endverbatim
! 128: *>
! 129: *> \param[in,out] EF
! 130: *> \verbatim
! 131: *> EF is or output) COMPLEX*16 array, dimension (N-1)
! 132: *> If FACT = 'F', then EF is an input argument and on entry
! 133: *> contains the (n-1) subdiagonal elements of the unit
! 134: *> bidiagonal factor L from the L*D*L**H factorization of A.
! 135: *> If FACT = 'N', then EF is an output argument and on exit
! 136: *> contains the (n-1) subdiagonal elements of the unit
! 137: *> bidiagonal factor L from the L*D*L**H factorization of A.
! 138: *> \endverbatim
! 139: *>
! 140: *> \param[in] B
! 141: *> \verbatim
! 142: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
! 143: *> The N-by-NRHS right hand side matrix B.
! 144: *> \endverbatim
! 145: *>
! 146: *> \param[in] LDB
! 147: *> \verbatim
! 148: *> LDB is INTEGER
! 149: *> The leading dimension of the array B. LDB >= max(1,N).
! 150: *> \endverbatim
! 151: *>
! 152: *> \param[out] X
! 153: *> \verbatim
! 154: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
! 155: *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
! 156: *> \endverbatim
! 157: *>
! 158: *> \param[in] LDX
! 159: *> \verbatim
! 160: *> LDX is INTEGER
! 161: *> The leading dimension of the array X. LDX >= max(1,N).
! 162: *> \endverbatim
! 163: *>
! 164: *> \param[out] RCOND
! 165: *> \verbatim
! 166: *> RCOND is DOUBLE PRECISION
! 167: *> The reciprocal condition number of the matrix A. If RCOND
! 168: *> is less than the machine precision (in particular, if
! 169: *> RCOND = 0), the matrix is singular to working precision.
! 170: *> This condition is indicated by a return code of INFO > 0.
! 171: *> \endverbatim
! 172: *>
! 173: *> \param[out] FERR
! 174: *> \verbatim
! 175: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
! 176: *> The forward error bound for each solution vector
! 177: *> X(j) (the j-th column of the solution matrix X).
! 178: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
! 179: *> is an estimated upper bound for the magnitude of the largest
! 180: *> element in (X(j) - XTRUE) divided by the magnitude of the
! 181: *> largest element in X(j).
! 182: *> \endverbatim
! 183: *>
! 184: *> \param[out] BERR
! 185: *> \verbatim
! 186: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 187: *> The componentwise relative backward error of each solution
! 188: *> vector X(j) (i.e., the smallest relative change in any
! 189: *> element of A or B that makes X(j) an exact solution).
! 190: *> \endverbatim
! 191: *>
! 192: *> \param[out] WORK
! 193: *> \verbatim
! 194: *> WORK is COMPLEX*16 array, dimension (N)
! 195: *> \endverbatim
! 196: *>
! 197: *> \param[out] RWORK
! 198: *> \verbatim
! 199: *> RWORK is DOUBLE PRECISION array, dimension (N)
! 200: *> \endverbatim
! 201: *>
! 202: *> \param[out] INFO
! 203: *> \verbatim
! 204: *> INFO is INTEGER
! 205: *> = 0: successful exit
! 206: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 207: *> > 0: if INFO = i, and i is
! 208: *> <= N: the leading minor of order i of A is
! 209: *> not positive definite, so the factorization
! 210: *> could not be completed, and the solution has not
! 211: *> been computed. RCOND = 0 is returned.
! 212: *> = N+1: U is nonsingular, but RCOND is less than machine
! 213: *> precision, meaning that the matrix is singular
! 214: *> to working precision. Nevertheless, the
! 215: *> solution and error bounds are computed because
! 216: *> there are a number of situations where the
! 217: *> computed solution can be more accurate than the
! 218: *> value of RCOND would suggest.
! 219: *> \endverbatim
! 220: *
! 221: * Authors:
! 222: * ========
! 223: *
! 224: *> \author Univ. of Tennessee
! 225: *> \author Univ. of California Berkeley
! 226: *> \author Univ. of Colorado Denver
! 227: *> \author NAG Ltd.
! 228: *
! 229: *> \date November 2011
! 230: *
! 231: *> \ingroup complex16OTHERcomputational
! 232: *
! 233: * =====================================================================
1.1 bertrand 234: SUBROUTINE ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
235: $ RCOND, FERR, BERR, WORK, RWORK, INFO )
236: *
1.9 ! bertrand 237: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 238: * -- LAPACK is a software package provided by Univ. of Tennessee, --
239: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 240: * November 2011
1.1 bertrand 241: *
242: * .. Scalar Arguments ..
243: CHARACTER FACT
244: INTEGER INFO, LDB, LDX, N, NRHS
245: DOUBLE PRECISION RCOND
246: * ..
247: * .. Array Arguments ..
248: DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
249: $ RWORK( * )
250: COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
251: $ X( LDX, * )
252: * ..
253: *
254: * =====================================================================
255: *
256: * .. Parameters ..
257: DOUBLE PRECISION ZERO
258: PARAMETER ( ZERO = 0.0D+0 )
259: * ..
260: * .. Local Scalars ..
261: LOGICAL NOFACT
262: DOUBLE PRECISION ANORM
263: * ..
264: * .. External Functions ..
265: LOGICAL LSAME
266: DOUBLE PRECISION DLAMCH, ZLANHT
267: EXTERNAL LSAME, DLAMCH, ZLANHT
268: * ..
269: * .. External Subroutines ..
270: EXTERNAL DCOPY, XERBLA, ZCOPY, ZLACPY, ZPTCON, ZPTRFS,
271: $ ZPTTRF, ZPTTRS
272: * ..
273: * .. Intrinsic Functions ..
274: INTRINSIC MAX
275: * ..
276: * .. Executable Statements ..
277: *
278: * Test the input parameters.
279: *
280: INFO = 0
281: NOFACT = LSAME( FACT, 'N' )
282: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
283: INFO = -1
284: ELSE IF( N.LT.0 ) THEN
285: INFO = -2
286: ELSE IF( NRHS.LT.0 ) THEN
287: INFO = -3
288: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
289: INFO = -9
290: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
291: INFO = -11
292: END IF
293: IF( INFO.NE.0 ) THEN
294: CALL XERBLA( 'ZPTSVX', -INFO )
295: RETURN
296: END IF
297: *
298: IF( NOFACT ) THEN
299: *
1.8 bertrand 300: * Compute the L*D*L**H (or U**H*D*U) factorization of A.
1.1 bertrand 301: *
302: CALL DCOPY( N, D, 1, DF, 1 )
303: IF( N.GT.1 )
304: $ CALL ZCOPY( N-1, E, 1, EF, 1 )
305: CALL ZPTTRF( N, DF, EF, INFO )
306: *
307: * Return if INFO is non-zero.
308: *
309: IF( INFO.GT.0 )THEN
310: RCOND = ZERO
311: RETURN
312: END IF
313: END IF
314: *
315: * Compute the norm of the matrix A.
316: *
317: ANORM = ZLANHT( '1', N, D, E )
318: *
319: * Compute the reciprocal of the condition number of A.
320: *
321: CALL ZPTCON( N, DF, EF, ANORM, RCOND, RWORK, INFO )
322: *
323: * Compute the solution vectors X.
324: *
325: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
326: CALL ZPTTRS( 'Lower', N, NRHS, DF, EF, X, LDX, INFO )
327: *
328: * Use iterative refinement to improve the computed solutions and
329: * compute error bounds and backward error estimates for them.
330: *
331: CALL ZPTRFS( 'Lower', N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
332: $ BERR, WORK, RWORK, INFO )
333: *
334: * Set INFO = N+1 if the matrix is singular to working precision.
335: *
336: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
337: $ INFO = N + 1
338: *
339: RETURN
340: *
341: * End of ZPTSVX
342: *
343: END
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