Annotation of rpl/lapack/lapack/dptsvx.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DPTSVX
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DPTSVX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptsvx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptsvx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptsvx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
! 22: * RCOND, FERR, BERR, WORK, 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 B( LDB, * ), BERR( * ), D( * ), DF( * ),
! 31: * $ E( * ), EF( * ), FERR( * ), WORK( * ),
! 32: * $ X( LDX, * )
! 33: * ..
! 34: *
! 35: *
! 36: *> \par Purpose:
! 37: * =============
! 38: *>
! 39: *> \verbatim
! 40: *>
! 41: *> DPTSVX uses the factorization A = L*D*L**T to compute the solution
! 42: *> to a real system of linear equations A*X = B, where A is an N-by-N
! 43: *> symmetric positive definite tridiagonal matrix and X and B are
! 44: *> N-by-NRHS matrices.
! 45: *>
! 46: *> Error bounds on the solution and a condition estimate are also
! 47: *> provided.
! 48: *> \endverbatim
! 49: *
! 50: *> \par Description:
! 51: * =================
! 52: *>
! 53: *> \verbatim
! 54: *>
! 55: *> The following steps are performed:
! 56: *>
! 57: *> 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
! 58: *> is a unit lower bidiagonal matrix and D is diagonal. The
! 59: *> factorization can also be regarded as having the form
! 60: *> A = U**T*D*U.
! 61: *>
! 62: *> 2. If the leading i-by-i principal minor is not positive definite,
! 63: *> then the routine returns with INFO = i. Otherwise, the factored
! 64: *> form of A is used to estimate the condition number of the matrix
! 65: *> A. If the reciprocal of the condition number is less than machine
! 66: *> precision, INFO = N+1 is returned as a warning, but the routine
! 67: *> still goes on to solve for X and compute error bounds as
! 68: *> described below.
! 69: *>
! 70: *> 3. The system of equations is solved for X using the factored form
! 71: *> of A.
! 72: *>
! 73: *> 4. Iterative refinement is applied to improve the computed solution
! 74: *> matrix and calculate error bounds and backward error estimates
! 75: *> for it.
! 76: *> \endverbatim
! 77: *
! 78: * Arguments:
! 79: * ==========
! 80: *
! 81: *> \param[in] FACT
! 82: *> \verbatim
! 83: *> FACT is CHARACTER*1
! 84: *> Specifies whether or not the factored form of A has been
! 85: *> supplied on entry.
! 86: *> = 'F': On entry, DF and EF contain the factored form of A.
! 87: *> D, E, DF, and EF will not be modified.
! 88: *> = 'N': The matrix A will be copied to DF and EF and
! 89: *> factored.
! 90: *> \endverbatim
! 91: *>
! 92: *> \param[in] N
! 93: *> \verbatim
! 94: *> N is INTEGER
! 95: *> The order of the matrix A. N >= 0.
! 96: *> \endverbatim
! 97: *>
! 98: *> \param[in] NRHS
! 99: *> \verbatim
! 100: *> NRHS is INTEGER
! 101: *> The number of right hand sides, i.e., the number of columns
! 102: *> of the matrices B and X. NRHS >= 0.
! 103: *> \endverbatim
! 104: *>
! 105: *> \param[in] D
! 106: *> \verbatim
! 107: *> D is DOUBLE PRECISION array, dimension (N)
! 108: *> The n diagonal elements of the tridiagonal matrix A.
! 109: *> \endverbatim
! 110: *>
! 111: *> \param[in] E
! 112: *> \verbatim
! 113: *> E is DOUBLE PRECISION array, dimension (N-1)
! 114: *> The (n-1) subdiagonal elements of the tridiagonal matrix A.
! 115: *> \endverbatim
! 116: *>
! 117: *> \param[in,out] DF
! 118: *> \verbatim
! 119: *> DF is or output) DOUBLE PRECISION array, dimension (N)
! 120: *> If FACT = 'F', then DF is an input argument and on entry
! 121: *> contains the n diagonal elements of the diagonal matrix D
! 122: *> from the L*D*L**T factorization of A.
! 123: *> If FACT = 'N', then DF is an output argument and on exit
! 124: *> contains the n diagonal elements of the diagonal matrix D
! 125: *> from the L*D*L**T factorization of A.
! 126: *> \endverbatim
! 127: *>
! 128: *> \param[in,out] EF
! 129: *> \verbatim
! 130: *> EF is or output) DOUBLE PRECISION array, dimension (N-1)
! 131: *> If FACT = 'F', then EF is an input argument and on entry
! 132: *> contains the (n-1) subdiagonal elements of the unit
! 133: *> bidiagonal factor L from the L*D*L**T factorization of A.
! 134: *> If FACT = 'N', then EF is an output argument and on exit
! 135: *> contains the (n-1) subdiagonal elements of the unit
! 136: *> bidiagonal factor L from the L*D*L**T factorization of A.
! 137: *> \endverbatim
! 138: *>
! 139: *> \param[in] B
! 140: *> \verbatim
! 141: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
! 142: *> The N-by-NRHS right hand side matrix B.
! 143: *> \endverbatim
! 144: *>
! 145: *> \param[in] LDB
! 146: *> \verbatim
! 147: *> LDB is INTEGER
! 148: *> The leading dimension of the array B. LDB >= max(1,N).
! 149: *> \endverbatim
! 150: *>
! 151: *> \param[out] X
! 152: *> \verbatim
! 153: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
! 154: *> If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.
! 155: *> \endverbatim
! 156: *>
! 157: *> \param[in] LDX
! 158: *> \verbatim
! 159: *> LDX is INTEGER
! 160: *> The leading dimension of the array X. LDX >= max(1,N).
! 161: *> \endverbatim
! 162: *>
! 163: *> \param[out] RCOND
! 164: *> \verbatim
! 165: *> RCOND is DOUBLE PRECISION
! 166: *> The reciprocal condition number of the matrix A. If RCOND
! 167: *> is less than the machine precision (in particular, if
! 168: *> RCOND = 0), the matrix is singular to working precision.
! 169: *> This condition is indicated by a return code of INFO > 0.
! 170: *> \endverbatim
! 171: *>
! 172: *> \param[out] FERR
! 173: *> \verbatim
! 174: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
! 175: *> The forward error bound for each solution vector
! 176: *> X(j) (the j-th column of the solution matrix X).
! 177: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
! 178: *> is an estimated upper bound for the magnitude of the largest
! 179: *> element in (X(j) - XTRUE) divided by the magnitude of the
! 180: *> largest element in X(j).
! 181: *> \endverbatim
! 182: *>
! 183: *> \param[out] BERR
! 184: *> \verbatim
! 185: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 186: *> The componentwise relative backward error of each solution
! 187: *> vector X(j) (i.e., the smallest relative change in any
! 188: *> element of A or B that makes X(j) an exact solution).
! 189: *> \endverbatim
! 190: *>
! 191: *> \param[out] WORK
! 192: *> \verbatim
! 193: *> WORK is DOUBLE PRECISION array, dimension (2*N)
! 194: *> \endverbatim
! 195: *>
! 196: *> \param[out] INFO
! 197: *> \verbatim
! 198: *> INFO is INTEGER
! 199: *> = 0: successful exit
! 200: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 201: *> > 0: if INFO = i, and i is
! 202: *> <= N: the leading minor of order i of A is
! 203: *> not positive definite, so the factorization
! 204: *> could not be completed, and the solution has not
! 205: *> been computed. RCOND = 0 is returned.
! 206: *> = N+1: U is nonsingular, but RCOND is less than machine
! 207: *> precision, meaning that the matrix is singular
! 208: *> to working precision. Nevertheless, the
! 209: *> solution and error bounds are computed because
! 210: *> there are a number of situations where the
! 211: *> computed solution can be more accurate than the
! 212: *> value of RCOND would suggest.
! 213: *> \endverbatim
! 214: *
! 215: * Authors:
! 216: * ========
! 217: *
! 218: *> \author Univ. of Tennessee
! 219: *> \author Univ. of California Berkeley
! 220: *> \author Univ. of Colorado Denver
! 221: *> \author NAG Ltd.
! 222: *
! 223: *> \date November 2011
! 224: *
! 225: *> \ingroup doubleOTHERcomputational
! 226: *
! 227: * =====================================================================
1.1 bertrand 228: SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
229: $ RCOND, FERR, BERR, WORK, INFO )
230: *
1.9 ! bertrand 231: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 232: * -- LAPACK is a software package provided by Univ. of Tennessee, --
233: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 234: * November 2011
1.1 bertrand 235: *
236: * .. Scalar Arguments ..
237: CHARACTER FACT
238: INTEGER INFO, LDB, LDX, N, NRHS
239: DOUBLE PRECISION RCOND
240: * ..
241: * .. Array Arguments ..
242: DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
243: $ E( * ), EF( * ), FERR( * ), WORK( * ),
244: $ X( LDX, * )
245: * ..
246: *
247: * =====================================================================
248: *
249: * .. Parameters ..
250: DOUBLE PRECISION ZERO
251: PARAMETER ( ZERO = 0.0D+0 )
252: * ..
253: * .. Local Scalars ..
254: LOGICAL NOFACT
255: DOUBLE PRECISION ANORM
256: * ..
257: * .. External Functions ..
258: LOGICAL LSAME
259: DOUBLE PRECISION DLAMCH, DLANST
260: EXTERNAL LSAME, DLAMCH, DLANST
261: * ..
262: * .. External Subroutines ..
263: EXTERNAL DCOPY, DLACPY, DPTCON, DPTRFS, DPTTRF, DPTTRS,
264: $ XERBLA
265: * ..
266: * .. Intrinsic Functions ..
267: INTRINSIC MAX
268: * ..
269: * .. Executable Statements ..
270: *
271: * Test the input parameters.
272: *
273: INFO = 0
274: NOFACT = LSAME( FACT, 'N' )
275: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
276: INFO = -1
277: ELSE IF( N.LT.0 ) THEN
278: INFO = -2
279: ELSE IF( NRHS.LT.0 ) THEN
280: INFO = -3
281: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
282: INFO = -9
283: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
284: INFO = -11
285: END IF
286: IF( INFO.NE.0 ) THEN
287: CALL XERBLA( 'DPTSVX', -INFO )
288: RETURN
289: END IF
290: *
291: IF( NOFACT ) THEN
292: *
1.8 bertrand 293: * Compute the L*D*L**T (or U**T*D*U) factorization of A.
1.1 bertrand 294: *
295: CALL DCOPY( N, D, 1, DF, 1 )
296: IF( N.GT.1 )
297: $ CALL DCOPY( N-1, E, 1, EF, 1 )
298: CALL DPTTRF( N, DF, EF, INFO )
299: *
300: * Return if INFO is non-zero.
301: *
302: IF( INFO.GT.0 )THEN
303: RCOND = ZERO
304: RETURN
305: END IF
306: END IF
307: *
308: * Compute the norm of the matrix A.
309: *
310: ANORM = DLANST( '1', N, D, E )
311: *
312: * Compute the reciprocal of the condition number of A.
313: *
314: CALL DPTCON( N, DF, EF, ANORM, RCOND, WORK, INFO )
315: *
316: * Compute the solution vectors X.
317: *
318: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
319: CALL DPTTRS( N, NRHS, DF, EF, X, LDX, INFO )
320: *
321: * Use iterative refinement to improve the computed solutions and
322: * compute error bounds and backward error estimates for them.
323: *
324: CALL DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR,
325: $ WORK, INFO )
326: *
327: * Set INFO = N+1 if the matrix is singular to working precision.
328: *
329: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
330: $ INFO = N + 1
331: *
332: RETURN
333: *
334: * End of DPTSVX
335: *
336: END
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