Annotation of rpl/lapack/lapack/dspsvx.f, revision 1.9
1.9 ! bertrand 1: *> \brief <b> DSPSVX 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 DSPSVX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspsvx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspsvx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspsvx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
! 22: * LDX, RCOND, FERR, BERR, WORK, IWORK, 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( * ), IWORK( * )
! 31: * DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
! 32: * $ FERR( * ), WORK( * ), X( LDX, * )
! 33: * ..
! 34: *
! 35: *
! 36: *> \par Purpose:
! 37: * =============
! 38: *>
! 39: *> \verbatim
! 40: *>
! 41: *> DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
! 42: *> A = L*D*L**T to compute the solution to a real system of linear
! 43: *> equations A * X = B, where A is an N-by-N symmetric matrix stored
! 44: *> in packed format and X and B are 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 diagonal pivoting method is used to factor A as
! 58: *> A = U * D * U**T, if UPLO = 'U', or
! 59: *> A = L * D * L**T, if UPLO = 'L',
! 60: *> where U (or L) is a product of permutation and unit upper (lower)
! 61: *> triangular matrices and D is symmetric and block diagonal with
! 62: *> 1-by-1 and 2-by-2 diagonal blocks.
! 63: *>
! 64: *> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
! 65: *> returns with INFO = i. Otherwise, the factored form of A is used
! 66: *> to estimate the condition number of the matrix A. If the
! 67: *> reciprocal of the condition number is less than machine precision,
! 68: *> INFO = N+1 is returned as a warning, but the routine still goes on
! 69: *> to solve for X and compute error bounds as 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 A has been
! 86: *> supplied on entry.
! 87: *> = 'F': On entry, AFP and IPIV contain the factored form of
! 88: *> A. AP, AFP and IPIV will not be modified.
! 89: *> = 'N': The matrix A will be copied to AFP and factored.
! 90: *> \endverbatim
! 91: *>
! 92: *> \param[in] UPLO
! 93: *> \verbatim
! 94: *> UPLO is CHARACTER*1
! 95: *> = 'U': Upper triangle of A is stored;
! 96: *> = 'L': Lower triangle of A is stored.
! 97: *> \endverbatim
! 98: *>
! 99: *> \param[in] N
! 100: *> \verbatim
! 101: *> N is INTEGER
! 102: *> The number of linear equations, i.e., the order of the
! 103: *> matrix A. N >= 0.
! 104: *> \endverbatim
! 105: *>
! 106: *> \param[in] NRHS
! 107: *> \verbatim
! 108: *> NRHS is INTEGER
! 109: *> The number of right hand sides, i.e., the number of columns
! 110: *> of the matrices B and X. NRHS >= 0.
! 111: *> \endverbatim
! 112: *>
! 113: *> \param[in] AP
! 114: *> \verbatim
! 115: *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
! 116: *> The upper or lower triangle of the symmetric matrix A, packed
! 117: *> columnwise in a linear array. The j-th column of A is stored
! 118: *> in the array AP as follows:
! 119: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 120: *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
! 121: *> See below for further details.
! 122: *> \endverbatim
! 123: *>
! 124: *> \param[in,out] AFP
! 125: *> \verbatim
! 126: *> AFP is or output) DOUBLE PRECISION array, dimension
! 127: *> (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**T or A = L*D*L**T as computed by DSPTRF, 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**T or A = L*D*L**T as computed by DSPTRF, 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 DSPTRF.
! 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 DSPTRF.
! 158: *> \endverbatim
! 159: *>
! 160: *> \param[in] B
! 161: *> \verbatim
! 162: *> B is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (3*N)
! 218: *> \endverbatim
! 219: *>
! 220: *> \param[out] IWORK
! 221: *> \verbatim
! 222: *> IWORK is INTEGER 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 doubleOTHERsolve
! 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 symmetric matrix A:
! 265: *>
! 266: *> a11 a12 a13 a14
! 267: *> a22 a23 a24
! 268: *> a33 a34 (aij = 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 DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X,
278: $ LDX, RCOND, FERR, BERR, WORK, IWORK, 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( * ), IWORK( * )
292: DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
293: $ FERR( * ), WORK( * ), X( LDX, * )
294: * ..
295: *
296: * =====================================================================
297: *
298: * .. Parameters ..
299: DOUBLE PRECISION ZERO
300: PARAMETER ( ZERO = 0.0D+0 )
301: * ..
302: * .. Local Scalars ..
303: LOGICAL NOFACT
304: DOUBLE PRECISION ANORM
305: * ..
306: * .. External Functions ..
307: LOGICAL LSAME
308: DOUBLE PRECISION DLAMCH, DLANSP
309: EXTERNAL LSAME, DLAMCH, DLANSP
310: * ..
311: * .. External Subroutines ..
312: EXTERNAL DCOPY, DLACPY, DSPCON, DSPRFS, DSPTRF, DSPTRS,
313: $ XERBLA
314: * ..
315: * .. Intrinsic Functions ..
316: INTRINSIC MAX
317: * ..
318: * .. Executable Statements ..
319: *
320: * Test the input parameters.
321: *
322: INFO = 0
323: NOFACT = LSAME( FACT, 'N' )
324: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
325: INFO = -1
326: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
327: $ THEN
328: INFO = -2
329: ELSE IF( N.LT.0 ) THEN
330: INFO = -3
331: ELSE IF( NRHS.LT.0 ) THEN
332: INFO = -4
333: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
334: INFO = -9
335: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
336: INFO = -11
337: END IF
338: IF( INFO.NE.0 ) THEN
339: CALL XERBLA( 'DSPSVX', -INFO )
340: RETURN
341: END IF
342: *
343: IF( NOFACT ) THEN
344: *
1.8 bertrand 345: * Compute the factorization A = U*D*U**T or A = L*D*L**T.
1.1 bertrand 346: *
347: CALL DCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
348: CALL DSPTRF( UPLO, N, AFP, IPIV, INFO )
349: *
350: * Return if INFO is non-zero.
351: *
352: IF( INFO.GT.0 )THEN
353: RCOND = ZERO
354: RETURN
355: END IF
356: END IF
357: *
358: * Compute the norm of the matrix A.
359: *
360: ANORM = DLANSP( 'I', UPLO, N, AP, WORK )
361: *
362: * Compute the reciprocal of the condition number of A.
363: *
364: CALL DSPCON( UPLO, N, AFP, IPIV, ANORM, RCOND, WORK, IWORK, INFO )
365: *
366: * Compute the solution vectors X.
367: *
368: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
369: CALL DSPTRS( UPLO, N, NRHS, AFP, IPIV, X, LDX, INFO )
370: *
371: * Use iterative refinement to improve the computed solutions and
372: * compute error bounds and backward error estimates for them.
373: *
374: CALL DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR,
375: $ BERR, WORK, IWORK, INFO )
376: *
377: * Set INFO = N+1 if the matrix is singular to working precision.
378: *
379: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
380: $ INFO = N + 1
381: *
382: RETURN
383: *
384: * End of DSPSVX
385: *
386: END
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