Annotation of rpl/lapack/lapack/dsysvx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
! 2: $ LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
! 3: $ IWORK, INFO )
! 4: *
! 5: * -- LAPACK driver routine (version 3.2) --
! 6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 8: * November 2006
! 9: *
! 10: * .. Scalar Arguments ..
! 11: CHARACTER FACT, UPLO
! 12: INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
! 13: DOUBLE PRECISION RCOND
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IPIV( * ), IWORK( * )
! 17: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
! 18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * DSYSVX uses the diagonal pivoting factorization to compute the
! 25: * solution to a real system of linear equations A * X = B,
! 26: * where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
! 27: * matrices.
! 28: *
! 29: * Error bounds on the solution and a condition estimate are also
! 30: * provided.
! 31: *
! 32: * Description
! 33: * ===========
! 34: *
! 35: * The following steps are performed:
! 36: *
! 37: * 1. If FACT = 'N', the diagonal pivoting method is used to factor A.
! 38: * The form of the factorization is
! 39: * A = U * D * U**T, if UPLO = 'U', or
! 40: * A = L * D * L**T, if UPLO = 'L',
! 41: * where U (or L) is a product of permutation and unit upper (lower)
! 42: * triangular matrices, and D is symmetric and block diagonal with
! 43: * 1-by-1 and 2-by-2 diagonal blocks.
! 44: *
! 45: * 2. If some D(i,i)=0, so that D is exactly singular, then the routine
! 46: * returns with INFO = i. Otherwise, the factored form of A is used
! 47: * to estimate the condition number of the matrix A. If the
! 48: * reciprocal of the condition number is less than machine precision,
! 49: * INFO = N+1 is returned as a warning, but the routine still goes on
! 50: * to solve for X and compute error bounds as described below.
! 51: *
! 52: * 3. The system of equations is solved for X using the factored form
! 53: * of A.
! 54: *
! 55: * 4. Iterative refinement is applied to improve the computed solution
! 56: * matrix and calculate error bounds and backward error estimates
! 57: * for it.
! 58: *
! 59: * Arguments
! 60: * =========
! 61: *
! 62: * FACT (input) CHARACTER*1
! 63: * Specifies whether or not the factored form of A has been
! 64: * supplied on entry.
! 65: * = 'F': On entry, AF and IPIV contain the factored form of
! 66: * A. AF and IPIV will not be modified.
! 67: * = 'N': The matrix A will be copied to AF and factored.
! 68: *
! 69: * UPLO (input) CHARACTER*1
! 70: * = 'U': Upper triangle of A is stored;
! 71: * = 'L': Lower triangle of A is stored.
! 72: *
! 73: * N (input) INTEGER
! 74: * The number of linear equations, i.e., the order of the
! 75: * matrix A. N >= 0.
! 76: *
! 77: * NRHS (input) INTEGER
! 78: * The number of right hand sides, i.e., the number of columns
! 79: * of the matrices B and X. NRHS >= 0.
! 80: *
! 81: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
! 82: * The symmetric matrix A. If UPLO = 'U', the leading N-by-N
! 83: * upper triangular part of A contains the upper triangular part
! 84: * of the matrix A, and the strictly lower triangular part of A
! 85: * is not referenced. If UPLO = 'L', the leading N-by-N lower
! 86: * triangular part of A contains the lower triangular part of
! 87: * the matrix A, and the strictly upper triangular part of A is
! 88: * not referenced.
! 89: *
! 90: * LDA (input) INTEGER
! 91: * The leading dimension of the array A. LDA >= max(1,N).
! 92: *
! 93: * AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
! 94: * If FACT = 'F', then AF is an input argument and on entry
! 95: * contains the block diagonal matrix D and the multipliers used
! 96: * to obtain the factor U or L from the factorization
! 97: * A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.
! 98: *
! 99: * If FACT = 'N', then AF is an output argument and on exit
! 100: * returns the block diagonal matrix D and the multipliers used
! 101: * to obtain the factor U or L from the factorization
! 102: * A = U*D*U**T or A = L*D*L**T.
! 103: *
! 104: * LDAF (input) INTEGER
! 105: * The leading dimension of the array AF. LDAF >= max(1,N).
! 106: *
! 107: * IPIV (input or output) INTEGER array, dimension (N)
! 108: * If FACT = 'F', then IPIV is an input argument and on entry
! 109: * contains details of the interchanges and the block structure
! 110: * of D, as determined by DSYTRF.
! 111: * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
! 112: * interchanged and D(k,k) is a 1-by-1 diagonal block.
! 113: * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
! 114: * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
! 115: * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
! 116: * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
! 117: * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
! 118: *
! 119: * If FACT = 'N', then IPIV is an output argument and on exit
! 120: * contains details of the interchanges and the block structure
! 121: * of D, as determined by DSYTRF.
! 122: *
! 123: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 124: * The N-by-NRHS right hand side matrix B.
! 125: *
! 126: * LDB (input) INTEGER
! 127: * The leading dimension of the array B. LDB >= max(1,N).
! 128: *
! 129: * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 130: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
! 131: *
! 132: * LDX (input) INTEGER
! 133: * The leading dimension of the array X. LDX >= max(1,N).
! 134: *
! 135: * RCOND (output) DOUBLE PRECISION
! 136: * The estimate of the reciprocal condition number of the matrix
! 137: * A. If RCOND is less than the machine precision (in
! 138: * particular, if RCOND = 0), the matrix is singular to working
! 139: * precision. This condition is indicated by a return code of
! 140: * INFO > 0.
! 141: *
! 142: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 143: * The estimated forward error bound for each solution vector
! 144: * X(j) (the j-th column of the solution matrix X).
! 145: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 146: * is an estimated upper bound for the magnitude of the largest
! 147: * element in (X(j) - XTRUE) divided by the magnitude of the
! 148: * largest element in X(j). The estimate is as reliable as
! 149: * the estimate for RCOND, and is almost always a slight
! 150: * overestimate of the true error.
! 151: *
! 152: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 153: * The componentwise relative backward error of each solution
! 154: * vector X(j) (i.e., the smallest relative change in
! 155: * any element of A or B that makes X(j) an exact solution).
! 156: *
! 157: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 158: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 159: *
! 160: * LWORK (input) INTEGER
! 161: * The length of WORK. LWORK >= max(1,3*N), and for best
! 162: * performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where
! 163: * NB is the optimal blocksize for DSYTRF.
! 164: *
! 165: * If LWORK = -1, then a workspace query is assumed; the routine
! 166: * only calculates the optimal size of the WORK array, returns
! 167: * this value as the first entry of the WORK array, and no error
! 168: * message related to LWORK is issued by XERBLA.
! 169: *
! 170: * IWORK (workspace) INTEGER array, dimension (N)
! 171: *
! 172: * INFO (output) INTEGER
! 173: * = 0: successful exit
! 174: * < 0: if INFO = -i, the i-th argument had an illegal value
! 175: * > 0: if INFO = i, and i is
! 176: * <= N: D(i,i) is exactly zero. The factorization
! 177: * has been completed but the factor D is exactly
! 178: * singular, so the solution and error bounds could
! 179: * not be computed. RCOND = 0 is returned.
! 180: * = N+1: D is nonsingular, but RCOND is less than machine
! 181: * precision, meaning that the matrix is singular
! 182: * to working precision. Nevertheless, the
! 183: * solution and error bounds are computed because
! 184: * there are a number of situations where the
! 185: * computed solution can be more accurate than the
! 186: * value of RCOND would suggest.
! 187: *
! 188: * =====================================================================
! 189: *
! 190: * .. Parameters ..
! 191: DOUBLE PRECISION ZERO
! 192: PARAMETER ( ZERO = 0.0D+0 )
! 193: * ..
! 194: * .. Local Scalars ..
! 195: LOGICAL LQUERY, NOFACT
! 196: INTEGER LWKOPT, NB
! 197: DOUBLE PRECISION ANORM
! 198: * ..
! 199: * .. External Functions ..
! 200: LOGICAL LSAME
! 201: INTEGER ILAENV
! 202: DOUBLE PRECISION DLAMCH, DLANSY
! 203: EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
! 204: * ..
! 205: * .. External Subroutines ..
! 206: EXTERNAL DLACPY, DSYCON, DSYRFS, DSYTRF, DSYTRS, XERBLA
! 207: * ..
! 208: * .. Intrinsic Functions ..
! 209: INTRINSIC MAX
! 210: * ..
! 211: * .. Executable Statements ..
! 212: *
! 213: * Test the input parameters.
! 214: *
! 215: INFO = 0
! 216: NOFACT = LSAME( FACT, 'N' )
! 217: LQUERY = ( LWORK.EQ.-1 )
! 218: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
! 219: INFO = -1
! 220: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
! 221: $ THEN
! 222: INFO = -2
! 223: ELSE IF( N.LT.0 ) THEN
! 224: INFO = -3
! 225: ELSE IF( NRHS.LT.0 ) THEN
! 226: INFO = -4
! 227: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 228: INFO = -6
! 229: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
! 230: INFO = -8
! 231: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 232: INFO = -11
! 233: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 234: INFO = -13
! 235: ELSE IF( LWORK.LT.MAX( 1, 3*N ) .AND. .NOT.LQUERY ) THEN
! 236: INFO = -18
! 237: END IF
! 238: *
! 239: IF( INFO.EQ.0 ) THEN
! 240: LWKOPT = MAX( 1, 3*N )
! 241: IF( NOFACT ) THEN
! 242: NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 )
! 243: LWKOPT = MAX( LWKOPT, N*NB )
! 244: END IF
! 245: WORK( 1 ) = LWKOPT
! 246: END IF
! 247: *
! 248: IF( INFO.NE.0 ) THEN
! 249: CALL XERBLA( 'DSYSVX', -INFO )
! 250: RETURN
! 251: ELSE IF( LQUERY ) THEN
! 252: RETURN
! 253: END IF
! 254: *
! 255: IF( NOFACT ) THEN
! 256: *
! 257: * Compute the factorization A = U*D*U' or A = L*D*L'.
! 258: *
! 259: CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
! 260: CALL DSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO )
! 261: *
! 262: * Return if INFO is non-zero.
! 263: *
! 264: IF( INFO.GT.0 )THEN
! 265: RCOND = ZERO
! 266: RETURN
! 267: END IF
! 268: END IF
! 269: *
! 270: * Compute the norm of the matrix A.
! 271: *
! 272: ANORM = DLANSY( 'I', UPLO, N, A, LDA, WORK )
! 273: *
! 274: * Compute the reciprocal of the condition number of A.
! 275: *
! 276: CALL DSYCON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, IWORK,
! 277: $ INFO )
! 278: *
! 279: * Compute the solution vectors X.
! 280: *
! 281: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
! 282: CALL DSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
! 283: *
! 284: * Use iterative refinement to improve the computed solutions and
! 285: * compute error bounds and backward error estimates for them.
! 286: *
! 287: CALL DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
! 288: $ LDX, FERR, BERR, WORK, IWORK, INFO )
! 289: *
! 290: * Set INFO = N+1 if the matrix is singular to working precision.
! 291: *
! 292: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
! 293: $ INFO = N + 1
! 294: *
! 295: WORK( 1 ) = LWKOPT
! 296: *
! 297: RETURN
! 298: *
! 299: * End of DSYSVX
! 300: *
! 301: END
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