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