Annotation of rpl/lapack/lapack/zptsvx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
! 2: $ RCOND, FERR, BERR, WORK, RWORK, INFO )
! 3: *
! 4: * -- LAPACK routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * .. Scalar Arguments ..
! 10: CHARACTER FACT
! 11: INTEGER INFO, LDB, LDX, N, NRHS
! 12: DOUBLE PRECISION RCOND
! 13: * ..
! 14: * .. Array Arguments ..
! 15: DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
! 16: $ RWORK( * )
! 17: COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
! 18: $ X( LDX, * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * ZPTSVX uses the factorization A = L*D*L**H to compute the solution
! 25: * to a complex system of linear equations A*X = B, where A is an
! 26: * N-by-N Hermitian positive definite tridiagonal matrix and X and B
! 27: * are N-by-NRHS 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 matrix A is factored as A = L*D*L**H, where L
! 38: * is a unit lower bidiagonal matrix and D is diagonal. The
! 39: * factorization can also be regarded as having the form
! 40: * A = U**H*D*U.
! 41: *
! 42: * 2. If the leading i-by-i principal minor is not positive definite,
! 43: * then the routine returns with INFO = i. Otherwise, the factored
! 44: * form of A is used to estimate the condition number of the matrix
! 45: * A. If the reciprocal of the condition number is less than machine
! 46: * precision, INFO = N+1 is returned as a warning, but the routine
! 47: * still goes on to solve for X and compute error bounds as
! 48: * described below.
! 49: *
! 50: * 3. The system of equations is solved for X using the factored form
! 51: * of A.
! 52: *
! 53: * 4. Iterative refinement is applied to improve the computed solution
! 54: * matrix and calculate error bounds and backward error estimates
! 55: * for it.
! 56: *
! 57: * Arguments
! 58: * =========
! 59: *
! 60: * FACT (input) CHARACTER*1
! 61: * Specifies whether or not the factored form of the matrix
! 62: * A is supplied on entry.
! 63: * = 'F': On entry, DF and EF contain the factored form of A.
! 64: * D, E, DF, and EF will not be modified.
! 65: * = 'N': The matrix A will be copied to DF and EF and
! 66: * factored.
! 67: *
! 68: * N (input) INTEGER
! 69: * The order of the matrix A. N >= 0.
! 70: *
! 71: * NRHS (input) INTEGER
! 72: * The number of right hand sides, i.e., the number of columns
! 73: * of the matrices B and X. NRHS >= 0.
! 74: *
! 75: * D (input) DOUBLE PRECISION array, dimension (N)
! 76: * The n diagonal elements of the tridiagonal matrix A.
! 77: *
! 78: * E (input) COMPLEX*16 array, dimension (N-1)
! 79: * The (n-1) subdiagonal elements of the tridiagonal matrix A.
! 80: *
! 81: * DF (input or output) DOUBLE PRECISION array, dimension (N)
! 82: * If FACT = 'F', then DF is an input argument and on entry
! 83: * contains the n diagonal elements of the diagonal matrix D
! 84: * from the L*D*L**H factorization of A.
! 85: * If FACT = 'N', then DF is an output argument and on exit
! 86: * contains the n diagonal elements of the diagonal matrix D
! 87: * from the L*D*L**H factorization of A.
! 88: *
! 89: * EF (input or output) COMPLEX*16 array, dimension (N-1)
! 90: * If FACT = 'F', then EF is an input argument and on entry
! 91: * contains the (n-1) subdiagonal elements of the unit
! 92: * bidiagonal factor L from the L*D*L**H factorization of A.
! 93: * If FACT = 'N', then EF is an output argument and on exit
! 94: * contains the (n-1) subdiagonal elements of the unit
! 95: * bidiagonal factor L from the L*D*L**H factorization of A.
! 96: *
! 97: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
! 98: * The N-by-NRHS right hand side matrix B.
! 99: *
! 100: * LDB (input) INTEGER
! 101: * The leading dimension of the array B. LDB >= max(1,N).
! 102: *
! 103: * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
! 104: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
! 105: *
! 106: * LDX (input) INTEGER
! 107: * The leading dimension of the array X. LDX >= max(1,N).
! 108: *
! 109: * RCOND (output) DOUBLE PRECISION
! 110: * The reciprocal condition number of the matrix A. If RCOND
! 111: * is less than the machine precision (in particular, if
! 112: * RCOND = 0), the matrix is singular to working precision.
! 113: * This condition is indicated by a return code of INFO > 0.
! 114: *
! 115: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 116: * The forward error bound for each solution vector
! 117: * X(j) (the j-th column of the solution matrix X).
! 118: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 119: * is an estimated upper bound for the magnitude of the largest
! 120: * element in (X(j) - XTRUE) divided by the magnitude of the
! 121: * largest element in X(j).
! 122: *
! 123: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 124: * The componentwise relative backward error of each solution
! 125: * vector X(j) (i.e., the smallest relative change in any
! 126: * element of A or B that makes X(j) an exact solution).
! 127: *
! 128: * WORK (workspace) COMPLEX*16 array, dimension (N)
! 129: *
! 130: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
! 131: *
! 132: * INFO (output) INTEGER
! 133: * = 0: successful exit
! 134: * < 0: if INFO = -i, the i-th argument had an illegal value
! 135: * > 0: if INFO = i, and i is
! 136: * <= N: the leading minor of order i of A is
! 137: * not positive definite, so the factorization
! 138: * could not be completed, and the solution has not
! 139: * been computed. RCOND = 0 is returned.
! 140: * = N+1: U is nonsingular, but RCOND is less than machine
! 141: * precision, meaning that the matrix is singular
! 142: * to working precision. Nevertheless, the
! 143: * solution and error bounds are computed because
! 144: * there are a number of situations where the
! 145: * computed solution can be more accurate than the
! 146: * value of RCOND would suggest.
! 147: *
! 148: * =====================================================================
! 149: *
! 150: * .. Parameters ..
! 151: DOUBLE PRECISION ZERO
! 152: PARAMETER ( ZERO = 0.0D+0 )
! 153: * ..
! 154: * .. Local Scalars ..
! 155: LOGICAL NOFACT
! 156: DOUBLE PRECISION ANORM
! 157: * ..
! 158: * .. External Functions ..
! 159: LOGICAL LSAME
! 160: DOUBLE PRECISION DLAMCH, ZLANHT
! 161: EXTERNAL LSAME, DLAMCH, ZLANHT
! 162: * ..
! 163: * .. External Subroutines ..
! 164: EXTERNAL DCOPY, XERBLA, ZCOPY, ZLACPY, ZPTCON, ZPTRFS,
! 165: $ ZPTTRF, ZPTTRS
! 166: * ..
! 167: * .. Intrinsic Functions ..
! 168: INTRINSIC MAX
! 169: * ..
! 170: * .. Executable Statements ..
! 171: *
! 172: * Test the input parameters.
! 173: *
! 174: INFO = 0
! 175: NOFACT = LSAME( FACT, 'N' )
! 176: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
! 177: INFO = -1
! 178: ELSE IF( N.LT.0 ) THEN
! 179: INFO = -2
! 180: ELSE IF( NRHS.LT.0 ) THEN
! 181: INFO = -3
! 182: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 183: INFO = -9
! 184: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 185: INFO = -11
! 186: END IF
! 187: IF( INFO.NE.0 ) THEN
! 188: CALL XERBLA( 'ZPTSVX', -INFO )
! 189: RETURN
! 190: END IF
! 191: *
! 192: IF( NOFACT ) THEN
! 193: *
! 194: * Compute the L*D*L' (or U'*D*U) factorization of A.
! 195: *
! 196: CALL DCOPY( N, D, 1, DF, 1 )
! 197: IF( N.GT.1 )
! 198: $ CALL ZCOPY( N-1, E, 1, EF, 1 )
! 199: CALL ZPTTRF( N, DF, EF, INFO )
! 200: *
! 201: * Return if INFO is non-zero.
! 202: *
! 203: IF( INFO.GT.0 )THEN
! 204: RCOND = ZERO
! 205: RETURN
! 206: END IF
! 207: END IF
! 208: *
! 209: * Compute the norm of the matrix A.
! 210: *
! 211: ANORM = ZLANHT( '1', N, D, E )
! 212: *
! 213: * Compute the reciprocal of the condition number of A.
! 214: *
! 215: CALL ZPTCON( N, DF, EF, ANORM, RCOND, RWORK, INFO )
! 216: *
! 217: * Compute the solution vectors X.
! 218: *
! 219: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
! 220: CALL ZPTTRS( 'Lower', N, NRHS, DF, EF, X, LDX, INFO )
! 221: *
! 222: * Use iterative refinement to improve the computed solutions and
! 223: * compute error bounds and backward error estimates for them.
! 224: *
! 225: CALL ZPTRFS( 'Lower', N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
! 226: $ BERR, WORK, RWORK, INFO )
! 227: *
! 228: * Set INFO = N+1 if the matrix is singular to working precision.
! 229: *
! 230: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
! 231: $ INFO = N + 1
! 232: *
! 233: RETURN
! 234: *
! 235: * End of ZPTSVX
! 236: *
! 237: END
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