Annotation of rpl/lapack/lapack/dptsvx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
! 2: $ RCOND, FERR, BERR, WORK, 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 B( LDB, * ), BERR( * ), D( * ), DF( * ),
! 16: $ E( * ), EF( * ), FERR( * ), WORK( * ),
! 17: $ X( LDX, * )
! 18: * ..
! 19: *
! 20: * Purpose
! 21: * =======
! 22: *
! 23: * DPTSVX uses the factorization A = L*D*L**T to compute the solution
! 24: * to a real system of linear equations A*X = B, where A is an N-by-N
! 25: * symmetric positive definite tridiagonal matrix and X and B are
! 26: * N-by-NRHS matrices.
! 27: *
! 28: * Error bounds on the solution and a condition estimate are also
! 29: * provided.
! 30: *
! 31: * Description
! 32: * ===========
! 33: *
! 34: * The following steps are performed:
! 35: *
! 36: * 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
! 37: * is a unit lower bidiagonal matrix and D is diagonal. The
! 38: * factorization can also be regarded as having the form
! 39: * A = U**T*D*U.
! 40: *
! 41: * 2. If the leading i-by-i principal minor is not positive definite,
! 42: * then the routine returns with INFO = i. Otherwise, the factored
! 43: * form of A is used to estimate the condition number of the matrix
! 44: * A. If the reciprocal of the condition number is less than machine
! 45: * precision, INFO = N+1 is returned as a warning, but the routine
! 46: * still goes on to solve for X and compute error bounds as
! 47: * described below.
! 48: *
! 49: * 3. The system of equations is solved for X using the factored form
! 50: * of A.
! 51: *
! 52: * 4. Iterative refinement is applied to improve the computed solution
! 53: * matrix and calculate error bounds and backward error estimates
! 54: * for it.
! 55: *
! 56: * Arguments
! 57: * =========
! 58: *
! 59: * FACT (input) CHARACTER*1
! 60: * Specifies whether or not the factored form of A has been
! 61: * supplied on entry.
! 62: * = 'F': On entry, DF and EF contain the factored form of A.
! 63: * D, E, DF, and EF will not be modified.
! 64: * = 'N': The matrix A will be copied to DF and EF and
! 65: * factored.
! 66: *
! 67: * N (input) INTEGER
! 68: * The order of the matrix A. N >= 0.
! 69: *
! 70: * NRHS (input) INTEGER
! 71: * The number of right hand sides, i.e., the number of columns
! 72: * of the matrices B and X. NRHS >= 0.
! 73: *
! 74: * D (input) DOUBLE PRECISION array, dimension (N)
! 75: * The n diagonal elements of the tridiagonal matrix A.
! 76: *
! 77: * E (input) DOUBLE PRECISION array, dimension (N-1)
! 78: * The (n-1) subdiagonal elements of the tridiagonal matrix A.
! 79: *
! 80: * DF (input or output) DOUBLE PRECISION array, dimension (N)
! 81: * If FACT = 'F', then DF is an input argument and on entry
! 82: * contains the n diagonal elements of the diagonal matrix D
! 83: * from the L*D*L**T factorization of A.
! 84: * If FACT = 'N', then DF is an output argument and on exit
! 85: * contains the n diagonal elements of the diagonal matrix D
! 86: * from the L*D*L**T factorization of A.
! 87: *
! 88: * EF (input or output) DOUBLE PRECISION array, dimension (N-1)
! 89: * If FACT = 'F', then EF is an input argument and on entry
! 90: * contains the (n-1) subdiagonal elements of the unit
! 91: * bidiagonal factor L from the L*D*L**T factorization of A.
! 92: * If FACT = 'N', then EF is an output argument and on exit
! 93: * contains the (n-1) subdiagonal elements of the unit
! 94: * bidiagonal factor L from the L*D*L**T factorization of A.
! 95: *
! 96: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 97: * The N-by-NRHS right hand side matrix B.
! 98: *
! 99: * LDB (input) INTEGER
! 100: * The leading dimension of the array B. LDB >= max(1,N).
! 101: *
! 102: * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 103: * If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.
! 104: *
! 105: * LDX (input) INTEGER
! 106: * The leading dimension of the array X. LDX >= max(1,N).
! 107: *
! 108: * RCOND (output) DOUBLE PRECISION
! 109: * The reciprocal condition number of the matrix A. If RCOND
! 110: * is less than the machine precision (in particular, if
! 111: * RCOND = 0), the matrix is singular to working precision.
! 112: * This condition is indicated by a return code of INFO > 0.
! 113: *
! 114: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 115: * The forward error bound for each solution vector
! 116: * X(j) (the j-th column of the solution matrix X).
! 117: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 118: * is an estimated upper bound for the magnitude of the largest
! 119: * element in (X(j) - XTRUE) divided by the magnitude of the
! 120: * largest element in X(j).
! 121: *
! 122: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 123: * The componentwise relative backward error of each solution
! 124: * vector X(j) (i.e., the smallest relative change in any
! 125: * element of A or B that makes X(j) an exact solution).
! 126: *
! 127: * WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
! 128: *
! 129: * INFO (output) INTEGER
! 130: * = 0: successful exit
! 131: * < 0: if INFO = -i, the i-th argument had an illegal value
! 132: * > 0: if INFO = i, and i is
! 133: * <= N: the leading minor of order i of A is
! 134: * not positive definite, so the factorization
! 135: * could not be completed, and the solution has not
! 136: * been computed. RCOND = 0 is returned.
! 137: * = N+1: U is nonsingular, but RCOND is less than machine
! 138: * precision, meaning that the matrix is singular
! 139: * to working precision. Nevertheless, the
! 140: * solution and error bounds are computed because
! 141: * there are a number of situations where the
! 142: * computed solution can be more accurate than the
! 143: * value of RCOND would suggest.
! 144: *
! 145: * =====================================================================
! 146: *
! 147: * .. Parameters ..
! 148: DOUBLE PRECISION ZERO
! 149: PARAMETER ( ZERO = 0.0D+0 )
! 150: * ..
! 151: * .. Local Scalars ..
! 152: LOGICAL NOFACT
! 153: DOUBLE PRECISION ANORM
! 154: * ..
! 155: * .. External Functions ..
! 156: LOGICAL LSAME
! 157: DOUBLE PRECISION DLAMCH, DLANST
! 158: EXTERNAL LSAME, DLAMCH, DLANST
! 159: * ..
! 160: * .. External Subroutines ..
! 161: EXTERNAL DCOPY, DLACPY, DPTCON, DPTRFS, DPTTRF, DPTTRS,
! 162: $ XERBLA
! 163: * ..
! 164: * .. Intrinsic Functions ..
! 165: INTRINSIC MAX
! 166: * ..
! 167: * .. Executable Statements ..
! 168: *
! 169: * Test the input parameters.
! 170: *
! 171: INFO = 0
! 172: NOFACT = LSAME( FACT, 'N' )
! 173: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
! 174: INFO = -1
! 175: ELSE IF( N.LT.0 ) THEN
! 176: INFO = -2
! 177: ELSE IF( NRHS.LT.0 ) THEN
! 178: INFO = -3
! 179: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 180: INFO = -9
! 181: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 182: INFO = -11
! 183: END IF
! 184: IF( INFO.NE.0 ) THEN
! 185: CALL XERBLA( 'DPTSVX', -INFO )
! 186: RETURN
! 187: END IF
! 188: *
! 189: IF( NOFACT ) THEN
! 190: *
! 191: * Compute the L*D*L' (or U'*D*U) factorization of A.
! 192: *
! 193: CALL DCOPY( N, D, 1, DF, 1 )
! 194: IF( N.GT.1 )
! 195: $ CALL DCOPY( N-1, E, 1, EF, 1 )
! 196: CALL DPTTRF( N, DF, EF, INFO )
! 197: *
! 198: * Return if INFO is non-zero.
! 199: *
! 200: IF( INFO.GT.0 )THEN
! 201: RCOND = ZERO
! 202: RETURN
! 203: END IF
! 204: END IF
! 205: *
! 206: * Compute the norm of the matrix A.
! 207: *
! 208: ANORM = DLANST( '1', N, D, E )
! 209: *
! 210: * Compute the reciprocal of the condition number of A.
! 211: *
! 212: CALL DPTCON( N, DF, EF, ANORM, RCOND, WORK, INFO )
! 213: *
! 214: * Compute the solution vectors X.
! 215: *
! 216: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
! 217: CALL DPTTRS( N, NRHS, DF, EF, X, LDX, INFO )
! 218: *
! 219: * Use iterative refinement to improve the computed solutions and
! 220: * compute error bounds and backward error estimates for them.
! 221: *
! 222: CALL DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR,
! 223: $ WORK, INFO )
! 224: *
! 225: * Set INFO = N+1 if the matrix is singular to working precision.
! 226: *
! 227: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
! 228: $ INFO = N + 1
! 229: *
! 230: RETURN
! 231: *
! 232: * End of DPTSVX
! 233: *
! 234: END
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