Annotation of rpl/lapack/lapack/dgtsvx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
! 2: $ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
! 3: $ WORK, IWORK, INFO )
! 4: *
! 5: * -- LAPACK 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, TRANS
! 12: INTEGER INFO, LDB, LDX, N, NRHS
! 13: DOUBLE PRECISION RCOND
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IPIV( * ), IWORK( * )
! 17: DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
! 18: $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
! 19: $ FERR( * ), WORK( * ), X( LDX, * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * DGTSVX uses the LU factorization to compute the solution to a real
! 26: * system of linear equations A * X = B or A**T * X = B,
! 27: * where A is a tridiagonal matrix of order N 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 LU decomposition is used to factor the matrix A
! 39: * as A = L * U, where L is a product of permutation and unit lower
! 40: * bidiagonal matrices and U is upper triangular with nonzeros in
! 41: * only the main diagonal and first two superdiagonals.
! 42: *
! 43: * 2. If some U(i,i)=0, so that U is exactly singular, then the routine
! 44: * returns with INFO = i. Otherwise, the factored form of A is used
! 45: * to estimate the condition number of the matrix A. If the
! 46: * reciprocal of the condition number is less than machine precision,
! 47: * INFO = N+1 is returned as a warning, but the routine still goes on
! 48: * to solve for X and compute error bounds as 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 A has been
! 62: * supplied on entry.
! 63: * = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored
! 64: * form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
! 65: * will not be modified.
! 66: * = 'N': The matrix will be copied to DLF, DF, and DUF
! 67: * and factored.
! 68: *
! 69: * TRANS (input) CHARACTER*1
! 70: * Specifies the form of the system of equations:
! 71: * = 'N': A * X = B (No transpose)
! 72: * = 'T': A**T * X = B (Transpose)
! 73: * = 'C': A**H * X = B (Conjugate transpose = Transpose)
! 74: *
! 75: * N (input) INTEGER
! 76: * The order of the 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 matrix B. NRHS >= 0.
! 81: *
! 82: * DL (input) DOUBLE PRECISION array, dimension (N-1)
! 83: * The (n-1) subdiagonal elements of A.
! 84: *
! 85: * D (input) DOUBLE PRECISION array, dimension (N)
! 86: * The n diagonal elements of A.
! 87: *
! 88: * DU (input) DOUBLE PRECISION array, dimension (N-1)
! 89: * The (n-1) superdiagonal elements of A.
! 90: *
! 91: * DLF (input or output) DOUBLE PRECISION array, dimension (N-1)
! 92: * If FACT = 'F', then DLF is an input argument and on entry
! 93: * contains the (n-1) multipliers that define the matrix L from
! 94: * the LU factorization of A as computed by DGTTRF.
! 95: *
! 96: * If FACT = 'N', then DLF is an output argument and on exit
! 97: * contains the (n-1) multipliers that define the matrix L from
! 98: * the LU factorization of A.
! 99: *
! 100: * DF (input or output) DOUBLE PRECISION array, dimension (N)
! 101: * If FACT = 'F', then DF is an input argument and on entry
! 102: * contains the n diagonal elements of the upper triangular
! 103: * matrix U from the LU factorization of A.
! 104: *
! 105: * If FACT = 'N', then DF is an output argument and on exit
! 106: * contains the n diagonal elements of the upper triangular
! 107: * matrix U from the LU factorization of A.
! 108: *
! 109: * DUF (input or output) DOUBLE PRECISION array, dimension (N-1)
! 110: * If FACT = 'F', then DUF is an input argument and on entry
! 111: * contains the (n-1) elements of the first superdiagonal of U.
! 112: *
! 113: * If FACT = 'N', then DUF is an output argument and on exit
! 114: * contains the (n-1) elements of the first superdiagonal of U.
! 115: *
! 116: * DU2 (input or output) DOUBLE PRECISION array, dimension (N-2)
! 117: * If FACT = 'F', then DU2 is an input argument and on entry
! 118: * contains the (n-2) elements of the second superdiagonal of
! 119: * U.
! 120: *
! 121: * If FACT = 'N', then DU2 is an output argument and on exit
! 122: * contains the (n-2) elements of the second superdiagonal of
! 123: * U.
! 124: *
! 125: * IPIV (input or output) INTEGER array, dimension (N)
! 126: * If FACT = 'F', then IPIV is an input argument and on entry
! 127: * contains the pivot indices from the LU factorization of A as
! 128: * computed by DGTTRF.
! 129: *
! 130: * If FACT = 'N', then IPIV is an output argument and on exit
! 131: * contains the pivot indices from the LU factorization of A;
! 132: * row i of the matrix was interchanged with row IPIV(i).
! 133: * IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
! 134: * a row interchange was not required.
! 135: *
! 136: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 137: * The N-by-NRHS right hand side matrix B.
! 138: *
! 139: * LDB (input) INTEGER
! 140: * The leading dimension of the array B. LDB >= max(1,N).
! 141: *
! 142: * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 143: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
! 144: *
! 145: * LDX (input) INTEGER
! 146: * The leading dimension of the array X. LDX >= max(1,N).
! 147: *
! 148: * RCOND (output) DOUBLE PRECISION
! 149: * The estimate of the reciprocal condition number of the matrix
! 150: * A. If RCOND is less than the machine precision (in
! 151: * particular, if RCOND = 0), the matrix is singular to working
! 152: * precision. This condition is indicated by a return code of
! 153: * INFO > 0.
! 154: *
! 155: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 156: * The estimated forward error bound for each solution vector
! 157: * X(j) (the j-th column of the solution matrix X).
! 158: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 159: * is an estimated upper bound for the magnitude of the largest
! 160: * element in (X(j) - XTRUE) divided by the magnitude of the
! 161: * largest element in X(j). The estimate is as reliable as
! 162: * the estimate for RCOND, and is almost always a slight
! 163: * overestimate of the true error.
! 164: *
! 165: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 166: * The componentwise relative backward error of each solution
! 167: * vector X(j) (i.e., the smallest relative change in
! 168: * any element of A or B that makes X(j) an exact solution).
! 169: *
! 170: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
! 171: *
! 172: * IWORK (workspace) INTEGER array, dimension (N)
! 173: *
! 174: * INFO (output) INTEGER
! 175: * = 0: successful exit
! 176: * < 0: if INFO = -i, the i-th argument had an illegal value
! 177: * > 0: if INFO = i, and i is
! 178: * <= N: U(i,i) is exactly zero. The factorization
! 179: * has not been completed unless i = N, but the
! 180: * factor U is exactly singular, so the solution
! 181: * and error bounds could not be computed.
! 182: * RCOND = 0 is returned.
! 183: * = N+1: U is nonsingular, but RCOND is less than machine
! 184: * precision, meaning that the matrix is singular
! 185: * to working precision. Nevertheless, the
! 186: * solution and error bounds are computed because
! 187: * there are a number of situations where the
! 188: * computed solution can be more accurate than the
! 189: * value of RCOND would suggest.
! 190: *
! 191: * =====================================================================
! 192: *
! 193: * .. Parameters ..
! 194: DOUBLE PRECISION ZERO
! 195: PARAMETER ( ZERO = 0.0D+0 )
! 196: * ..
! 197: * .. Local Scalars ..
! 198: LOGICAL NOFACT, NOTRAN
! 199: CHARACTER NORM
! 200: DOUBLE PRECISION ANORM
! 201: * ..
! 202: * .. External Functions ..
! 203: LOGICAL LSAME
! 204: DOUBLE PRECISION DLAMCH, DLANGT
! 205: EXTERNAL LSAME, DLAMCH, DLANGT
! 206: * ..
! 207: * .. External Subroutines ..
! 208: EXTERNAL DCOPY, DGTCON, DGTRFS, DGTTRF, DGTTRS, DLACPY,
! 209: $ XERBLA
! 210: * ..
! 211: * .. Intrinsic Functions ..
! 212: INTRINSIC MAX
! 213: * ..
! 214: * .. Executable Statements ..
! 215: *
! 216: INFO = 0
! 217: NOFACT = LSAME( FACT, 'N' )
! 218: NOTRAN = LSAME( TRANS, 'N' )
! 219: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
! 220: INFO = -1
! 221: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
! 222: $ LSAME( TRANS, 'C' ) ) 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( LDB.LT.MAX( 1, N ) ) THEN
! 229: INFO = -14
! 230: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 231: INFO = -16
! 232: END IF
! 233: IF( INFO.NE.0 ) THEN
! 234: CALL XERBLA( 'DGTSVX', -INFO )
! 235: RETURN
! 236: END IF
! 237: *
! 238: IF( NOFACT ) THEN
! 239: *
! 240: * Compute the LU factorization of A.
! 241: *
! 242: CALL DCOPY( N, D, 1, DF, 1 )
! 243: IF( N.GT.1 ) THEN
! 244: CALL DCOPY( N-1, DL, 1, DLF, 1 )
! 245: CALL DCOPY( N-1, DU, 1, DUF, 1 )
! 246: END IF
! 247: CALL DGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO )
! 248: *
! 249: * Return if INFO is non-zero.
! 250: *
! 251: IF( INFO.GT.0 )THEN
! 252: RCOND = ZERO
! 253: RETURN
! 254: END IF
! 255: END IF
! 256: *
! 257: * Compute the norm of the matrix A.
! 258: *
! 259: IF( NOTRAN ) THEN
! 260: NORM = '1'
! 261: ELSE
! 262: NORM = 'I'
! 263: END IF
! 264: ANORM = DLANGT( NORM, N, DL, D, DU )
! 265: *
! 266: * Compute the reciprocal of the condition number of A.
! 267: *
! 268: CALL DGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
! 269: $ IWORK, INFO )
! 270: *
! 271: * Compute the solution vectors X.
! 272: *
! 273: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
! 274: CALL DGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
! 275: $ INFO )
! 276: *
! 277: * Use iterative refinement to improve the computed solutions and
! 278: * compute error bounds and backward error estimates for them.
! 279: *
! 280: CALL DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
! 281: $ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
! 282: *
! 283: * Set INFO = N+1 if the matrix is singular to working precision.
! 284: *
! 285: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
! 286: $ INFO = N + 1
! 287: *
! 288: RETURN
! 289: *
! 290: * End of DGTSVX
! 291: *
! 292: END
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