Annotation of rpl/lapack/lapack/zgtsvx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
! 2: $ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
! 3: $ WORK, RWORK, 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( * )
! 17: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
! 18: COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
! 19: $ DLF( * ), DU( * ), DU2( * ), DUF( * ),
! 20: $ WORK( * ), X( LDX, * )
! 21: * ..
! 22: *
! 23: * Purpose
! 24: * =======
! 25: *
! 26: * ZGTSVX uses the LU factorization to compute the solution to a complex
! 27: * system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
! 28: * where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
! 29: * matrices.
! 30: *
! 31: * Error bounds on the solution and a condition estimate are also
! 32: * provided.
! 33: *
! 34: * Description
! 35: * ===========
! 36: *
! 37: * The following steps are performed:
! 38: *
! 39: * 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
! 40: * as A = L * U, where L is a product of permutation and unit lower
! 41: * bidiagonal matrices and U is upper triangular with nonzeros in
! 42: * only the main diagonal and first two superdiagonals.
! 43: *
! 44: * 2. If some U(i,i)=0, so that U is exactly singular, then the routine
! 45: * returns with INFO = i. Otherwise, the factored form of A is used
! 46: * to estimate the condition number of the matrix A. If the
! 47: * reciprocal of the condition number is less than machine precision,
! 48: * INFO = N+1 is returned as a warning, but the routine still goes on
! 49: * to solve for X and compute error bounds as described below.
! 50: *
! 51: * 3. The system of equations is solved for X using the factored form
! 52: * of A.
! 53: *
! 54: * 4. Iterative refinement is applied to improve the computed solution
! 55: * matrix and calculate error bounds and backward error estimates
! 56: * for it.
! 57: *
! 58: * Arguments
! 59: * =========
! 60: *
! 61: * FACT (input) CHARACTER*1
! 62: * Specifies whether or not the factored form of A has been
! 63: * supplied on entry.
! 64: * = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored form
! 65: * of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
! 66: * be modified.
! 67: * = 'N': The matrix will be copied to DLF, DF, and DUF
! 68: * and factored.
! 69: *
! 70: * TRANS (input) CHARACTER*1
! 71: * Specifies the form of the system of equations:
! 72: * = 'N': A * X = B (No transpose)
! 73: * = 'T': A**T * X = B (Transpose)
! 74: * = 'C': A**H * X = B (Conjugate transpose)
! 75: *
! 76: * N (input) INTEGER
! 77: * The order of the matrix A. N >= 0.
! 78: *
! 79: * NRHS (input) INTEGER
! 80: * The number of right hand sides, i.e., the number of columns
! 81: * of the matrix B. NRHS >= 0.
! 82: *
! 83: * DL (input) COMPLEX*16 array, dimension (N-1)
! 84: * The (n-1) subdiagonal elements of A.
! 85: *
! 86: * D (input) COMPLEX*16 array, dimension (N)
! 87: * The n diagonal elements of A.
! 88: *
! 89: * DU (input) COMPLEX*16 array, dimension (N-1)
! 90: * The (n-1) superdiagonal elements of A.
! 91: *
! 92: * DLF (input or output) COMPLEX*16 array, dimension (N-1)
! 93: * If FACT = 'F', then DLF is an input argument and on entry
! 94: * contains the (n-1) multipliers that define the matrix L from
! 95: * the LU factorization of A as computed by ZGTTRF.
! 96: *
! 97: * If FACT = 'N', then DLF is an output argument and on exit
! 98: * contains the (n-1) multipliers that define the matrix L from
! 99: * the LU factorization of A.
! 100: *
! 101: * DF (input or output) COMPLEX*16 array, dimension (N)
! 102: * If FACT = 'F', then DF is an input argument and on entry
! 103: * contains the n diagonal elements of the upper triangular
! 104: * matrix U from the LU factorization of A.
! 105: *
! 106: * If FACT = 'N', then DF is an output argument and on exit
! 107: * contains the n diagonal elements of the upper triangular
! 108: * matrix U from the LU factorization of A.
! 109: *
! 110: * DUF (input or output) COMPLEX*16 array, dimension (N-1)
! 111: * If FACT = 'F', then DUF is an input argument and on entry
! 112: * contains the (n-1) elements of the first superdiagonal of U.
! 113: *
! 114: * If FACT = 'N', then DUF is an output argument and on exit
! 115: * contains the (n-1) elements of the first superdiagonal of U.
! 116: *
! 117: * DU2 (input or output) COMPLEX*16 array, dimension (N-2)
! 118: * If FACT = 'F', then DU2 is an input argument and on entry
! 119: * contains the (n-2) elements of the second superdiagonal of
! 120: * U.
! 121: *
! 122: * If FACT = 'N', then DU2 is an output argument and on exit
! 123: * contains the (n-2) elements of the second superdiagonal of
! 124: * U.
! 125: *
! 126: * IPIV (input or output) INTEGER array, dimension (N)
! 127: * If FACT = 'F', then IPIV is an input argument and on entry
! 128: * contains the pivot indices from the LU factorization of A as
! 129: * computed by ZGTTRF.
! 130: *
! 131: * If FACT = 'N', then IPIV is an output argument and on exit
! 132: * contains the pivot indices from the LU factorization of A;
! 133: * row i of the matrix was interchanged with row IPIV(i).
! 134: * IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
! 135: * a row interchange was not required.
! 136: *
! 137: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
! 138: * The N-by-NRHS right hand side matrix B.
! 139: *
! 140: * LDB (input) INTEGER
! 141: * The leading dimension of the array B. LDB >= max(1,N).
! 142: *
! 143: * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
! 144: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
! 145: *
! 146: * LDX (input) INTEGER
! 147: * The leading dimension of the array X. LDX >= max(1,N).
! 148: *
! 149: * RCOND (output) DOUBLE PRECISION
! 150: * The estimate of the reciprocal condition number of the matrix
! 151: * A. If RCOND is less than the machine precision (in
! 152: * particular, if RCOND = 0), the matrix is singular to working
! 153: * precision. This condition is indicated by a return code of
! 154: * INFO > 0.
! 155: *
! 156: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 157: * The estimated forward error bound for each solution vector
! 158: * X(j) (the j-th column of the solution matrix X).
! 159: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 160: * is an estimated upper bound for the magnitude of the largest
! 161: * element in (X(j) - XTRUE) divided by the magnitude of the
! 162: * largest element in X(j). The estimate is as reliable as
! 163: * the estimate for RCOND, and is almost always a slight
! 164: * overestimate of the true error.
! 165: *
! 166: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 167: * The componentwise relative backward error of each solution
! 168: * vector X(j) (i.e., the smallest relative change in
! 169: * any element of A or B that makes X(j) an exact solution).
! 170: *
! 171: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
! 172: *
! 173: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
! 174: *
! 175: * INFO (output) INTEGER
! 176: * = 0: successful exit
! 177: * < 0: if INFO = -i, the i-th argument had an illegal value
! 178: * > 0: if INFO = i, and i is
! 179: * <= N: U(i,i) is exactly zero. The factorization
! 180: * has not been completed unless i = N, but the
! 181: * factor U is exactly singular, so the solution
! 182: * and error bounds could not be computed.
! 183: * RCOND = 0 is returned.
! 184: * = N+1: U is nonsingular, but RCOND is less than machine
! 185: * precision, meaning that the matrix is singular
! 186: * to working precision. Nevertheless, the
! 187: * solution and error bounds are computed because
! 188: * there are a number of situations where the
! 189: * computed solution can be more accurate than the
! 190: * value of RCOND would suggest.
! 191: *
! 192: * =====================================================================
! 193: *
! 194: * .. Parameters ..
! 195: DOUBLE PRECISION ZERO
! 196: PARAMETER ( ZERO = 0.0D+0 )
! 197: * ..
! 198: * .. Local Scalars ..
! 199: LOGICAL NOFACT, NOTRAN
! 200: CHARACTER NORM
! 201: DOUBLE PRECISION ANORM
! 202: * ..
! 203: * .. External Functions ..
! 204: LOGICAL LSAME
! 205: DOUBLE PRECISION DLAMCH, ZLANGT
! 206: EXTERNAL LSAME, DLAMCH, ZLANGT
! 207: * ..
! 208: * .. External Subroutines ..
! 209: EXTERNAL XERBLA, ZCOPY, ZGTCON, ZGTRFS, ZGTTRF, ZGTTRS,
! 210: $ ZLACPY
! 211: * ..
! 212: * .. Intrinsic Functions ..
! 213: INTRINSIC MAX
! 214: * ..
! 215: * .. Executable Statements ..
! 216: *
! 217: INFO = 0
! 218: NOFACT = LSAME( FACT, 'N' )
! 219: NOTRAN = LSAME( TRANS, 'N' )
! 220: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
! 221: INFO = -1
! 222: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
! 223: $ LSAME( TRANS, 'C' ) ) THEN
! 224: INFO = -2
! 225: ELSE IF( N.LT.0 ) THEN
! 226: INFO = -3
! 227: ELSE IF( NRHS.LT.0 ) THEN
! 228: INFO = -4
! 229: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 230: INFO = -14
! 231: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 232: INFO = -16
! 233: END IF
! 234: IF( INFO.NE.0 ) THEN
! 235: CALL XERBLA( 'ZGTSVX', -INFO )
! 236: RETURN
! 237: END IF
! 238: *
! 239: IF( NOFACT ) THEN
! 240: *
! 241: * Compute the LU factorization of A.
! 242: *
! 243: CALL ZCOPY( N, D, 1, DF, 1 )
! 244: IF( N.GT.1 ) THEN
! 245: CALL ZCOPY( N-1, DL, 1, DLF, 1 )
! 246: CALL ZCOPY( N-1, DU, 1, DUF, 1 )
! 247: END IF
! 248: CALL ZGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO )
! 249: *
! 250: * Return if INFO is non-zero.
! 251: *
! 252: IF( INFO.GT.0 )THEN
! 253: RCOND = ZERO
! 254: RETURN
! 255: END IF
! 256: END IF
! 257: *
! 258: * Compute the norm of the matrix A.
! 259: *
! 260: IF( NOTRAN ) THEN
! 261: NORM = '1'
! 262: ELSE
! 263: NORM = 'I'
! 264: END IF
! 265: ANORM = ZLANGT( NORM, N, DL, D, DU )
! 266: *
! 267: * Compute the reciprocal of the condition number of A.
! 268: *
! 269: CALL ZGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
! 270: $ INFO )
! 271: *
! 272: * Compute the solution vectors X.
! 273: *
! 274: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
! 275: CALL ZGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
! 276: $ INFO )
! 277: *
! 278: * Use iterative refinement to improve the computed solutions and
! 279: * compute error bounds and backward error estimates for them.
! 280: *
! 281: CALL ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
! 282: $ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
! 283: *
! 284: * Set INFO = N+1 if the matrix is singular to working precision.
! 285: *
! 286: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
! 287: $ INFO = N + 1
! 288: *
! 289: RETURN
! 290: *
! 291: * End of ZGTSVX
! 292: *
! 293: END
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