Annotation of rpl/lapack/lapack/dgtrfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
! 2: $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
! 3: $ 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: * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
! 11: *
! 12: * .. Scalar Arguments ..
! 13: CHARACTER TRANS
! 14: INTEGER INFO, LDB, LDX, N, NRHS
! 15: * ..
! 16: * .. Array Arguments ..
! 17: INTEGER IPIV( * ), IWORK( * )
! 18: DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
! 19: $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
! 20: $ FERR( * ), WORK( * ), X( LDX, * )
! 21: * ..
! 22: *
! 23: * Purpose
! 24: * =======
! 25: *
! 26: * DGTRFS improves the computed solution to a system of linear
! 27: * equations when the coefficient matrix is tridiagonal, and provides
! 28: * error bounds and backward error estimates for the solution.
! 29: *
! 30: * Arguments
! 31: * =========
! 32: *
! 33: * TRANS (input) CHARACTER*1
! 34: * Specifies the form of the system of equations:
! 35: * = 'N': A * X = B (No transpose)
! 36: * = 'T': A**T * X = B (Transpose)
! 37: * = 'C': A**H * X = B (Conjugate transpose = Transpose)
! 38: *
! 39: * N (input) INTEGER
! 40: * The order of the matrix A. N >= 0.
! 41: *
! 42: * NRHS (input) INTEGER
! 43: * The number of right hand sides, i.e., the number of columns
! 44: * of the matrix B. NRHS >= 0.
! 45: *
! 46: * DL (input) DOUBLE PRECISION array, dimension (N-1)
! 47: * The (n-1) subdiagonal elements of A.
! 48: *
! 49: * D (input) DOUBLE PRECISION array, dimension (N)
! 50: * The diagonal elements of A.
! 51: *
! 52: * DU (input) DOUBLE PRECISION array, dimension (N-1)
! 53: * The (n-1) superdiagonal elements of A.
! 54: *
! 55: * DLF (input) DOUBLE PRECISION array, dimension (N-1)
! 56: * The (n-1) multipliers that define the matrix L from the
! 57: * LU factorization of A as computed by DGTTRF.
! 58: *
! 59: * DF (input) DOUBLE PRECISION array, dimension (N)
! 60: * The n diagonal elements of the upper triangular matrix U from
! 61: * the LU factorization of A.
! 62: *
! 63: * DUF (input) DOUBLE PRECISION array, dimension (N-1)
! 64: * The (n-1) elements of the first superdiagonal of U.
! 65: *
! 66: * DU2 (input) DOUBLE PRECISION array, dimension (N-2)
! 67: * The (n-2) elements of the second superdiagonal of U.
! 68: *
! 69: * IPIV (input) INTEGER array, dimension (N)
! 70: * The pivot indices; for 1 <= i <= n, row i of the matrix was
! 71: * interchanged with row IPIV(i). IPIV(i) will always be either
! 72: * i or i+1; IPIV(i) = i indicates a row interchange was not
! 73: * required.
! 74: *
! 75: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 76: * The right hand side matrix B.
! 77: *
! 78: * LDB (input) INTEGER
! 79: * The leading dimension of the array B. LDB >= max(1,N).
! 80: *
! 81: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 82: * On entry, the solution matrix X, as computed by DGTTRS.
! 83: * On exit, the improved solution matrix X.
! 84: *
! 85: * LDX (input) INTEGER
! 86: * The leading dimension of the array X. LDX >= max(1,N).
! 87: *
! 88: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 89: * The estimated forward error bound for each solution vector
! 90: * X(j) (the j-th column of the solution matrix X).
! 91: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 92: * is an estimated upper bound for the magnitude of the largest
! 93: * element in (X(j) - XTRUE) divided by the magnitude of the
! 94: * largest element in X(j). The estimate is as reliable as
! 95: * the estimate for RCOND, and is almost always a slight
! 96: * overestimate of the true error.
! 97: *
! 98: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 99: * The componentwise relative backward error of each solution
! 100: * vector X(j) (i.e., the smallest relative change in
! 101: * any element of A or B that makes X(j) an exact solution).
! 102: *
! 103: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
! 104: *
! 105: * IWORK (workspace) INTEGER array, dimension (N)
! 106: *
! 107: * INFO (output) INTEGER
! 108: * = 0: successful exit
! 109: * < 0: if INFO = -i, the i-th argument had an illegal value
! 110: *
! 111: * Internal Parameters
! 112: * ===================
! 113: *
! 114: * ITMAX is the maximum number of steps of iterative refinement.
! 115: *
! 116: * =====================================================================
! 117: *
! 118: * .. Parameters ..
! 119: INTEGER ITMAX
! 120: PARAMETER ( ITMAX = 5 )
! 121: DOUBLE PRECISION ZERO, ONE
! 122: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 123: DOUBLE PRECISION TWO
! 124: PARAMETER ( TWO = 2.0D+0 )
! 125: DOUBLE PRECISION THREE
! 126: PARAMETER ( THREE = 3.0D+0 )
! 127: * ..
! 128: * .. Local Scalars ..
! 129: LOGICAL NOTRAN
! 130: CHARACTER TRANSN, TRANST
! 131: INTEGER COUNT, I, J, KASE, NZ
! 132: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
! 133: * ..
! 134: * .. Local Arrays ..
! 135: INTEGER ISAVE( 3 )
! 136: * ..
! 137: * .. External Subroutines ..
! 138: EXTERNAL DAXPY, DCOPY, DGTTRS, DLACN2, DLAGTM, XERBLA
! 139: * ..
! 140: * .. Intrinsic Functions ..
! 141: INTRINSIC ABS, MAX
! 142: * ..
! 143: * .. External Functions ..
! 144: LOGICAL LSAME
! 145: DOUBLE PRECISION DLAMCH
! 146: EXTERNAL LSAME, DLAMCH
! 147: * ..
! 148: * .. Executable Statements ..
! 149: *
! 150: * Test the input parameters.
! 151: *
! 152: INFO = 0
! 153: NOTRAN = LSAME( TRANS, 'N' )
! 154: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
! 155: $ LSAME( TRANS, 'C' ) ) THEN
! 156: INFO = -1
! 157: ELSE IF( N.LT.0 ) THEN
! 158: INFO = -2
! 159: ELSE IF( NRHS.LT.0 ) THEN
! 160: INFO = -3
! 161: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 162: INFO = -13
! 163: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 164: INFO = -15
! 165: END IF
! 166: IF( INFO.NE.0 ) THEN
! 167: CALL XERBLA( 'DGTRFS', -INFO )
! 168: RETURN
! 169: END IF
! 170: *
! 171: * Quick return if possible
! 172: *
! 173: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 174: DO 10 J = 1, NRHS
! 175: FERR( J ) = ZERO
! 176: BERR( J ) = ZERO
! 177: 10 CONTINUE
! 178: RETURN
! 179: END IF
! 180: *
! 181: IF( NOTRAN ) THEN
! 182: TRANSN = 'N'
! 183: TRANST = 'T'
! 184: ELSE
! 185: TRANSN = 'T'
! 186: TRANST = 'N'
! 187: END IF
! 188: *
! 189: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 190: *
! 191: NZ = 4
! 192: EPS = DLAMCH( 'Epsilon' )
! 193: SAFMIN = DLAMCH( 'Safe minimum' )
! 194: SAFE1 = NZ*SAFMIN
! 195: SAFE2 = SAFE1 / EPS
! 196: *
! 197: * Do for each right hand side
! 198: *
! 199: DO 110 J = 1, NRHS
! 200: *
! 201: COUNT = 1
! 202: LSTRES = THREE
! 203: 20 CONTINUE
! 204: *
! 205: * Loop until stopping criterion is satisfied.
! 206: *
! 207: * Compute residual R = B - op(A) * X,
! 208: * where op(A) = A, A**T, or A**H, depending on TRANS.
! 209: *
! 210: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
! 211: CALL DLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE,
! 212: $ WORK( N+1 ), N )
! 213: *
! 214: * Compute abs(op(A))*abs(x) + abs(b) for use in the backward
! 215: * error bound.
! 216: *
! 217: IF( NOTRAN ) THEN
! 218: IF( N.EQ.1 ) THEN
! 219: WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) )
! 220: ELSE
! 221: WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) +
! 222: $ ABS( DU( 1 )*X( 2, J ) )
! 223: DO 30 I = 2, N - 1
! 224: WORK( I ) = ABS( B( I, J ) ) +
! 225: $ ABS( DL( I-1 )*X( I-1, J ) ) +
! 226: $ ABS( D( I )*X( I, J ) ) +
! 227: $ ABS( DU( I )*X( I+1, J ) )
! 228: 30 CONTINUE
! 229: WORK( N ) = ABS( B( N, J ) ) +
! 230: $ ABS( DL( N-1 )*X( N-1, J ) ) +
! 231: $ ABS( D( N )*X( N, J ) )
! 232: END IF
! 233: ELSE
! 234: IF( N.EQ.1 ) THEN
! 235: WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) )
! 236: ELSE
! 237: WORK( 1 ) = ABS( B( 1, J ) ) + ABS( D( 1 )*X( 1, J ) ) +
! 238: $ ABS( DL( 1 )*X( 2, J ) )
! 239: DO 40 I = 2, N - 1
! 240: WORK( I ) = ABS( B( I, J ) ) +
! 241: $ ABS( DU( I-1 )*X( I-1, J ) ) +
! 242: $ ABS( D( I )*X( I, J ) ) +
! 243: $ ABS( DL( I )*X( I+1, J ) )
! 244: 40 CONTINUE
! 245: WORK( N ) = ABS( B( N, J ) ) +
! 246: $ ABS( DU( N-1 )*X( N-1, J ) ) +
! 247: $ ABS( D( N )*X( N, J ) )
! 248: END IF
! 249: END IF
! 250: *
! 251: * Compute componentwise relative backward error from formula
! 252: *
! 253: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
! 254: *
! 255: * where abs(Z) is the componentwise absolute value of the matrix
! 256: * or vector Z. If the i-th component of the denominator is less
! 257: * than SAFE2, then SAFE1 is added to the i-th components of the
! 258: * numerator and denominator before dividing.
! 259: *
! 260: S = ZERO
! 261: DO 50 I = 1, N
! 262: IF( WORK( I ).GT.SAFE2 ) THEN
! 263: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
! 264: ELSE
! 265: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
! 266: $ ( WORK( I )+SAFE1 ) )
! 267: END IF
! 268: 50 CONTINUE
! 269: BERR( J ) = S
! 270: *
! 271: * Test stopping criterion. Continue iterating if
! 272: * 1) The residual BERR(J) is larger than machine epsilon, and
! 273: * 2) BERR(J) decreased by at least a factor of 2 during the
! 274: * last iteration, and
! 275: * 3) At most ITMAX iterations tried.
! 276: *
! 277: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
! 278: $ COUNT.LE.ITMAX ) THEN
! 279: *
! 280: * Update solution and try again.
! 281: *
! 282: CALL DGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV,
! 283: $ WORK( N+1 ), N, INFO )
! 284: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
! 285: LSTRES = BERR( J )
! 286: COUNT = COUNT + 1
! 287: GO TO 20
! 288: END IF
! 289: *
! 290: * Bound error from formula
! 291: *
! 292: * norm(X - XTRUE) / norm(X) .le. FERR =
! 293: * norm( abs(inv(op(A)))*
! 294: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
! 295: *
! 296: * where
! 297: * norm(Z) is the magnitude of the largest component of Z
! 298: * inv(op(A)) is the inverse of op(A)
! 299: * abs(Z) is the componentwise absolute value of the matrix or
! 300: * vector Z
! 301: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 302: * EPS is machine epsilon
! 303: *
! 304: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
! 305: * is incremented by SAFE1 if the i-th component of
! 306: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
! 307: *
! 308: * Use DLACN2 to estimate the infinity-norm of the matrix
! 309: * inv(op(A)) * diag(W),
! 310: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
! 311: *
! 312: DO 60 I = 1, N
! 313: IF( WORK( I ).GT.SAFE2 ) THEN
! 314: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
! 315: ELSE
! 316: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
! 317: END IF
! 318: 60 CONTINUE
! 319: *
! 320: KASE = 0
! 321: 70 CONTINUE
! 322: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
! 323: $ KASE, ISAVE )
! 324: IF( KASE.NE.0 ) THEN
! 325: IF( KASE.EQ.1 ) THEN
! 326: *
! 327: * Multiply by diag(W)*inv(op(A)**T).
! 328: *
! 329: CALL DGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV,
! 330: $ WORK( N+1 ), N, INFO )
! 331: DO 80 I = 1, N
! 332: WORK( N+I ) = WORK( I )*WORK( N+I )
! 333: 80 CONTINUE
! 334: ELSE
! 335: *
! 336: * Multiply by inv(op(A))*diag(W).
! 337: *
! 338: DO 90 I = 1, N
! 339: WORK( N+I ) = WORK( I )*WORK( N+I )
! 340: 90 CONTINUE
! 341: CALL DGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV,
! 342: $ WORK( N+1 ), N, INFO )
! 343: END IF
! 344: GO TO 70
! 345: END IF
! 346: *
! 347: * Normalize error.
! 348: *
! 349: LSTRES = ZERO
! 350: DO 100 I = 1, N
! 351: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
! 352: 100 CONTINUE
! 353: IF( LSTRES.NE.ZERO )
! 354: $ FERR( J ) = FERR( J ) / LSTRES
! 355: *
! 356: 110 CONTINUE
! 357: *
! 358: RETURN
! 359: *
! 360: * End of DGTRFS
! 361: *
! 362: END
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