Annotation of rpl/lapack/lapack/zgtrfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
! 2: $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
! 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 ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
! 11: *
! 12: * .. Scalar Arguments ..
! 13: CHARACTER TRANS
! 14: INTEGER INFO, LDB, LDX, N, NRHS
! 15: * ..
! 16: * .. Array Arguments ..
! 17: INTEGER IPIV( * )
! 18: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
! 19: COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
! 20: $ DLF( * ), DU( * ), DU2( * ), DUF( * ),
! 21: $ WORK( * ), X( LDX, * )
! 22: * ..
! 23: *
! 24: * Purpose
! 25: * =======
! 26: *
! 27: * ZGTRFS improves the computed solution to a system of linear
! 28: * equations when the coefficient matrix is tridiagonal, and provides
! 29: * error bounds and backward error estimates for the solution.
! 30: *
! 31: * Arguments
! 32: * =========
! 33: *
! 34: * TRANS (input) CHARACTER*1
! 35: * Specifies the form of the system of equations:
! 36: * = 'N': A * X = B (No transpose)
! 37: * = 'T': A**T * X = B (Transpose)
! 38: * = 'C': A**H * X = B (Conjugate transpose)
! 39: *
! 40: * N (input) INTEGER
! 41: * The order of the matrix A. N >= 0.
! 42: *
! 43: * NRHS (input) INTEGER
! 44: * The number of right hand sides, i.e., the number of columns
! 45: * of the matrix B. NRHS >= 0.
! 46: *
! 47: * DL (input) COMPLEX*16 array, dimension (N-1)
! 48: * The (n-1) subdiagonal elements of A.
! 49: *
! 50: * D (input) COMPLEX*16 array, dimension (N)
! 51: * The diagonal elements of A.
! 52: *
! 53: * DU (input) COMPLEX*16 array, dimension (N-1)
! 54: * The (n-1) superdiagonal elements of A.
! 55: *
! 56: * DLF (input) COMPLEX*16 array, dimension (N-1)
! 57: * The (n-1) multipliers that define the matrix L from the
! 58: * LU factorization of A as computed by ZGTTRF.
! 59: *
! 60: * DF (input) COMPLEX*16 array, dimension (N)
! 61: * The n diagonal elements of the upper triangular matrix U from
! 62: * the LU factorization of A.
! 63: *
! 64: * DUF (input) COMPLEX*16 array, dimension (N-1)
! 65: * The (n-1) elements of the first superdiagonal of U.
! 66: *
! 67: * DU2 (input) COMPLEX*16 array, dimension (N-2)
! 68: * The (n-2) elements of the second superdiagonal of U.
! 69: *
! 70: * IPIV (input) INTEGER array, dimension (N)
! 71: * The pivot indices; for 1 <= i <= n, row i of the matrix was
! 72: * interchanged with row IPIV(i). IPIV(i) will always be either
! 73: * i or i+1; IPIV(i) = i indicates a row interchange was not
! 74: * required.
! 75: *
! 76: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
! 77: * The right hand side matrix B.
! 78: *
! 79: * LDB (input) INTEGER
! 80: * The leading dimension of the array B. LDB >= max(1,N).
! 81: *
! 82: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
! 83: * On entry, the solution matrix X, as computed by ZGTTRS.
! 84: * On exit, the improved solution matrix X.
! 85: *
! 86: * LDX (input) INTEGER
! 87: * The leading dimension of the array X. LDX >= max(1,N).
! 88: *
! 89: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 90: * The estimated forward error bound for each solution vector
! 91: * X(j) (the j-th column of the solution matrix X).
! 92: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 93: * is an estimated upper bound for the magnitude of the largest
! 94: * element in (X(j) - XTRUE) divided by the magnitude of the
! 95: * largest element in X(j). The estimate is as reliable as
! 96: * the estimate for RCOND, and is almost always a slight
! 97: * overestimate of the true error.
! 98: *
! 99: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 100: * The componentwise relative backward error of each solution
! 101: * vector X(j) (i.e., the smallest relative change in
! 102: * any element of A or B that makes X(j) an exact solution).
! 103: *
! 104: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
! 105: *
! 106: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
! 107: *
! 108: * INFO (output) INTEGER
! 109: * = 0: successful exit
! 110: * < 0: if INFO = -i, the i-th argument had an illegal value
! 111: *
! 112: * Internal Parameters
! 113: * ===================
! 114: *
! 115: * ITMAX is the maximum number of steps of iterative refinement.
! 116: *
! 117: * =====================================================================
! 118: *
! 119: * .. Parameters ..
! 120: INTEGER ITMAX
! 121: PARAMETER ( ITMAX = 5 )
! 122: DOUBLE PRECISION ZERO, ONE
! 123: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 124: DOUBLE PRECISION TWO
! 125: PARAMETER ( TWO = 2.0D+0 )
! 126: DOUBLE PRECISION THREE
! 127: PARAMETER ( THREE = 3.0D+0 )
! 128: * ..
! 129: * .. Local Scalars ..
! 130: LOGICAL NOTRAN
! 131: CHARACTER TRANSN, TRANST
! 132: INTEGER COUNT, I, J, KASE, NZ
! 133: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
! 134: COMPLEX*16 ZDUM
! 135: * ..
! 136: * .. Local Arrays ..
! 137: INTEGER ISAVE( 3 )
! 138: * ..
! 139: * .. External Subroutines ..
! 140: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGTTRS, ZLACN2, ZLAGTM
! 141: * ..
! 142: * .. Intrinsic Functions ..
! 143: INTRINSIC ABS, DBLE, DCMPLX, DIMAG, MAX
! 144: * ..
! 145: * .. External Functions ..
! 146: LOGICAL LSAME
! 147: DOUBLE PRECISION DLAMCH
! 148: EXTERNAL LSAME, DLAMCH
! 149: * ..
! 150: * .. Statement Functions ..
! 151: DOUBLE PRECISION CABS1
! 152: * ..
! 153: * .. Statement Function definitions ..
! 154: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
! 155: * ..
! 156: * .. Executable Statements ..
! 157: *
! 158: * Test the input parameters.
! 159: *
! 160: INFO = 0
! 161: NOTRAN = LSAME( TRANS, 'N' )
! 162: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
! 163: $ LSAME( TRANS, 'C' ) ) THEN
! 164: INFO = -1
! 165: ELSE IF( N.LT.0 ) THEN
! 166: INFO = -2
! 167: ELSE IF( NRHS.LT.0 ) THEN
! 168: INFO = -3
! 169: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 170: INFO = -13
! 171: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 172: INFO = -15
! 173: END IF
! 174: IF( INFO.NE.0 ) THEN
! 175: CALL XERBLA( 'ZGTRFS', -INFO )
! 176: RETURN
! 177: END IF
! 178: *
! 179: * Quick return if possible
! 180: *
! 181: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 182: DO 10 J = 1, NRHS
! 183: FERR( J ) = ZERO
! 184: BERR( J ) = ZERO
! 185: 10 CONTINUE
! 186: RETURN
! 187: END IF
! 188: *
! 189: IF( NOTRAN ) THEN
! 190: TRANSN = 'N'
! 191: TRANST = 'C'
! 192: ELSE
! 193: TRANSN = 'C'
! 194: TRANST = 'N'
! 195: END IF
! 196: *
! 197: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 198: *
! 199: NZ = 4
! 200: EPS = DLAMCH( 'Epsilon' )
! 201: SAFMIN = DLAMCH( 'Safe minimum' )
! 202: SAFE1 = NZ*SAFMIN
! 203: SAFE2 = SAFE1 / EPS
! 204: *
! 205: * Do for each right hand side
! 206: *
! 207: DO 110 J = 1, NRHS
! 208: *
! 209: COUNT = 1
! 210: LSTRES = THREE
! 211: 20 CONTINUE
! 212: *
! 213: * Loop until stopping criterion is satisfied.
! 214: *
! 215: * Compute residual R = B - op(A) * X,
! 216: * where op(A) = A, A**T, or A**H, depending on TRANS.
! 217: *
! 218: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
! 219: CALL ZLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE,
! 220: $ WORK, N )
! 221: *
! 222: * Compute abs(op(A))*abs(x) + abs(b) for use in the backward
! 223: * error bound.
! 224: *
! 225: IF( NOTRAN ) THEN
! 226: IF( N.EQ.1 ) THEN
! 227: RWORK( 1 ) = CABS1( B( 1, J ) ) +
! 228: $ CABS1( D( 1 ) )*CABS1( X( 1, J ) )
! 229: ELSE
! 230: RWORK( 1 ) = CABS1( B( 1, J ) ) +
! 231: $ CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
! 232: $ CABS1( DU( 1 ) )*CABS1( X( 2, J ) )
! 233: DO 30 I = 2, N - 1
! 234: RWORK( I ) = CABS1( B( I, J ) ) +
! 235: $ CABS1( DL( I-1 ) )*CABS1( X( I-1, J ) ) +
! 236: $ CABS1( D( I ) )*CABS1( X( I, J ) ) +
! 237: $ CABS1( DU( I ) )*CABS1( X( I+1, J ) )
! 238: 30 CONTINUE
! 239: RWORK( N ) = CABS1( B( N, J ) ) +
! 240: $ CABS1( DL( N-1 ) )*CABS1( X( N-1, J ) ) +
! 241: $ CABS1( D( N ) )*CABS1( X( N, J ) )
! 242: END IF
! 243: ELSE
! 244: IF( N.EQ.1 ) THEN
! 245: RWORK( 1 ) = CABS1( B( 1, J ) ) +
! 246: $ CABS1( D( 1 ) )*CABS1( X( 1, J ) )
! 247: ELSE
! 248: RWORK( 1 ) = CABS1( B( 1, J ) ) +
! 249: $ CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
! 250: $ CABS1( DL( 1 ) )*CABS1( X( 2, J ) )
! 251: DO 40 I = 2, N - 1
! 252: RWORK( I ) = CABS1( B( I, J ) ) +
! 253: $ CABS1( DU( I-1 ) )*CABS1( X( I-1, J ) ) +
! 254: $ CABS1( D( I ) )*CABS1( X( I, J ) ) +
! 255: $ CABS1( DL( I ) )*CABS1( X( I+1, J ) )
! 256: 40 CONTINUE
! 257: RWORK( N ) = CABS1( B( N, J ) ) +
! 258: $ CABS1( DU( N-1 ) )*CABS1( X( N-1, J ) ) +
! 259: $ CABS1( D( N ) )*CABS1( X( N, J ) )
! 260: END IF
! 261: END IF
! 262: *
! 263: * Compute componentwise relative backward error from formula
! 264: *
! 265: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
! 266: *
! 267: * where abs(Z) is the componentwise absolute value of the matrix
! 268: * or vector Z. If the i-th component of the denominator is less
! 269: * than SAFE2, then SAFE1 is added to the i-th components of the
! 270: * numerator and denominator before dividing.
! 271: *
! 272: S = ZERO
! 273: DO 50 I = 1, N
! 274: IF( RWORK( I ).GT.SAFE2 ) THEN
! 275: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
! 276: ELSE
! 277: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
! 278: $ ( RWORK( I )+SAFE1 ) )
! 279: END IF
! 280: 50 CONTINUE
! 281: BERR( J ) = S
! 282: *
! 283: * Test stopping criterion. Continue iterating if
! 284: * 1) The residual BERR(J) is larger than machine epsilon, and
! 285: * 2) BERR(J) decreased by at least a factor of 2 during the
! 286: * last iteration, and
! 287: * 3) At most ITMAX iterations tried.
! 288: *
! 289: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
! 290: $ COUNT.LE.ITMAX ) THEN
! 291: *
! 292: * Update solution and try again.
! 293: *
! 294: CALL ZGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, WORK, N,
! 295: $ INFO )
! 296: CALL ZAXPY( N, DCMPLX( ONE ), WORK, 1, X( 1, J ), 1 )
! 297: LSTRES = BERR( J )
! 298: COUNT = COUNT + 1
! 299: GO TO 20
! 300: END IF
! 301: *
! 302: * Bound error from formula
! 303: *
! 304: * norm(X - XTRUE) / norm(X) .le. FERR =
! 305: * norm( abs(inv(op(A)))*
! 306: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
! 307: *
! 308: * where
! 309: * norm(Z) is the magnitude of the largest component of Z
! 310: * inv(op(A)) is the inverse of op(A)
! 311: * abs(Z) is the componentwise absolute value of the matrix or
! 312: * vector Z
! 313: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 314: * EPS is machine epsilon
! 315: *
! 316: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
! 317: * is incremented by SAFE1 if the i-th component of
! 318: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
! 319: *
! 320: * Use ZLACN2 to estimate the infinity-norm of the matrix
! 321: * inv(op(A)) * diag(W),
! 322: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
! 323: *
! 324: DO 60 I = 1, N
! 325: IF( RWORK( I ).GT.SAFE2 ) THEN
! 326: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
! 327: ELSE
! 328: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
! 329: $ SAFE1
! 330: END IF
! 331: 60 CONTINUE
! 332: *
! 333: KASE = 0
! 334: 70 CONTINUE
! 335: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
! 336: IF( KASE.NE.0 ) THEN
! 337: IF( KASE.EQ.1 ) THEN
! 338: *
! 339: * Multiply by diag(W)*inv(op(A)**H).
! 340: *
! 341: CALL ZGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
! 342: $ N, INFO )
! 343: DO 80 I = 1, N
! 344: WORK( I ) = RWORK( I )*WORK( I )
! 345: 80 CONTINUE
! 346: ELSE
! 347: *
! 348: * Multiply by inv(op(A))*diag(W).
! 349: *
! 350: DO 90 I = 1, N
! 351: WORK( I ) = RWORK( I )*WORK( I )
! 352: 90 CONTINUE
! 353: CALL ZGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
! 354: $ N, INFO )
! 355: END IF
! 356: GO TO 70
! 357: END IF
! 358: *
! 359: * Normalize error.
! 360: *
! 361: LSTRES = ZERO
! 362: DO 100 I = 1, N
! 363: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
! 364: 100 CONTINUE
! 365: IF( LSTRES.NE.ZERO )
! 366: $ FERR( J ) = FERR( J ) / LSTRES
! 367: *
! 368: 110 CONTINUE
! 369: *
! 370: RETURN
! 371: *
! 372: * End of ZGTRFS
! 373: *
! 374: END
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