Annotation of rpl/lapack/lapack/dgerfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
! 2: $ X, LDX, FERR, BERR, WORK, IWORK, 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: * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
! 10: *
! 11: * .. Scalar Arguments ..
! 12: CHARACTER TRANS
! 13: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IPIV( * ), IWORK( * )
! 17: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
! 18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * DGERFS improves the computed solution to a system of linear
! 25: * equations and provides error bounds and backward error estimates for
! 26: * the solution.
! 27: *
! 28: * Arguments
! 29: * =========
! 30: *
! 31: * TRANS (input) CHARACTER*1
! 32: * Specifies the form of the system of equations:
! 33: * = 'N': A * X = B (No transpose)
! 34: * = 'T': A**T * X = B (Transpose)
! 35: * = 'C': A**H * X = B (Conjugate transpose = Transpose)
! 36: *
! 37: * N (input) INTEGER
! 38: * The order of the matrix A. N >= 0.
! 39: *
! 40: * NRHS (input) INTEGER
! 41: * The number of right hand sides, i.e., the number of columns
! 42: * of the matrices B and X. NRHS >= 0.
! 43: *
! 44: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
! 45: * The original N-by-N matrix A.
! 46: *
! 47: * LDA (input) INTEGER
! 48: * The leading dimension of the array A. LDA >= max(1,N).
! 49: *
! 50: * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
! 51: * The factors L and U from the factorization A = P*L*U
! 52: * as computed by DGETRF.
! 53: *
! 54: * LDAF (input) INTEGER
! 55: * The leading dimension of the array AF. LDAF >= max(1,N).
! 56: *
! 57: * IPIV (input) INTEGER array, dimension (N)
! 58: * The pivot indices from DGETRF; for 1<=i<=N, row i of the
! 59: * matrix was interchanged with row IPIV(i).
! 60: *
! 61: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 62: * The right hand side matrix B.
! 63: *
! 64: * LDB (input) INTEGER
! 65: * The leading dimension of the array B. LDB >= max(1,N).
! 66: *
! 67: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 68: * On entry, the solution matrix X, as computed by DGETRS.
! 69: * On exit, the improved solution matrix X.
! 70: *
! 71: * LDX (input) INTEGER
! 72: * The leading dimension of the array X. LDX >= max(1,N).
! 73: *
! 74: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 75: * The estimated forward error bound for each solution vector
! 76: * X(j) (the j-th column of the solution matrix X).
! 77: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 78: * is an estimated upper bound for the magnitude of the largest
! 79: * element in (X(j) - XTRUE) divided by the magnitude of the
! 80: * largest element in X(j). The estimate is as reliable as
! 81: * the estimate for RCOND, and is almost always a slight
! 82: * overestimate of the true error.
! 83: *
! 84: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 85: * The componentwise relative backward error of each solution
! 86: * vector X(j) (i.e., the smallest relative change in
! 87: * any element of A or B that makes X(j) an exact solution).
! 88: *
! 89: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
! 90: *
! 91: * IWORK (workspace) INTEGER array, dimension (N)
! 92: *
! 93: * INFO (output) INTEGER
! 94: * = 0: successful exit
! 95: * < 0: if INFO = -i, the i-th argument had an illegal value
! 96: *
! 97: * Internal Parameters
! 98: * ===================
! 99: *
! 100: * ITMAX is the maximum number of steps of iterative refinement.
! 101: *
! 102: * =====================================================================
! 103: *
! 104: * .. Parameters ..
! 105: INTEGER ITMAX
! 106: PARAMETER ( ITMAX = 5 )
! 107: DOUBLE PRECISION ZERO
! 108: PARAMETER ( ZERO = 0.0D+0 )
! 109: DOUBLE PRECISION ONE
! 110: PARAMETER ( ONE = 1.0D+0 )
! 111: DOUBLE PRECISION TWO
! 112: PARAMETER ( TWO = 2.0D+0 )
! 113: DOUBLE PRECISION THREE
! 114: PARAMETER ( THREE = 3.0D+0 )
! 115: * ..
! 116: * .. Local Scalars ..
! 117: LOGICAL NOTRAN
! 118: CHARACTER TRANST
! 119: INTEGER COUNT, I, J, K, KASE, NZ
! 120: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
! 121: * ..
! 122: * .. Local Arrays ..
! 123: INTEGER ISAVE( 3 )
! 124: * ..
! 125: * .. External Subroutines ..
! 126: EXTERNAL DAXPY, DCOPY, DGEMV, DGETRS, DLACN2, XERBLA
! 127: * ..
! 128: * .. Intrinsic Functions ..
! 129: INTRINSIC ABS, MAX
! 130: * ..
! 131: * .. External Functions ..
! 132: LOGICAL LSAME
! 133: DOUBLE PRECISION DLAMCH
! 134: EXTERNAL LSAME, DLAMCH
! 135: * ..
! 136: * .. Executable Statements ..
! 137: *
! 138: * Test the input parameters.
! 139: *
! 140: INFO = 0
! 141: NOTRAN = LSAME( TRANS, 'N' )
! 142: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
! 143: $ LSAME( TRANS, 'C' ) ) THEN
! 144: INFO = -1
! 145: ELSE IF( N.LT.0 ) THEN
! 146: INFO = -2
! 147: ELSE IF( NRHS.LT.0 ) THEN
! 148: INFO = -3
! 149: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 150: INFO = -5
! 151: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
! 152: INFO = -7
! 153: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 154: INFO = -10
! 155: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 156: INFO = -12
! 157: END IF
! 158: IF( INFO.NE.0 ) THEN
! 159: CALL XERBLA( 'DGERFS', -INFO )
! 160: RETURN
! 161: END IF
! 162: *
! 163: * Quick return if possible
! 164: *
! 165: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 166: DO 10 J = 1, NRHS
! 167: FERR( J ) = ZERO
! 168: BERR( J ) = ZERO
! 169: 10 CONTINUE
! 170: RETURN
! 171: END IF
! 172: *
! 173: IF( NOTRAN ) THEN
! 174: TRANST = 'T'
! 175: ELSE
! 176: TRANST = 'N'
! 177: END IF
! 178: *
! 179: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 180: *
! 181: NZ = N + 1
! 182: EPS = DLAMCH( 'Epsilon' )
! 183: SAFMIN = DLAMCH( 'Safe minimum' )
! 184: SAFE1 = NZ*SAFMIN
! 185: SAFE2 = SAFE1 / EPS
! 186: *
! 187: * Do for each right hand side
! 188: *
! 189: DO 140 J = 1, NRHS
! 190: *
! 191: COUNT = 1
! 192: LSTRES = THREE
! 193: 20 CONTINUE
! 194: *
! 195: * Loop until stopping criterion is satisfied.
! 196: *
! 197: * Compute residual R = B - op(A) * X,
! 198: * where op(A) = A, A**T, or A**H, depending on TRANS.
! 199: *
! 200: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
! 201: CALL DGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
! 202: $ WORK( N+1 ), 1 )
! 203: *
! 204: * Compute componentwise relative backward error from formula
! 205: *
! 206: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
! 207: *
! 208: * where abs(Z) is the componentwise absolute value of the matrix
! 209: * or vector Z. If the i-th component of the denominator is less
! 210: * than SAFE2, then SAFE1 is added to the i-th components of the
! 211: * numerator and denominator before dividing.
! 212: *
! 213: DO 30 I = 1, N
! 214: WORK( I ) = ABS( B( I, J ) )
! 215: 30 CONTINUE
! 216: *
! 217: * Compute abs(op(A))*abs(X) + abs(B).
! 218: *
! 219: IF( NOTRAN ) THEN
! 220: DO 50 K = 1, N
! 221: XK = ABS( X( K, J ) )
! 222: DO 40 I = 1, N
! 223: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
! 224: 40 CONTINUE
! 225: 50 CONTINUE
! 226: ELSE
! 227: DO 70 K = 1, N
! 228: S = ZERO
! 229: DO 60 I = 1, N
! 230: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
! 231: 60 CONTINUE
! 232: WORK( K ) = WORK( K ) + S
! 233: 70 CONTINUE
! 234: END IF
! 235: S = ZERO
! 236: DO 80 I = 1, N
! 237: IF( WORK( I ).GT.SAFE2 ) THEN
! 238: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
! 239: ELSE
! 240: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
! 241: $ ( WORK( I )+SAFE1 ) )
! 242: END IF
! 243: 80 CONTINUE
! 244: BERR( J ) = S
! 245: *
! 246: * Test stopping criterion. Continue iterating if
! 247: * 1) The residual BERR(J) is larger than machine epsilon, and
! 248: * 2) BERR(J) decreased by at least a factor of 2 during the
! 249: * last iteration, and
! 250: * 3) At most ITMAX iterations tried.
! 251: *
! 252: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
! 253: $ COUNT.LE.ITMAX ) THEN
! 254: *
! 255: * Update solution and try again.
! 256: *
! 257: CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
! 258: $ INFO )
! 259: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
! 260: LSTRES = BERR( J )
! 261: COUNT = COUNT + 1
! 262: GO TO 20
! 263: END IF
! 264: *
! 265: * Bound error from formula
! 266: *
! 267: * norm(X - XTRUE) / norm(X) .le. FERR =
! 268: * norm( abs(inv(op(A)))*
! 269: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
! 270: *
! 271: * where
! 272: * norm(Z) is the magnitude of the largest component of Z
! 273: * inv(op(A)) is the inverse of op(A)
! 274: * abs(Z) is the componentwise absolute value of the matrix or
! 275: * vector Z
! 276: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 277: * EPS is machine epsilon
! 278: *
! 279: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
! 280: * is incremented by SAFE1 if the i-th component of
! 281: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
! 282: *
! 283: * Use DLACN2 to estimate the infinity-norm of the matrix
! 284: * inv(op(A)) * diag(W),
! 285: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
! 286: *
! 287: DO 90 I = 1, N
! 288: IF( WORK( I ).GT.SAFE2 ) THEN
! 289: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
! 290: ELSE
! 291: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
! 292: END IF
! 293: 90 CONTINUE
! 294: *
! 295: KASE = 0
! 296: 100 CONTINUE
! 297: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
! 298: $ KASE, ISAVE )
! 299: IF( KASE.NE.0 ) THEN
! 300: IF( KASE.EQ.1 ) THEN
! 301: *
! 302: * Multiply by diag(W)*inv(op(A)**T).
! 303: *
! 304: CALL DGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
! 305: $ N, INFO )
! 306: DO 110 I = 1, N
! 307: WORK( N+I ) = WORK( I )*WORK( N+I )
! 308: 110 CONTINUE
! 309: ELSE
! 310: *
! 311: * Multiply by inv(op(A))*diag(W).
! 312: *
! 313: DO 120 I = 1, N
! 314: WORK( N+I ) = WORK( I )*WORK( N+I )
! 315: 120 CONTINUE
! 316: CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
! 317: $ INFO )
! 318: END IF
! 319: GO TO 100
! 320: END IF
! 321: *
! 322: * Normalize error.
! 323: *
! 324: LSTRES = ZERO
! 325: DO 130 I = 1, N
! 326: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
! 327: 130 CONTINUE
! 328: IF( LSTRES.NE.ZERO )
! 329: $ FERR( J ) = FERR( J ) / LSTRES
! 330: *
! 331: 140 CONTINUE
! 332: *
! 333: RETURN
! 334: *
! 335: * End of DGERFS
! 336: *
! 337: END
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