Annotation of rpl/lapack/lapack/zherfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZHERFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
! 2: $ X, LDX, FERR, BERR, WORK, RWORK, 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 ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
! 10: *
! 11: * .. Scalar Arguments ..
! 12: CHARACTER UPLO
! 13: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IPIV( * )
! 17: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
! 18: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
! 19: $ WORK( * ), X( LDX, * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * ZHERFS improves the computed solution to a system of linear
! 26: * equations when the coefficient matrix is Hermitian indefinite, and
! 27: * provides error bounds and backward error estimates for the solution.
! 28: *
! 29: * Arguments
! 30: * =========
! 31: *
! 32: * UPLO (input) CHARACTER*1
! 33: * = 'U': Upper triangle of A is stored;
! 34: * = 'L': Lower triangle of A is stored.
! 35: *
! 36: * N (input) INTEGER
! 37: * The order of the matrix A. N >= 0.
! 38: *
! 39: * NRHS (input) INTEGER
! 40: * The number of right hand sides, i.e., the number of columns
! 41: * of the matrices B and X. NRHS >= 0.
! 42: *
! 43: * A (input) COMPLEX*16 array, dimension (LDA,N)
! 44: * The Hermitian matrix A. If UPLO = 'U', the leading N-by-N
! 45: * upper triangular part of A contains the upper triangular part
! 46: * of the matrix A, and the strictly lower triangular part of A
! 47: * is not referenced. If UPLO = 'L', the leading N-by-N lower
! 48: * triangular part of A contains the lower triangular part of
! 49: * the matrix A, and the strictly upper triangular part of A is
! 50: * not referenced.
! 51: *
! 52: * LDA (input) INTEGER
! 53: * The leading dimension of the array A. LDA >= max(1,N).
! 54: *
! 55: * AF (input) COMPLEX*16 array, dimension (LDAF,N)
! 56: * The factored form of the matrix A. AF contains the block
! 57: * diagonal matrix D and the multipliers used to obtain the
! 58: * factor U or L from the factorization A = U*D*U**H or
! 59: * A = L*D*L**H as computed by ZHETRF.
! 60: *
! 61: * LDAF (input) INTEGER
! 62: * The leading dimension of the array AF. LDAF >= max(1,N).
! 63: *
! 64: * IPIV (input) INTEGER array, dimension (N)
! 65: * Details of the interchanges and the block structure of D
! 66: * as determined by ZHETRF.
! 67: *
! 68: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
! 69: * The right hand side matrix B.
! 70: *
! 71: * LDB (input) INTEGER
! 72: * The leading dimension of the array B. LDB >= max(1,N).
! 73: *
! 74: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
! 75: * On entry, the solution matrix X, as computed by ZHETRS.
! 76: * On exit, the improved solution matrix X.
! 77: *
! 78: * LDX (input) INTEGER
! 79: * The leading dimension of the array X. LDX >= max(1,N).
! 80: *
! 81: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 82: * The estimated forward error bound for each solution vector
! 83: * X(j) (the j-th column of the solution matrix X).
! 84: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 85: * is an estimated upper bound for the magnitude of the largest
! 86: * element in (X(j) - XTRUE) divided by the magnitude of the
! 87: * largest element in X(j). The estimate is as reliable as
! 88: * the estimate for RCOND, and is almost always a slight
! 89: * overestimate of the true error.
! 90: *
! 91: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 92: * The componentwise relative backward error of each solution
! 93: * vector X(j) (i.e., the smallest relative change in
! 94: * any element of A or B that makes X(j) an exact solution).
! 95: *
! 96: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
! 97: *
! 98: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
! 99: *
! 100: * INFO (output) INTEGER
! 101: * = 0: successful exit
! 102: * < 0: if INFO = -i, the i-th argument had an illegal value
! 103: *
! 104: * Internal Parameters
! 105: * ===================
! 106: *
! 107: * ITMAX is the maximum number of steps of iterative refinement.
! 108: *
! 109: * =====================================================================
! 110: *
! 111: * .. Parameters ..
! 112: INTEGER ITMAX
! 113: PARAMETER ( ITMAX = 5 )
! 114: DOUBLE PRECISION ZERO
! 115: PARAMETER ( ZERO = 0.0D+0 )
! 116: COMPLEX*16 ONE
! 117: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
! 118: DOUBLE PRECISION TWO
! 119: PARAMETER ( TWO = 2.0D+0 )
! 120: DOUBLE PRECISION THREE
! 121: PARAMETER ( THREE = 3.0D+0 )
! 122: * ..
! 123: * .. Local Scalars ..
! 124: LOGICAL UPPER
! 125: INTEGER COUNT, I, J, K, KASE, NZ
! 126: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
! 127: COMPLEX*16 ZDUM
! 128: * ..
! 129: * .. Local Arrays ..
! 130: INTEGER ISAVE( 3 )
! 131: * ..
! 132: * .. External Subroutines ..
! 133: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZHETRS, ZLACN2
! 134: * ..
! 135: * .. Intrinsic Functions ..
! 136: INTRINSIC ABS, DBLE, DIMAG, MAX
! 137: * ..
! 138: * .. External Functions ..
! 139: LOGICAL LSAME
! 140: DOUBLE PRECISION DLAMCH
! 141: EXTERNAL LSAME, DLAMCH
! 142: * ..
! 143: * .. Statement Functions ..
! 144: DOUBLE PRECISION CABS1
! 145: * ..
! 146: * .. Statement Function definitions ..
! 147: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
! 148: * ..
! 149: * .. Executable Statements ..
! 150: *
! 151: * Test the input parameters.
! 152: *
! 153: INFO = 0
! 154: UPPER = LSAME( UPLO, 'U' )
! 155: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) 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( LDA.LT.MAX( 1, N ) ) THEN
! 162: INFO = -5
! 163: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
! 164: INFO = -7
! 165: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 166: INFO = -10
! 167: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 168: INFO = -12
! 169: END IF
! 170: IF( INFO.NE.0 ) THEN
! 171: CALL XERBLA( 'ZHERFS', -INFO )
! 172: RETURN
! 173: END IF
! 174: *
! 175: * Quick return if possible
! 176: *
! 177: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 178: DO 10 J = 1, NRHS
! 179: FERR( J ) = ZERO
! 180: BERR( J ) = ZERO
! 181: 10 CONTINUE
! 182: RETURN
! 183: END IF
! 184: *
! 185: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 186: *
! 187: NZ = N + 1
! 188: EPS = DLAMCH( 'Epsilon' )
! 189: SAFMIN = DLAMCH( 'Safe minimum' )
! 190: SAFE1 = NZ*SAFMIN
! 191: SAFE2 = SAFE1 / EPS
! 192: *
! 193: * Do for each right hand side
! 194: *
! 195: DO 140 J = 1, NRHS
! 196: *
! 197: COUNT = 1
! 198: LSTRES = THREE
! 199: 20 CONTINUE
! 200: *
! 201: * Loop until stopping criterion is satisfied.
! 202: *
! 203: * Compute residual R = B - A * X
! 204: *
! 205: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
! 206: CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
! 207: *
! 208: * Compute componentwise relative backward error from formula
! 209: *
! 210: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
! 211: *
! 212: * where abs(Z) is the componentwise absolute value of the matrix
! 213: * or vector Z. If the i-th component of the denominator is less
! 214: * than SAFE2, then SAFE1 is added to the i-th components of the
! 215: * numerator and denominator before dividing.
! 216: *
! 217: DO 30 I = 1, N
! 218: RWORK( I ) = CABS1( B( I, J ) )
! 219: 30 CONTINUE
! 220: *
! 221: * Compute abs(A)*abs(X) + abs(B).
! 222: *
! 223: IF( UPPER ) THEN
! 224: DO 50 K = 1, N
! 225: S = ZERO
! 226: XK = CABS1( X( K, J ) )
! 227: DO 40 I = 1, K - 1
! 228: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
! 229: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
! 230: 40 CONTINUE
! 231: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S
! 232: 50 CONTINUE
! 233: ELSE
! 234: DO 70 K = 1, N
! 235: S = ZERO
! 236: XK = CABS1( X( K, J ) )
! 237: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK
! 238: DO 60 I = K + 1, N
! 239: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
! 240: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
! 241: 60 CONTINUE
! 242: RWORK( K ) = RWORK( K ) + S
! 243: 70 CONTINUE
! 244: END IF
! 245: S = ZERO
! 246: DO 80 I = 1, N
! 247: IF( RWORK( I ).GT.SAFE2 ) THEN
! 248: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
! 249: ELSE
! 250: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
! 251: $ ( RWORK( I )+SAFE1 ) )
! 252: END IF
! 253: 80 CONTINUE
! 254: BERR( J ) = S
! 255: *
! 256: * Test stopping criterion. Continue iterating if
! 257: * 1) The residual BERR(J) is larger than machine epsilon, and
! 258: * 2) BERR(J) decreased by at least a factor of 2 during the
! 259: * last iteration, and
! 260: * 3) At most ITMAX iterations tried.
! 261: *
! 262: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
! 263: $ COUNT.LE.ITMAX ) THEN
! 264: *
! 265: * Update solution and try again.
! 266: *
! 267: CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
! 268: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
! 269: LSTRES = BERR( J )
! 270: COUNT = COUNT + 1
! 271: GO TO 20
! 272: END IF
! 273: *
! 274: * Bound error from formula
! 275: *
! 276: * norm(X - XTRUE) / norm(X) .le. FERR =
! 277: * norm( abs(inv(A))*
! 278: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
! 279: *
! 280: * where
! 281: * norm(Z) is the magnitude of the largest component of Z
! 282: * inv(A) is the inverse of A
! 283: * abs(Z) is the componentwise absolute value of the matrix or
! 284: * vector Z
! 285: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 286: * EPS is machine epsilon
! 287: *
! 288: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
! 289: * is incremented by SAFE1 if the i-th component of
! 290: * abs(A)*abs(X) + abs(B) is less than SAFE2.
! 291: *
! 292: * Use ZLACN2 to estimate the infinity-norm of the matrix
! 293: * inv(A) * diag(W),
! 294: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
! 295: *
! 296: DO 90 I = 1, N
! 297: IF( RWORK( I ).GT.SAFE2 ) THEN
! 298: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
! 299: ELSE
! 300: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
! 301: $ SAFE1
! 302: END IF
! 303: 90 CONTINUE
! 304: *
! 305: KASE = 0
! 306: 100 CONTINUE
! 307: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
! 308: IF( KASE.NE.0 ) THEN
! 309: IF( KASE.EQ.1 ) THEN
! 310: *
! 311: * Multiply by diag(W)*inv(A').
! 312: *
! 313: CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
! 314: DO 110 I = 1, N
! 315: WORK( I ) = RWORK( I )*WORK( I )
! 316: 110 CONTINUE
! 317: ELSE IF( KASE.EQ.2 ) THEN
! 318: *
! 319: * Multiply by inv(A)*diag(W).
! 320: *
! 321: DO 120 I = 1, N
! 322: WORK( I ) = RWORK( I )*WORK( I )
! 323: 120 CONTINUE
! 324: CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
! 325: END IF
! 326: GO TO 100
! 327: END IF
! 328: *
! 329: * Normalize error.
! 330: *
! 331: LSTRES = ZERO
! 332: DO 130 I = 1, N
! 333: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
! 334: 130 CONTINUE
! 335: IF( LSTRES.NE.ZERO )
! 336: $ FERR( J ) = FERR( J ) / LSTRES
! 337: *
! 338: 140 CONTINUE
! 339: *
! 340: RETURN
! 341: *
! 342: * End of ZHERFS
! 343: *
! 344: END
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