LAPACK 3.3.1 Linear Algebra PACKage

# zgerfs.f

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```00001       SUBROUTINE ZGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
00002      \$                   X, LDX, FERR, BERR, WORK, RWORK, INFO )
00003 *
00004 *  -- LAPACK routine (version 3.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     November 2006
00008 *
00009 *     Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
00010 *
00011 *     .. Scalar Arguments ..
00012       CHARACTER          TRANS
00013       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS
00014 *     ..
00015 *     .. Array Arguments ..
00016       INTEGER            IPIV( * )
00017       DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * )
00018       COMPLEX*16         A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
00019      \$                   WORK( * ), X( LDX, * )
00020 *     ..
00021 *
00022 *  Purpose
00023 *  =======
00024 *
00025 *  ZGERFS improves the computed solution to a system of linear
00026 *  equations and provides error bounds and backward error estimates for
00027 *  the solution.
00028 *
00029 *  Arguments
00030 *  =========
00031 *
00032 *  TRANS   (input) CHARACTER*1
00033 *          Specifies the form of the system of equations:
00034 *          = 'N':  A * X = B     (No transpose)
00035 *          = 'T':  A**T * X = B  (Transpose)
00036 *          = 'C':  A**H * X = B  (Conjugate transpose)
00037 *
00038 *  N       (input) INTEGER
00039 *          The order of the matrix A.  N >= 0.
00040 *
00041 *  NRHS    (input) INTEGER
00042 *          The number of right hand sides, i.e., the number of columns
00043 *          of the matrices B and X.  NRHS >= 0.
00044 *
00045 *  A       (input) COMPLEX*16 array, dimension (LDA,N)
00046 *          The original N-by-N matrix A.
00047 *
00048 *  LDA     (input) INTEGER
00049 *          The leading dimension of the array A.  LDA >= max(1,N).
00050 *
00051 *  AF      (input) COMPLEX*16 array, dimension (LDAF,N)
00052 *          The factors L and U from the factorization A = P*L*U
00053 *          as computed by ZGETRF.
00054 *
00055 *  LDAF    (input) INTEGER
00056 *          The leading dimension of the array AF.  LDAF >= max(1,N).
00057 *
00058 *  IPIV    (input) INTEGER array, dimension (N)
00059 *          The pivot indices from ZGETRF; for 1<=i<=N, row i of the
00060 *          matrix was interchanged with row IPIV(i).
00061 *
00062 *  B       (input) COMPLEX*16 array, dimension (LDB,NRHS)
00063 *          The right hand side matrix B.
00064 *
00065 *  LDB     (input) INTEGER
00066 *          The leading dimension of the array B.  LDB >= max(1,N).
00067 *
00068 *  X       (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
00069 *          On entry, the solution matrix X, as computed by ZGETRS.
00070 *          On exit, the improved solution matrix X.
00071 *
00072 *  LDX     (input) INTEGER
00073 *          The leading dimension of the array X.  LDX >= max(1,N).
00074 *
00075 *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
00076 *          The estimated forward error bound for each solution vector
00077 *          X(j) (the j-th column of the solution matrix X).
00078 *          If XTRUE is the true solution corresponding to X(j), FERR(j)
00079 *          is an estimated upper bound for the magnitude of the largest
00080 *          element in (X(j) - XTRUE) divided by the magnitude of the
00081 *          largest element in X(j).  The estimate is as reliable as
00082 *          the estimate for RCOND, and is almost always a slight
00083 *          overestimate of the true error.
00084 *
00085 *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
00086 *          The componentwise relative backward error of each solution
00087 *          vector X(j) (i.e., the smallest relative change in
00088 *          any element of A or B that makes X(j) an exact solution).
00089 *
00090 *  WORK    (workspace) COMPLEX*16 array, dimension (2*N)
00091 *
00092 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
00093 *
00094 *  INFO    (output) INTEGER
00095 *          = 0:  successful exit
00096 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00097 *
00098 *  Internal Parameters
00099 *  ===================
00100 *
00101 *  ITMAX is the maximum number of steps of iterative refinement.
00102 *
00103 *  =====================================================================
00104 *
00105 *     .. Parameters ..
00106       INTEGER            ITMAX
00107       PARAMETER          ( ITMAX = 5 )
00108       DOUBLE PRECISION   ZERO
00109       PARAMETER          ( ZERO = 0.0D+0 )
00110       COMPLEX*16         ONE
00111       PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ) )
00112       DOUBLE PRECISION   TWO
00113       PARAMETER          ( TWO = 2.0D+0 )
00114       DOUBLE PRECISION   THREE
00115       PARAMETER          ( THREE = 3.0D+0 )
00116 *     ..
00117 *     .. Local Scalars ..
00118       LOGICAL            NOTRAN
00119       CHARACTER          TRANSN, TRANST
00120       INTEGER            COUNT, I, J, K, KASE, NZ
00121       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
00122       COMPLEX*16         ZDUM
00123 *     ..
00124 *     .. Local Arrays ..
00125       INTEGER            ISAVE( 3 )
00126 *     ..
00127 *     .. External Functions ..
00128       LOGICAL            LSAME
00129       DOUBLE PRECISION   DLAMCH
00130       EXTERNAL           LSAME, DLAMCH
00131 *     ..
00132 *     .. External Subroutines ..
00133       EXTERNAL           XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGETRS, ZLACN2
00134 *     ..
00135 *     .. Intrinsic Functions ..
00136       INTRINSIC          ABS, DBLE, DIMAG, MAX
00137 *     ..
00138 *     .. Statement Functions ..
00139       DOUBLE PRECISION   CABS1
00140 *     ..
00141 *     .. Statement Function definitions ..
00142       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
00143 *     ..
00144 *     .. Executable Statements ..
00145 *
00146 *     Test the input parameters.
00147 *
00148       INFO = 0
00149       NOTRAN = LSAME( TRANS, 'N' )
00150       IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
00151      \$    LSAME( TRANS, 'C' ) ) THEN
00152          INFO = -1
00153       ELSE IF( N.LT.0 ) THEN
00154          INFO = -2
00155       ELSE IF( NRHS.LT.0 ) THEN
00156          INFO = -3
00157       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00158          INFO = -5
00159       ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
00160          INFO = -7
00161       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00162          INFO = -10
00163       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00164          INFO = -12
00165       END IF
00166       IF( INFO.NE.0 ) THEN
00167          CALL XERBLA( 'ZGERFS', -INFO )
00168          RETURN
00169       END IF
00170 *
00171 *     Quick return if possible
00172 *
00173       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00174          DO 10 J = 1, NRHS
00175             FERR( J ) = ZERO
00176             BERR( J ) = ZERO
00177    10    CONTINUE
00178          RETURN
00179       END IF
00180 *
00181       IF( NOTRAN ) THEN
00182          TRANSN = 'N'
00183          TRANST = 'C'
00184       ELSE
00185          TRANSN = 'C'
00186          TRANST = 'N'
00187       END IF
00188 *
00189 *     NZ = maximum number of nonzero elements in each row of A, plus 1
00190 *
00191       NZ = N + 1
00192       EPS = DLAMCH( 'Epsilon' )
00193       SAFMIN = DLAMCH( 'Safe minimum' )
00194       SAFE1 = NZ*SAFMIN
00195       SAFE2 = SAFE1 / EPS
00196 *
00197 *     Do for each right hand side
00198 *
00199       DO 140 J = 1, NRHS
00200 *
00201          COUNT = 1
00202          LSTRES = THREE
00203    20    CONTINUE
00204 *
00205 *        Loop until stopping criterion is satisfied.
00206 *
00207 *        Compute residual R = B - op(A) * X,
00208 *        where op(A) = A, A**T, or A**H, depending on TRANS.
00209 *
00210          CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
00211          CALL ZGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK,
00212      \$               1 )
00213 *
00214 *        Compute componentwise relative backward error from formula
00215 *
00216 *        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
00217 *
00218 *        where abs(Z) is the componentwise absolute value of the matrix
00219 *        or vector Z.  If the i-th component of the denominator is less
00220 *        than SAFE2, then SAFE1 is added to the i-th components of the
00221 *        numerator and denominator before dividing.
00222 *
00223          DO 30 I = 1, N
00224             RWORK( I ) = CABS1( B( I, J ) )
00225    30    CONTINUE
00226 *
00227 *        Compute abs(op(A))*abs(X) + abs(B).
00228 *
00229          IF( NOTRAN ) THEN
00230             DO 50 K = 1, N
00231                XK = CABS1( X( K, J ) )
00232                DO 40 I = 1, N
00233                   RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
00234    40          CONTINUE
00235    50       CONTINUE
00236          ELSE
00237             DO 70 K = 1, N
00238                S = ZERO
00239                DO 60 I = 1, N
00240                   S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
00241    60          CONTINUE
00242                RWORK( K ) = RWORK( K ) + S
00243    70       CONTINUE
00244          END IF
00245          S = ZERO
00246          DO 80 I = 1, N
00247             IF( RWORK( I ).GT.SAFE2 ) THEN
00248                S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
00249             ELSE
00250                S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
00251      \$             ( RWORK( I )+SAFE1 ) )
00252             END IF
00253    80    CONTINUE
00254          BERR( J ) = S
00255 *
00256 *        Test stopping criterion. Continue iterating if
00257 *           1) The residual BERR(J) is larger than machine epsilon, and
00258 *           2) BERR(J) decreased by at least a factor of 2 during the
00259 *              last iteration, and
00260 *           3) At most ITMAX iterations tried.
00261 *
00262          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
00263      \$       COUNT.LE.ITMAX ) THEN
00264 *
00265 *           Update solution and try again.
00266 *
00267             CALL ZGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
00268             CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
00269             LSTRES = BERR( J )
00270             COUNT = COUNT + 1
00271             GO TO 20
00272          END IF
00273 *
00274 *        Bound error from formula
00275 *
00276 *        norm(X - XTRUE) / norm(X) .le. FERR =
00277 *        norm( abs(inv(op(A)))*
00278 *           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
00279 *
00280 *        where
00281 *          norm(Z) is the magnitude of the largest component of Z
00282 *          inv(op(A)) is the inverse of op(A)
00283 *          abs(Z) is the componentwise absolute value of the matrix or
00284 *             vector Z
00285 *          NZ is the maximum number of nonzeros in any row of A, plus 1
00286 *          EPS is machine epsilon
00287 *
00288 *        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
00289 *        is incremented by SAFE1 if the i-th component of
00290 *        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
00291 *
00292 *        Use ZLACN2 to estimate the infinity-norm of the matrix
00293 *           inv(op(A)) * diag(W),
00294 *        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
00295 *
00296          DO 90 I = 1, N
00297             IF( RWORK( I ).GT.SAFE2 ) THEN
00298                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
00299             ELSE
00300                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
00301      \$                      SAFE1
00302             END IF
00303    90    CONTINUE
00304 *
00305          KASE = 0
00306   100    CONTINUE
00307          CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
00308          IF( KASE.NE.0 ) THEN
00309             IF( KASE.EQ.1 ) THEN
00310 *
00311 *              Multiply by diag(W)*inv(op(A)**H).
00312 *
00313                CALL ZGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK, N,
00314      \$                      INFO )
00315                DO 110 I = 1, N
00316                   WORK( I ) = RWORK( I )*WORK( I )
00317   110          CONTINUE
00318             ELSE
00319 *
00320 *              Multiply by inv(op(A))*diag(W).
00321 *
00322                DO 120 I = 1, N
00323                   WORK( I ) = RWORK( I )*WORK( I )
00324   120          CONTINUE
00325                CALL ZGETRS( TRANSN, N, 1, AF, LDAF, IPIV, WORK, N,
00326      \$                      INFO )
00327             END IF
00328             GO TO 100
00329          END IF
00330 *
00331 *        Normalize error.
00332 *
00333          LSTRES = ZERO
00334          DO 130 I = 1, N
00335             LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
00336   130    CONTINUE
00337          IF( LSTRES.NE.ZERO )
00338      \$      FERR( J ) = FERR( J ) / LSTRES
00339 *
00340   140 CONTINUE
00341 *
00342       RETURN
00343 *
00344 *     End of ZGERFS
00345 *
00346       END
```