LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
sebchvxx.f
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00001 *> \brief \b SEBCHVXX
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *  Definition:
00009 *  ===========
00010 *
00011 *      SUBROUTINE SEBCHVXX( THRESH, PATH )
00012 *
00013 *     .. Scalar Arguments ..
00014 *      REAL               THRESH
00015 *      CHARACTER*3        PATH
00016 *       ..
00017 *
00018 *
00019 *> \par Purpose:
00020 *  =============
00021 *>
00022 *> \verbatim
00023 *>
00024 *>  SEBCHVXX will run S**SVXX on a series of Hilbert matrices and then
00025 *>  compare the error bounds returned by SGESVXX to see if the returned
00026 *>  answer indeed falls within those bounds.
00027 *>
00028 *>  Eight test ratios will be computed.  The tests will pass if they are .LT.
00029 *>  THRESH.  There are two cases that are determined by 1 / (SQRT( N ) * EPS).
00030 *>  If that value is .LE. to the component wise reciprocal condition number,
00031 *>  it uses the guaranteed case, other wise it uses the unguaranteed case.
00032 *>
00033 *>  Test ratios:
00034 *>     Let Xc be X_computed and Xt be X_truth.
00035 *>     The norm used is the infinity norm.
00036 *>
00037 *>     Let A be the guaranteed case and B be the unguaranteed case.
00038 *>
00039 *>       1. Normwise guaranteed forward error bound.
00040 *>       A: norm ( abs( Xc - Xt ) / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ) and
00041 *>          ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N),10) * EPS.
00042 *>          If these conditions are met, the test ratio is set to be
00043 *>          ERRBND( *, nwise_i, bnd_i ) / MAX(SQRT(N), 10).  Otherwise it is 1/EPS.
00044 *>       B: For this case, SGESVXX should just return 1.  If it is less than
00045 *>          one, treat it the same as in 1A.  Otherwise it fails. (Set test
00046 *>          ratio to ERRBND( *, nwise_i, bnd_i ) * THRESH?)
00047 *>
00048 *>       2. Componentwise guaranteed forward error bound.
00049 *>       A: norm ( abs( Xc(j) - Xt(j) ) ) / norm (Xt(j)) .LE. ERRBND( *, cwise_i, bnd_i )
00050 *>          for all j .AND. ERRBND( *, cwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS.
00051 *>          If these conditions are met, the test ratio is set to be
00052 *>          ERRBND( *, cwise_i, bnd_i ) / MAX(SQRT(N), 10).  Otherwise it is 1/EPS.
00053 *>       B: Same as normwise test ratio.
00054 *>
00055 *>       3. Backwards error.
00056 *>       A: The test ratio is set to BERR/EPS.
00057 *>       B: Same test ratio.
00058 *>
00059 *>       4. Reciprocal condition number.
00060 *>       A: A condition number is computed with Xt and compared with the one
00061 *>          returned from SGESVXX.  Let RCONDc be the RCOND returned by SGESVXX
00062 *>          and RCONDt be the RCOND from the truth value.  Test ratio is set to
00063 *>          MAX(RCONDc/RCONDt, RCONDt/RCONDc).
00064 *>       B: Test ratio is set to 1 / (EPS * RCONDc).
00065 *>
00066 *>       5. Reciprocal normwise condition number.
00067 *>       A: The test ratio is set to
00068 *>          MAX(ERRBND( *, nwise_i, cond_i ) / NCOND, NCOND / ERRBND( *, nwise_i, cond_i )).
00069 *>       B: Test ratio is set to 1 / (EPS * ERRBND( *, nwise_i, cond_i )).
00070 *>
00071 *>       7. Reciprocal componentwise condition number.
00072 *>       A: Test ratio is set to
00073 *>          MAX(ERRBND( *, cwise_i, cond_i ) / CCOND, CCOND / ERRBND( *, cwise_i, cond_i )).
00074 *>       B: Test ratio is set to 1 / (EPS * ERRBND( *, cwise_i, cond_i )).
00075 *>
00076 *>     .. Parameters ..
00077 *>     NMAX is determined by the largest number in the inverse of the Hilbert
00078 *>     matrix.  Precision is exhausted when the largest entry in it is greater
00079 *>     than 2 to the power of the number of bits in the fraction of the data
00080 *>     type used plus one, which is 24 for single precision.
00081 *>     NMAX should be 6 for single and 11 for double.
00082 *> \endverbatim
00083 *
00084 *  Authors:
00085 *  ========
00086 *
00087 *> \author Univ. of Tennessee 
00088 *> \author Univ. of California Berkeley 
00089 *> \author Univ. of Colorado Denver 
00090 *> \author NAG Ltd. 
00091 *
00092 *> \date November 2011
00093 *
00094 *> \ingroup single_lin
00095 *
00096 *  =====================================================================
00097       SUBROUTINE SEBCHVXX( THRESH, PATH )
00098       IMPLICIT NONE
00099 *     .. Scalar Arguments ..
00100       REAL               THRESH
00101       CHARACTER*3        PATH
00102 
00103       INTEGER            NMAX, NPARAMS, NERRBND, NTESTS, KL, KU
00104       PARAMETER          (NMAX = 6, NPARAMS = 2, NERRBND = 3,
00105      $                    NTESTS = 6)
00106 
00107 *     .. Local Scalars ..
00108       INTEGER            N, NRHS, INFO, I ,J, k, NFAIL, LDA, LDAB,
00109      $                   LDAFB, N_AUX_TESTS
00110       CHARACTER          FACT, TRANS, UPLO, EQUED
00111       CHARACTER*2        C2
00112       CHARACTER(3)       NGUAR, CGUAR
00113       LOGICAL            printed_guide
00114       REAL               NCOND, CCOND, M, NORMDIF, NORMT, RCOND,
00115      $                   RNORM, RINORM, SUMR, SUMRI, EPS,
00116      $                   BERR(NMAX), RPVGRW, ORCOND,
00117      $                   CWISE_ERR, NWISE_ERR, CWISE_BND, NWISE_BND,
00118      $                   CWISE_RCOND, NWISE_RCOND,
00119      $                   CONDTHRESH, ERRTHRESH
00120 
00121 *     .. Local Arrays ..
00122       REAL               TSTRAT(NTESTS), RINV(NMAX), PARAMS(NPARAMS),
00123      $                   A(NMAX, NMAX), ACOPY(NMAX, NMAX),
00124      $                   INVHILB(NMAX, NMAX), R(NMAX), C(NMAX), S(NMAX),
00125      $                   WORK(NMAX * 5), B(NMAX, NMAX), X(NMAX, NMAX),
00126      $                   DIFF(NMAX, NMAX), AF(NMAX, NMAX),
00127      $                   AB( (NMAX-1)+(NMAX-1)+1, NMAX ),
00128      $                   ABCOPY( (NMAX-1)+(NMAX-1)+1, NMAX ),
00129      $                   AFB( 2*(NMAX-1)+(NMAX-1)+1, NMAX ),
00130      $                   ERRBND_N(NMAX*3), ERRBND_C(NMAX*3)
00131       INTEGER            IWORK(NMAX), IPIV(NMAX)
00132 
00133 *     .. External Functions ..
00134       REAL               SLAMCH
00135 
00136 *     .. External Subroutines ..
00137       EXTERNAL           SLAHILB, SGESVXX, SSYSVXX, SPOSVXX, SGBSVXX,
00138      $                   SLACPY, LSAMEN
00139       LOGICAL            LSAMEN
00140 
00141 *     .. Intrinsic Functions ..
00142       INTRINSIC          SQRT, MAX, ABS
00143 
00144 *     .. Parameters ..
00145       INTEGER            NWISE_I, CWISE_I
00146       PARAMETER          (NWISE_I = 1, CWISE_I = 1)
00147       INTEGER            BND_I, COND_I
00148       PARAMETER          (BND_I = 2, COND_I = 3)
00149 
00150 *     Create the loop to test out the Hilbert matrices
00151 
00152       FACT = 'E'
00153       UPLO = 'U'
00154       TRANS = 'N'
00155       EQUED = 'N'
00156       EPS = SLAMCH('Epsilon')
00157       NFAIL = 0
00158       N_AUX_TESTS = 0
00159       LDA = NMAX
00160       LDAB = (NMAX-1)+(NMAX-1)+1
00161       LDAFB = 2*(NMAX-1)+(NMAX-1)+1
00162       C2 = PATH( 2: 3 )
00163 
00164 *     Main loop to test the different Hilbert Matrices.
00165 
00166       printed_guide = .false.
00167 
00168       DO N = 1 , NMAX
00169          PARAMS(1) = -1
00170          PARAMS(2) = -1
00171 
00172          KL = N-1
00173          KU = N-1
00174          NRHS = n
00175          M = MAX(SQRT(REAL(N)), 10.0)
00176 
00177 *        Generate the Hilbert matrix, its inverse, and the
00178 *        right hand side, all scaled by the LCM(1,..,2N-1).
00179          CALL SLAHILB(N, N, A, LDA, INVHILB, LDA, B, LDA, WORK, INFO)
00180 
00181 *        Copy A into ACOPY.
00182          CALL SLACPY('ALL', N, N, A, NMAX, ACOPY, NMAX)
00183 
00184 *        Store A in band format for GB tests
00185          DO J = 1, N
00186             DO I = 1, KL+KU+1
00187                AB( I, J ) = 0.0E+0
00188             END DO
00189          END DO
00190          DO J = 1, N
00191             DO I = MAX( 1, J-KU ), MIN( N, J+KL )
00192                AB( KU+1+I-J, J ) = A( I, J )
00193             END DO
00194          END DO
00195 
00196 *        Copy AB into ABCOPY.
00197          DO J = 1, N
00198             DO I = 1, KL+KU+1
00199                ABCOPY( I, J ) = 0.0E+0
00200             END DO
00201          END DO
00202          CALL SLACPY('ALL', KL+KU+1, N, AB, LDAB, ABCOPY, LDAB)
00203 
00204 *        Call S**SVXX with default PARAMS and N_ERR_BND = 3.
00205          IF ( LSAMEN( 2, C2, 'SY' ) ) THEN
00206             CALL SSYSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA,
00207      $           IPIV, EQUED, S, B, LDA, X, LDA, ORCOND,
00208      $           RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
00209      $           PARAMS, WORK, IWORK, INFO)
00210          ELSE IF ( LSAMEN( 2, C2, 'PO' ) ) THEN
00211             CALL SPOSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA,
00212      $           EQUED, S, B, LDA, X, LDA, ORCOND,
00213      $           RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
00214      $           PARAMS, WORK, IWORK, INFO)
00215          ELSE IF ( LSAMEN( 2, C2, 'GB' ) ) THEN
00216             CALL SGBSVXX(FACT, TRANS, N, KL, KU, NRHS, ABCOPY,
00217      $           LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B,
00218      $           LDA, X, LDA, ORCOND, RPVGRW, BERR, NERRBND,
00219      $           ERRBND_N, ERRBND_C, NPARAMS, PARAMS, WORK, IWORK,
00220      $           INFO)
00221          ELSE
00222             CALL SGESVXX(FACT, TRANS, N, NRHS, ACOPY, LDA, AF, LDA,
00223      $           IPIV, EQUED, R, C, B, LDA, X, LDA, ORCOND,
00224      $           RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
00225      $           PARAMS, WORK, IWORK, INFO)
00226          END IF
00227 
00228          N_AUX_TESTS = N_AUX_TESTS + 1
00229          IF (ORCOND .LT. EPS) THEN
00230 !        Either factorization failed or the matrix is flagged, and 1 <=
00231 !        INFO <= N+1. We don't decide based on rcond anymore.
00232 !            IF (INFO .EQ. 0 .OR. INFO .GT. N+1) THEN
00233 !               NFAIL = NFAIL + 1
00234 !               WRITE (*, FMT=8000) N, INFO, ORCOND, RCOND
00235 !            END IF
00236          ELSE
00237 !        Either everything succeeded (INFO == 0) or some solution failed
00238 !        to converge (INFO > N+1).
00239             IF (INFO .GT. 0 .AND. INFO .LE. N+1) THEN
00240                NFAIL = NFAIL + 1
00241                WRITE (*, FMT=8000) C2, N, INFO, ORCOND, RCOND
00242             END IF
00243          END IF
00244 
00245 *        Calculating the difference between S**SVXX's X and the true X.
00246          DO I = 1, N
00247             DO J = 1, NRHS
00248                DIFF( I, J ) = X( I, J ) - INVHILB( I, J )
00249             END DO
00250          END DO
00251 
00252 *        Calculating the RCOND
00253          RNORM = 0
00254          RINORM = 0
00255          IF ( LSAMEN( 2, C2, 'PO' ) .OR. LSAMEN( 2, C2, 'SY' ) ) THEN
00256             DO I = 1, N
00257                SUMR = 0
00258                SUMRI = 0
00259                DO J = 1, N
00260                   SUMR = SUMR + ABS(S(I) * A(I,J) * S(J))
00261                   SUMRI = SUMRI + ABS(INVHILB(I, J) / S(J) / S(I))
00262                END DO
00263                RNORM = MAX(RNORM,SUMR)
00264                RINORM = MAX(RINORM,SUMRI)
00265             END DO
00266          ELSE IF ( LSAMEN( 2, C2, 'GE' ) .OR. LSAMEN( 2, C2, 'GB' ) )
00267      $           THEN
00268             DO I = 1, N
00269                SUMR = 0
00270                SUMRI = 0
00271                DO J = 1, N
00272                   SUMR = SUMR + ABS(R(I) * A(I,J) * C(J))
00273                   SUMRI = SUMRI + ABS(INVHILB(I, J) / R(J) / C(I))
00274                END DO
00275                RNORM = MAX(RNORM,SUMR)
00276                RINORM = MAX(RINORM,SUMRI)
00277             END DO
00278          END IF
00279 
00280          RNORM = RNORM / A(1, 1)
00281          RCOND = 1.0/(RNORM * RINORM)
00282 
00283 *        Calculating the R for normwise rcond.
00284          DO I = 1, N
00285             RINV(I) = 0.0
00286          END DO
00287          DO J = 1, N
00288             DO I = 1, N
00289                RINV(I) = RINV(I) + ABS(A(I,J))
00290             END DO
00291          END DO
00292 
00293 *        Calculating the Normwise rcond.
00294          RINORM = 0.0
00295          DO I = 1, N
00296             SUMRI = 0.0
00297             DO J = 1, N
00298                SUMRI = SUMRI + ABS(INVHILB(I,J) * RINV(J))
00299             END DO
00300             RINORM = MAX(RINORM, SUMRI)
00301          END DO
00302 
00303 !        invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm
00304 !        by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix)
00305          NCOND = A(1,1) / RINORM
00306 
00307          CONDTHRESH = M * EPS
00308          ERRTHRESH = M * EPS
00309 
00310          DO K = 1, NRHS
00311             NORMT = 0.0
00312             NORMDIF = 0.0
00313             CWISE_ERR = 0.0
00314             DO I = 1, N
00315                NORMT = MAX(ABS(INVHILB(I, K)), NORMT)
00316                NORMDIF = MAX(ABS(X(I,K) - INVHILB(I,K)), NORMDIF)
00317                IF (INVHILB(I,K) .NE. 0.0) THEN
00318                   CWISE_ERR = MAX(ABS(X(I,K) - INVHILB(I,K))
00319      $                 /ABS(INVHILB(I,K)), CWISE_ERR)
00320                ELSE IF (X(I, K) .NE. 0.0) THEN
00321                   CWISE_ERR = SLAMCH('OVERFLOW')
00322                END IF
00323             END DO
00324             IF (NORMT .NE. 0.0) THEN
00325                NWISE_ERR = NORMDIF / NORMT
00326             ELSE IF (NORMDIF .NE. 0.0) THEN
00327                NWISE_ERR = SLAMCH('OVERFLOW')
00328             ELSE
00329                NWISE_ERR = 0.0
00330             ENDIF
00331 
00332             DO I = 1, N
00333                RINV(I) = 0.0
00334             END DO
00335             DO J = 1, N
00336                DO I = 1, N
00337                   RINV(I) = RINV(I) + ABS(A(I, J) * INVHILB(J, K))
00338                END DO
00339             END DO
00340             RINORM = 0.0
00341             DO I = 1, N
00342                SUMRI = 0.0
00343                DO J = 1, N
00344                   SUMRI = SUMRI
00345      $                 + ABS(INVHILB(I, J) * RINV(J) / INVHILB(I, K))
00346                END DO
00347                RINORM = MAX(RINORM, SUMRI)
00348             END DO
00349 !        invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm
00350 !        by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix)
00351             CCOND = A(1,1)/RINORM
00352 
00353 !        Forward error bound tests
00354             NWISE_BND = ERRBND_N(K + (BND_I-1)*NRHS)
00355             CWISE_BND = ERRBND_C(K + (BND_I-1)*NRHS)
00356             NWISE_RCOND = ERRBND_N(K + (COND_I-1)*NRHS)
00357             CWISE_RCOND = ERRBND_C(K + (COND_I-1)*NRHS)
00358 !            write (*,*) 'nwise : ', n, k, ncond, nwise_rcond,
00359 !     $           condthresh, ncond.ge.condthresh
00360 !            write (*,*) 'nwise2: ', k, nwise_bnd, nwise_err, errthresh
00361 
00362             IF (NCOND .GE. CONDTHRESH) THEN
00363                NGUAR = 'YES'
00364                IF (NWISE_BND .GT. ERRTHRESH) THEN
00365                   TSTRAT(1) = 1/(2.0*EPS)
00366                ELSE
00367 
00368                   IF (NWISE_BND .NE. 0.0) THEN
00369                      TSTRAT(1) = NWISE_ERR / NWISE_BND
00370                   ELSE IF (NWISE_ERR .NE. 0.0) THEN
00371                      TSTRAT(1) = 1/(16.0*EPS)
00372                   ELSE
00373                      TSTRAT(1) = 0.0
00374                   END IF
00375                   IF (TSTRAT(1) .GT. 1.0) THEN
00376                      TSTRAT(1) = 1/(4.0*EPS)
00377                   END IF
00378                END IF
00379             ELSE
00380                NGUAR = 'NO'
00381                IF (NWISE_BND .LT. 1.0) THEN
00382                   TSTRAT(1) = 1/(8.0*EPS)
00383                ELSE
00384                   TSTRAT(1) = 1.0
00385                END IF
00386             END IF
00387 !            write (*,*) 'cwise : ', n, k, ccond, cwise_rcond,
00388 !     $           condthresh, ccond.ge.condthresh
00389 !            write (*,*) 'cwise2: ', k, cwise_bnd, cwise_err, errthresh
00390             IF (CCOND .GE. CONDTHRESH) THEN
00391                CGUAR = 'YES'
00392 
00393                IF (CWISE_BND .GT. ERRTHRESH) THEN
00394                   TSTRAT(2) = 1/(2.0*EPS)
00395                ELSE
00396                   IF (CWISE_BND .NE. 0.0) THEN
00397                      TSTRAT(2) = CWISE_ERR / CWISE_BND
00398                   ELSE IF (CWISE_ERR .NE. 0.0) THEN
00399                      TSTRAT(2) = 1/(16.0*EPS)
00400                   ELSE
00401                      TSTRAT(2) = 0.0
00402                   END IF
00403                   IF (TSTRAT(2) .GT. 1.0) TSTRAT(2) = 1/(4.0*EPS)
00404                END IF
00405             ELSE
00406                CGUAR = 'NO'
00407                IF (CWISE_BND .LT. 1.0) THEN
00408                   TSTRAT(2) = 1/(8.0*EPS)
00409                ELSE
00410                   TSTRAT(2) = 1.0
00411                END IF
00412             END IF
00413 
00414 !     Backwards error test
00415             TSTRAT(3) = BERR(K)/EPS
00416 
00417 !     Condition number tests
00418             TSTRAT(4) = RCOND / ORCOND
00419             IF (RCOND .GE. CONDTHRESH .AND. TSTRAT(4) .LT. 1.0)
00420      $         TSTRAT(4) = 1.0 / TSTRAT(4)
00421 
00422             TSTRAT(5) = NCOND / NWISE_RCOND
00423             IF (NCOND .GE. CONDTHRESH .AND. TSTRAT(5) .LT. 1.0)
00424      $         TSTRAT(5) = 1.0 / TSTRAT(5)
00425 
00426             TSTRAT(6) = CCOND / NWISE_RCOND
00427             IF (CCOND .GE. CONDTHRESH .AND. TSTRAT(6) .LT. 1.0)
00428      $         TSTRAT(6) = 1.0 / TSTRAT(6)
00429 
00430             DO I = 1, NTESTS
00431                IF (TSTRAT(I) .GT. THRESH) THEN
00432                   IF (.NOT.PRINTED_GUIDE) THEN
00433                      WRITE(*,*)
00434                      WRITE( *, 9996) 1
00435                      WRITE( *, 9995) 2
00436                      WRITE( *, 9994) 3
00437                      WRITE( *, 9993) 4
00438                      WRITE( *, 9992) 5
00439                      WRITE( *, 9991) 6
00440                      WRITE( *, 9990) 7
00441                      WRITE( *, 9989) 8
00442                      WRITE(*,*)
00443                      PRINTED_GUIDE = .TRUE.
00444                   END IF
00445                   WRITE( *, 9999) C2, N, K, NGUAR, CGUAR, I, TSTRAT(I)
00446                   NFAIL = NFAIL + 1
00447                END IF
00448             END DO
00449       END DO
00450 
00451 c$$$         WRITE(*,*)
00452 c$$$         WRITE(*,*) 'Normwise Error Bounds'
00453 c$$$         WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,nwise_i,bnd_i)
00454 c$$$         WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,nwise_i,cond_i)
00455 c$$$         WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,nwise_i,rawbnd_i)
00456 c$$$         WRITE(*,*)
00457 c$$$         WRITE(*,*) 'Componentwise Error Bounds'
00458 c$$$         WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,cwise_i,bnd_i)
00459 c$$$         WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,cwise_i,cond_i)
00460 c$$$         WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,cwise_i,rawbnd_i)
00461 c$$$         print *, 'Info: ', info
00462 c$$$         WRITE(*,*)
00463 *         WRITE(*,*) 'TSTRAT: ',TSTRAT
00464 
00465       END DO
00466 
00467       WRITE(*,*)
00468       IF( NFAIL .GT. 0 ) THEN
00469          WRITE(*,9998) C2, NFAIL, NTESTS*N+N_AUX_TESTS
00470       ELSE
00471          WRITE(*,9997) C2
00472       END IF
00473  9999 FORMAT( ' S', A2, 'SVXX: N =', I2, ', RHS = ', I2,
00474      $     ', NWISE GUAR. = ', A, ', CWISE GUAR. = ', A,
00475      $     ' test(',I1,') =', G12.5 )
00476  9998 FORMAT( ' S', A2, 'SVXX: ', I6, ' out of ', I6,
00477      $     ' tests failed to pass the threshold' )
00478  9997 FORMAT( ' S', A2, 'SVXX passed the tests of error bounds' )
00479 *     Test ratios.
00480  9996 FORMAT( 3X, I2, ': Normwise guaranteed forward error', / 5X,
00481      $     'Guaranteed case: if norm ( abs( Xc - Xt )',
00482      $     ' / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ), then',
00483      $     / 5X,
00484      $     'ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS')
00485  9995 FORMAT( 3X, I2, ': Componentwise guaranteed forward error' )
00486  9994 FORMAT( 3X, I2, ': Backwards error' )
00487  9993 FORMAT( 3X, I2, ': Reciprocal condition number' )
00488  9992 FORMAT( 3X, I2, ': Reciprocal normwise condition number' )
00489  9991 FORMAT( 3X, I2, ': Raw normwise error estimate' )
00490  9990 FORMAT( 3X, I2, ': Reciprocal componentwise condition number' )
00491  9989 FORMAT( 3X, I2, ': Raw componentwise error estimate' )
00492 
00493  8000 FORMAT( ' S', A2, 'SVXX: N =', I2, ', INFO = ', I3,
00494      $     ', ORCOND = ', G12.5, ', real RCOND = ', G12.5 )
00495 
00496       END
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