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SUBROUTINE DDRGVX( NSIZE, THRESH, NIN, NOUT, A, LDA, B, AI, BI,
$ ALPHAR, ALPHAI, BETA, VL, VR, ILO, IHI, LSCALE, $ RSCALE, S, DTRU, DIF, DIFTRU, WORK, LWORK, $ IWORK, LIWORK, RESULT, BWORK, INFO ) * * -- LAPACK test routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER IHI, ILO, INFO, LDA, LIWORK, LWORK, NIN, NOUT, $ NSIZE DOUBLE PRECISION THRESH * .. * .. Array Arguments .. LOGICAL BWORK( * ) INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), AI( LDA, * ), ALPHAI( * ), $ ALPHAR( * ), B( LDA, * ), BETA( * ), $ BI( LDA, * ), DIF( * ), DIFTRU( * ), DTRU( * ), $ LSCALE( * ), RESULT( 4 ), RSCALE( * ), S( * ), $ VL( LDA, * ), VR( LDA, * ), WORK( * ) * .. * * Purpose * ======= * * DDRGVX checks the nonsymmetric generalized eigenvalue problem * expert driver DGGEVX. * * DGGEVX computes the generalized eigenvalues, (optionally) the left * and/or right eigenvectors, (optionally) computes a balancing * transformation to improve the conditioning, and (optionally) * reciprocal condition numbers for the eigenvalues and eigenvectors. * * When DDRGVX is called with NSIZE > 0, two types of test matrix pairs * are generated by the subroutine DLATM6 and test the driver DGGEVX. * The test matrices have the known exact condition numbers for * eigenvalues. For the condition numbers of the eigenvectors * corresponding the first and last eigenvalues are also know * ``exactly'' (see DLATM6). * * For each matrix pair, the following tests will be performed and * compared with the threshhold THRESH. * * (1) max over all left eigenvalue/-vector pairs (beta/alpha,l) of * * | l**H * (beta A - alpha B) | / ( ulp max( |beta A|, |alpha B| ) ) * * where l**H is the conjugate tranpose of l. * * (2) max over all right eigenvalue/-vector pairs (beta/alpha,r) of * * | (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) ) * * (3) The condition number S(i) of eigenvalues computed by DGGEVX * differs less than a factor THRESH from the exact S(i) (see * DLATM6). * * (4) DIF(i) computed by DTGSNA differs less than a factor 10*THRESH * from the exact value (for the 1st and 5th vectors only). * * Test Matrices * ============= * * Two kinds of test matrix pairs * * (A, B) = inverse(YH) * (Da, Db) * inverse(X) * * are used in the tests: * * 1: Da = 1+a 0 0 0 0 Db = 1 0 0 0 0 * 0 2+a 0 0 0 0 1 0 0 0 * 0 0 3+a 0 0 0 0 1 0 0 * 0 0 0 4+a 0 0 0 0 1 0 * 0 0 0 0 5+a , 0 0 0 0 1 , and * * 2: Da = 1 -1 0 0 0 Db = 1 0 0 0 0 * 1 1 0 0 0 0 1 0 0 0 * 0 0 1 0 0 0 0 1 0 0 * 0 0 0 1+a 1+b 0 0 0 1 0 * 0 0 0 -1-b 1+a , 0 0 0 0 1 . * * In both cases the same inverse(YH) and inverse(X) are used to compute * (A, B), giving the exact eigenvectors to (A,B) as (YH, X): * * YH: = 1 0 -y y -y X = 1 0 -x -x x * 0 1 -y y -y 0 1 x -x -x * 0 0 1 0 0 0 0 1 0 0 * 0 0 0 1 0 0 0 0 1 0 * 0 0 0 0 1, 0 0 0 0 1 , where * * a, b, x and y will have all values independently of each other from * { sqrt(sqrt(ULP)), 0.1, 1, 10, 1/sqrt(sqrt(ULP)) }. * * Arguments * ========= * * NSIZE (input) INTEGER * The number of sizes of matrices to use. NSIZE must be at * least zero. If it is zero, no randomly generated matrices * are tested, but any test matrices read from NIN will be * tested. * * THRESH (input) DOUBLE PRECISION * A test will count as "failed" if the "error", computed as * described above, exceeds THRESH. Note that the error * is scaled to be O(1), so THRESH should be a reasonably * small multiple of 1, e.g., 10 or 100. In particular, * it should not depend on the precision (single vs. double) * or the size of the matrix. It must be at least zero. * * NIN (input) INTEGER * The FORTRAN unit number for reading in the data file of * problems to solve. * * NOUT (input) INTEGER * The FORTRAN unit number for printing out error messages * (e.g., if a routine returns IINFO not equal to 0.) * * A (workspace) DOUBLE PRECISION array, dimension (LDA, NSIZE) * Used to hold the matrix whose eigenvalues are to be * computed. On exit, A contains the last matrix actually used. * * LDA (input) INTEGER * The leading dimension of A, B, AI, BI, Ao, and Bo. * It must be at least 1 and at least NSIZE. * * B (workspace) DOUBLE PRECISION array, dimension (LDA, NSIZE) * Used to hold the matrix whose eigenvalues are to be * computed. On exit, B contains the last matrix actually used. * * AI (workspace) DOUBLE PRECISION array, dimension (LDA, NSIZE) * Copy of A, modified by DGGEVX. * * BI (workspace) DOUBLE PRECISION array, dimension (LDA, NSIZE) * Copy of B, modified by DGGEVX. * * ALPHAR (workspace) DOUBLE PRECISION array, dimension (NSIZE) * ALPHAI (workspace) DOUBLE PRECISION array, dimension (NSIZE) * BETA (workspace) DOUBLE PRECISION array, dimension (NSIZE) * On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues. * * VL (workspace) DOUBLE PRECISION array, dimension (LDA, NSIZE) * VL holds the left eigenvectors computed by DGGEVX. * * VR (workspace) DOUBLE PRECISION array, dimension (LDA, NSIZE) * VR holds the right eigenvectors computed by DGGEVX. * * ILO (output/workspace) INTEGER * * IHI (output/workspace) INTEGER * * LSCALE (output/workspace) DOUBLE PRECISION array, dimension (N) * * RSCALE (output/workspace) DOUBLE PRECISION array, dimension (N) * * S (output/workspace) DOUBLE PRECISION array, dimension (N) * * DTRU (output/workspace) DOUBLE PRECISION array, dimension (N) * * DIF (output/workspace) DOUBLE PRECISION array, dimension (N) * * DIFTRU (output/workspace) DOUBLE PRECISION array, dimension (N) * * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK) * * LWORK (input) INTEGER * Leading dimension of WORK. LWORK >= 2*N*N+12*N+16. * * IWORK (workspace) INTEGER array, dimension (LIWORK) * * LIWORK (input) INTEGER * Leading dimension of IWORK. Must be at least N+6. * * RESULT (output/workspace) DOUBLE PRECISION array, dimension (4) * * BWORK (workspace) LOGICAL array, dimension (N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * > 0: A routine returned an error code. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE, TEN, TNTH PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 1.0D+1, $ TNTH = 1.0D-1 ) * .. * .. Local Scalars .. INTEGER I, IPTYPE, IWA, IWB, IWX, IWY, J, LINFO, $ MAXWRK, MINWRK, N, NERRS, NMAX, NPTKNT, NTESTT DOUBLE PRECISION ABNORM, ANORM, BNORM, RATIO1, RATIO2, THRSH2, $ ULP, ULPINV * .. * .. Local Arrays .. DOUBLE PRECISION WEIGHT( 5 ) * .. * .. External Functions .. INTEGER ILAENV DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL ILAENV, DLAMCH, DLANGE * .. * .. External Subroutines .. EXTERNAL ALASVM, DGET52, DGGEVX, DLACPY, DLATM6, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT * .. * .. Executable Statements .. * * Check for errors * INFO = 0 * NMAX = 5 * IF( NSIZE.LT.0 ) THEN INFO = -1 ELSE IF( THRESH.LT.ZERO ) THEN INFO = -2 ELSE IF( NIN.LE.0 ) THEN INFO = -3 ELSE IF( NOUT.LE.0 ) THEN INFO = -4 ELSE IF( LDA.LT.1 .OR. LDA.LT.NMAX ) THEN INFO = -6 ELSE IF( LIWORK.LT.NMAX+6 ) THEN INFO = -26 END IF * * Compute workspace * (Note: Comments in the code beginning "Workspace:" describe the * minimal amount of workspace needed at that point in the code, * as well as the preferred amount for good performance. * NB refers to the optimal block size for the immediately * following subroutine, as returned by ILAENV.) * MINWRK = 1 IF( INFO.EQ.0 .AND. LWORK.GE.1 ) THEN MINWRK = 2*NMAX*NMAX + 12*NMAX + 16 MAXWRK = 6*NMAX + NMAX*ILAENV( 1, 'DGEQRF', ' ', NMAX, 1, NMAX, $ 0 ) MAXWRK = MAX( MAXWRK, 2*NMAX*NMAX+12*NMAX+16 ) WORK( 1 ) = MAXWRK END IF * IF( LWORK.LT.MINWRK ) $ INFO = -24 * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DDRGVX', -INFO ) RETURN END IF * N = 5 ULP = DLAMCH( 'P' ) ULPINV = ONE / ULP THRSH2 = TEN*THRESH NERRS = 0 NPTKNT = 0 NTESTT = 0 * IF( NSIZE.EQ.0 ) $ GO TO 90 * * Parameters used for generating test matrices. * WEIGHT( 1 ) = SQRT( SQRT( ULP ) ) WEIGHT( 2 ) = TNTH WEIGHT( 3 ) = ONE WEIGHT( 4 ) = ONE / WEIGHT( 2 ) WEIGHT( 5 ) = ONE / WEIGHT( 1 ) * DO 80 IPTYPE = 1, 2 DO 70 IWA = 1, 5 DO 60 IWB = 1, 5 DO 50 IWX = 1, 5 DO 40 IWY = 1, 5 * * generated a test matrix pair * CALL DLATM6( IPTYPE, 5, A, LDA, B, VR, LDA, VL, $ LDA, WEIGHT( IWA ), WEIGHT( IWB ), $ WEIGHT( IWX ), WEIGHT( IWY ), DTRU, $ DIFTRU ) * * Compute eigenvalues/eigenvectors of (A, B). * Compute eigenvalue/eigenvector condition numbers * using computed eigenvectors. * CALL DLACPY( 'F', N, N, A, LDA, AI, LDA ) CALL DLACPY( 'F', N, N, B, LDA, BI, LDA ) * CALL DGGEVX( 'N', 'V', 'V', 'B', N, AI, LDA, BI, $ LDA, ALPHAR, ALPHAI, BETA, VL, LDA, $ VR, LDA, ILO, IHI, LSCALE, RSCALE, $ ANORM, BNORM, S, DIF, WORK, LWORK, $ IWORK, BWORK, LINFO ) IF( LINFO.NE.0 ) THEN RESULT( 1 ) = ULPINV WRITE( NOUT, FMT = 9999 )'DGGEVX', LINFO, N, $ IPTYPE GO TO 30 END IF * * Compute the norm(A, B) * CALL DLACPY( 'Full', N, N, AI, LDA, WORK, N ) CALL DLACPY( 'Full', N, N, BI, LDA, WORK( N*N+1 ), $ N ) ABNORM = DLANGE( 'Fro', N, 2*N, WORK, N, WORK ) * * Tests (1) and (2) * RESULT( 1 ) = ZERO CALL DGET52( .TRUE., N, A, LDA, B, LDA, VL, LDA, $ ALPHAR, ALPHAI, BETA, WORK, $ RESULT( 1 ) ) IF( RESULT( 2 ).GT.THRESH ) THEN WRITE( NOUT, FMT = 9998 )'Left', 'DGGEVX', $ RESULT( 2 ), N, IPTYPE, IWA, IWB, IWX, IWY END IF * RESULT( 2 ) = ZERO CALL DGET52( .FALSE., N, A, LDA, B, LDA, VR, LDA, $ ALPHAR, ALPHAI, BETA, WORK, $ RESULT( 2 ) ) IF( RESULT( 3 ).GT.THRESH ) THEN WRITE( NOUT, FMT = 9998 )'Right', 'DGGEVX', $ RESULT( 3 ), N, IPTYPE, IWA, IWB, IWX, IWY END IF * * Test (3) * RESULT( 3 ) = ZERO DO 10 I = 1, N IF( S( I ).EQ.ZERO ) THEN IF( DTRU( I ).GT.ABNORM*ULP ) $ RESULT( 3 ) = ULPINV ELSE IF( DTRU( I ).EQ.ZERO ) THEN IF( S( I ).GT.ABNORM*ULP ) $ RESULT( 3 ) = ULPINV ELSE WORK( I ) = MAX( ABS( DTRU( I ) / S( I ) ), $ ABS( S( I ) / DTRU( I ) ) ) RESULT( 3 ) = MAX( RESULT( 3 ), WORK( I ) ) END IF 10 CONTINUE * * Test (4) * RESULT( 4 ) = ZERO IF( DIF( 1 ).EQ.ZERO ) THEN IF( DIFTRU( 1 ).GT.ABNORM*ULP ) $ RESULT( 4 ) = ULPINV ELSE IF( DIFTRU( 1 ).EQ.ZERO ) THEN IF( DIF( 1 ).GT.ABNORM*ULP ) $ RESULT( 4 ) = ULPINV ELSE IF( DIF( 5 ).EQ.ZERO ) THEN IF( DIFTRU( 5 ).GT.ABNORM*ULP ) $ RESULT( 4 ) = ULPINV ELSE IF( DIFTRU( 5 ).EQ.ZERO ) THEN IF( DIF( 5 ).GT.ABNORM*ULP ) $ RESULT( 4 ) = ULPINV ELSE RATIO1 = MAX( ABS( DIFTRU( 1 ) / DIF( 1 ) ), $ ABS( DIF( 1 ) / DIFTRU( 1 ) ) ) RATIO2 = MAX( ABS( DIFTRU( 5 ) / DIF( 5 ) ), $ ABS( DIF( 5 ) / DIFTRU( 5 ) ) ) RESULT( 4 ) = MAX( RATIO1, RATIO2 ) END IF * NTESTT = NTESTT + 4 * * Print out tests which fail. * DO 20 J = 1, 4 IF( ( RESULT( J ).GE.THRSH2 .AND. J.GE.4 ) .OR. $ ( RESULT( J ).GE.THRESH .AND. J.LE.3 ) ) $ THEN * * If this is the first test to fail, * print a header to the data file. * IF( NERRS.EQ.0 ) THEN WRITE( NOUT, FMT = 9997 )'DXV' * * Print out messages for built-in examples * * Matrix types * WRITE( NOUT, FMT = 9995 ) WRITE( NOUT, FMT = 9994 ) WRITE( NOUT, FMT = 9993 ) * * Tests performed * WRITE( NOUT, FMT = 9992 )'''', $ 'transpose', '''' * END IF NERRS = NERRS + 1 IF( RESULT( J ).LT.10000.0D0 ) THEN WRITE( NOUT, FMT = 9991 )IPTYPE, IWA, $ IWB, IWX, IWY, J, RESULT( J ) ELSE WRITE( NOUT, FMT = 9990 )IPTYPE, IWA, $ IWB, IWX, IWY, J, RESULT( J ) END IF END IF 20 CONTINUE * 30 CONTINUE * 40 CONTINUE 50 CONTINUE 60 CONTINUE 70 CONTINUE 80 CONTINUE * GO TO 150 * 90 CONTINUE * * Read in data from file to check accuracy of condition estimation * Read input data until N=0 * READ( NIN, FMT = *, END = 150 )N IF( N.EQ.0 ) $ GO TO 150 DO 100 I = 1, N READ( NIN, FMT = * )( A( I, J ), J = 1, N ) 100 CONTINUE DO 110 I = 1, N READ( NIN, FMT = * )( B( I, J ), J = 1, N ) 110 CONTINUE READ( NIN, FMT = * )( DTRU( I ), I = 1, N ) READ( NIN, FMT = * )( DIFTRU( I ), I = 1, N ) * NPTKNT = NPTKNT + 1 * * Compute eigenvalues/eigenvectors of (A, B). * Compute eigenvalue/eigenvector condition numbers * using computed eigenvectors. * CALL DLACPY( 'F', N, N, A, LDA, AI, LDA ) CALL DLACPY( 'F', N, N, B, LDA, BI, LDA ) * CALL DGGEVX( 'N', 'V', 'V', 'B', N, AI, LDA, BI, LDA, ALPHAR, $ ALPHAI, BETA, VL, LDA, VR, LDA, ILO, IHI, LSCALE, $ RSCALE, ANORM, BNORM, S, DIF, WORK, LWORK, IWORK, $ BWORK, LINFO ) * IF( LINFO.NE.0 ) THEN RESULT( 1 ) = ULPINV WRITE( NOUT, FMT = 9987 )'DGGEVX', LINFO, N, NPTKNT GO TO 140 END IF * * Compute the norm(A, B) * CALL DLACPY( 'Full', N, N, AI, LDA, WORK, N ) CALL DLACPY( 'Full', N, N, BI, LDA, WORK( N*N+1 ), N ) ABNORM = DLANGE( 'Fro', N, 2*N, WORK, N, WORK ) * * Tests (1) and (2) * RESULT( 1 ) = ZERO CALL DGET52( .TRUE., N, A, LDA, B, LDA, VL, LDA, ALPHAR, ALPHAI, $ BETA, WORK, RESULT( 1 ) ) IF( RESULT( 2 ).GT.THRESH ) THEN WRITE( NOUT, FMT = 9986 )'Left', 'DGGEVX', RESULT( 2 ), N, $ NPTKNT END IF * RESULT( 2 ) = ZERO CALL DGET52( .FALSE., N, A, LDA, B, LDA, VR, LDA, ALPHAR, ALPHAI, $ BETA, WORK, RESULT( 2 ) ) IF( RESULT( 3 ).GT.THRESH ) THEN WRITE( NOUT, FMT = 9986 )'Right', 'DGGEVX', RESULT( 3 ), N, $ NPTKNT END IF * * Test (3) * RESULT( 3 ) = ZERO DO 120 I = 1, N IF( S( I ).EQ.ZERO ) THEN IF( DTRU( I ).GT.ABNORM*ULP ) $ RESULT( 3 ) = ULPINV ELSE IF( DTRU( I ).EQ.ZERO ) THEN IF( S( I ).GT.ABNORM*ULP ) $ RESULT( 3 ) = ULPINV ELSE WORK( I ) = MAX( ABS( DTRU( I ) / S( I ) ), $ ABS( S( I ) / DTRU( I ) ) ) RESULT( 3 ) = MAX( RESULT( 3 ), WORK( I ) ) END IF 120 CONTINUE * * Test (4) * RESULT( 4 ) = ZERO IF( DIF( 1 ).EQ.ZERO ) THEN IF( DIFTRU( 1 ).GT.ABNORM*ULP ) $ RESULT( 4 ) = ULPINV ELSE IF( DIFTRU( 1 ).EQ.ZERO ) THEN IF( DIF( 1 ).GT.ABNORM*ULP ) $ RESULT( 4 ) = ULPINV ELSE IF( DIF( 5 ).EQ.ZERO ) THEN IF( DIFTRU( 5 ).GT.ABNORM*ULP ) $ RESULT( 4 ) = ULPINV ELSE IF( DIFTRU( 5 ).EQ.ZERO ) THEN IF( DIF( 5 ).GT.ABNORM*ULP ) $ RESULT( 4 ) = ULPINV ELSE RATIO1 = MAX( ABS( DIFTRU( 1 ) / DIF( 1 ) ), $ ABS( DIF( 1 ) / DIFTRU( 1 ) ) ) RATIO2 = MAX( ABS( DIFTRU( 5 ) / DIF( 5 ) ), $ ABS( DIF( 5 ) / DIFTRU( 5 ) ) ) RESULT( 4 ) = MAX( RATIO1, RATIO2 ) END IF * NTESTT = NTESTT + 4 * * Print out tests which fail. * DO 130 J = 1, 4 IF( RESULT( J ).GE.THRSH2 ) THEN * * If this is the first test to fail, * print a header to the data file. * IF( NERRS.EQ.0 ) THEN WRITE( NOUT, FMT = 9997 )'DXV' * * Print out messages for built-in examples * * Matrix types * WRITE( NOUT, FMT = 9996 ) * * Tests performed * WRITE( NOUT, FMT = 9992 )'''', 'transpose', '''' * END IF NERRS = NERRS + 1 IF( RESULT( J ).LT.10000.0D0 ) THEN WRITE( NOUT, FMT = 9989 )NPTKNT, N, J, RESULT( J ) ELSE WRITE( NOUT, FMT = 9988 )NPTKNT, N, J, RESULT( J ) END IF END IF 130 CONTINUE * 140 CONTINUE * GO TO 90 150 CONTINUE * * Summary * CALL ALASVM( 'DXV', NOUT, NERRS, NTESTT, 0 ) * WORK( 1 ) = MAXWRK * RETURN * 9999 FORMAT( ' DDRGVX: ', A, ' returned INFO=', I6, '.', / 9X, 'N=', $ I6, ', JTYPE=', I6, ')' ) * 9998 FORMAT( ' DDRGVX: ', A, ' Eigenvectors from ', A, ' incorrectly ', $ 'normalized.', / ' Bits of error=', 0P, G10.3, ',', 9X, $ 'N=', I6, ', JTYPE=', I6, ', IWA=', I5, ', IWB=', I5, $ ', IWX=', I5, ', IWY=', I5 ) * 9997 FORMAT( / 1X, A3, ' -- Real Expert Eigenvalue/vector', $ ' problem driver' ) * 9996 FORMAT( ' Input Example' ) * 9995 FORMAT( ' Matrix types: ', / ) * 9994 FORMAT( ' TYPE 1: Da is diagonal, Db is identity, ', $ / ' A = Y^(-H) Da X^(-1), B = Y^(-H) Db X^(-1) ', $ / ' YH and X are left and right eigenvectors. ', / ) * 9993 FORMAT( ' TYPE 2: Da is quasi-diagonal, Db is identity, ', $ / ' A = Y^(-H) Da X^(-1), B = Y^(-H) Db X^(-1) ', $ / ' YH and X are left and right eigenvectors. ', / ) * 9992 FORMAT( / ' Tests performed: ', / 4X, $ ' a is alpha, b is beta, l is a left eigenvector, ', / 4X, $ ' r is a right eigenvector and ', A, ' means ', A, '.', $ / ' 1 = max | ( b A - a B )', A, ' l | / const.', $ / ' 2 = max | ( b A - a B ) r | / const.', $ / ' 3 = max ( Sest/Stru, Stru/Sest ) ', $ ' over all eigenvalues', / $ ' 4 = max( DIFest/DIFtru, DIFtru/DIFest ) ', $ ' over the 1st and 5th eigenvectors', / ) * 9991 FORMAT( ' Type=', I2, ',', ' IWA=', I2, ', IWB=', I2, ', IWX=', $ I2, ', IWY=', I2, ', result ', I2, ' is', 0P, F8.2 ) 9990 FORMAT( ' Type=', I2, ',', ' IWA=', I2, ', IWB=', I2, ', IWX=', $ I2, ', IWY=', I2, ', result ', I2, ' is', 1P, D10.3 ) 9989 FORMAT( ' Input example #', I2, ', matrix order=', I4, ',', $ ' result ', I2, ' is', 0P, F8.2 ) 9988 FORMAT( ' Input example #', I2, ', matrix order=', I4, ',', $ ' result ', I2, ' is', 1P, D10.3 ) 9987 FORMAT( ' DDRGVX: ', A, ' returned INFO=', I6, '.', / 9X, 'N=', $ I6, ', Input example #', I2, ')' ) * 9986 FORMAT( ' DDRGVX: ', A, ' Eigenvectors from ', A, ' incorrectly ', $ 'normalized.', / ' Bits of error=', 0P, G10.3, ',', 9X, $ 'N=', I6, ', Input Example #', I2, ')' ) * * * End of DDRGVX * END |