SUBROUTINE SCHKSB( NSIZES, NN, NWDTHS, KK, NTYPES, DOTYPE, ISEED, \$ THRESH, NOUNIT, A, LDA, SD, SE, U, LDU, WORK, \$ LWORK, RESULT, INFO ) * * -- LAPACK test routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDU, LWORK, NOUNIT, NSIZES, NTYPES, \$ NWDTHS REAL THRESH * .. * .. Array Arguments .. LOGICAL DOTYPE( * ) INTEGER ISEED( 4 ), KK( * ), NN( * ) REAL A( LDA, * ), RESULT( * ), SD( * ), SE( * ), \$ U( LDU, * ), WORK( * ) * .. * * Purpose * ======= * * SCHKSB tests the reduction of a symmetric band matrix to tridiagonal * form, used with the symmetric eigenvalue problem. * * SSBTRD factors a symmetric band matrix A as U S U' , where ' means * transpose, S is symmetric tridiagonal, and U is orthogonal. * SSBTRD can use either just the lower or just the upper triangle * of A; SCHKSB checks both cases. * * When SCHKSB is called, a number of matrix "sizes" ("n's"), a number * of bandwidths ("k's"), and a number of matrix "types" are * specified. For each size ("n"), each bandwidth ("k") less than or * equal to "n", and each type of matrix, one matrix will be generated * and used to test the symmetric banded reduction routine. For each * matrix, a number of tests will be performed: * * (1) | A - V S V' | / ( |A| n ulp ) computed by SSBTRD with * UPLO='U' * * (2) | I - UU' | / ( n ulp ) * * (3) | A - V S V' | / ( |A| n ulp ) computed by SSBTRD with * UPLO='L' * * (4) | I - UU' | / ( n ulp ) * * The "sizes" are specified by an array NN(1:NSIZES); the value of * each element NN(j) specifies one size. * The "types" are specified by a logical array DOTYPE( 1:NTYPES ); * if DOTYPE(j) is .TRUE., then matrix type "j" will be generated. * Currently, the list of possible types is: * * (1) The zero matrix. * (2) The identity matrix. * * (3) A diagonal matrix with evenly spaced entries * 1, ..., ULP and random signs. * (ULP = (first number larger than 1) - 1 ) * (4) A diagonal matrix with geometrically spaced entries * 1, ..., ULP and random signs. * (5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP * and random signs. * * (6) Same as (4), but multiplied by SQRT( overflow threshold ) * (7) Same as (4), but multiplied by SQRT( underflow threshold ) * * (8) A matrix of the form U' D U, where U is orthogonal and * D has evenly spaced entries 1, ..., ULP with random signs * on the diagonal. * * (9) A matrix of the form U' D U, where U is orthogonal and * D has geometrically spaced entries 1, ..., ULP with random * signs on the diagonal. * * (10) A matrix of the form U' D U, where U is orthogonal and * D has "clustered" entries 1, ULP,..., ULP with random * signs on the diagonal. * * (11) Same as (8), but multiplied by SQRT( overflow threshold ) * (12) Same as (8), but multiplied by SQRT( underflow threshold ) * * (13) Symmetric matrix with random entries chosen from (-1,1). * (14) Same as (13), but multiplied by SQRT( overflow threshold ) * (15) Same as (13), but multiplied by SQRT( underflow threshold ) * * Arguments * ========= * * NSIZES (input) INTEGER * The number of sizes of matrices to use. If it is zero, * SCHKSB does nothing. It must be at least zero. * * NN (input) INTEGER array, dimension (NSIZES) * An array containing the sizes to be used for the matrices. * Zero values will be skipped. The values must be at least * zero. * * NWDTHS (input) INTEGER * The number of bandwidths to use. If it is zero, * SCHKSB does nothing. It must be at least zero. * * KK (input) INTEGER array, dimension (NWDTHS) * An array containing the bandwidths to be used for the band * matrices. The values must be at least zero. * * NTYPES (input) INTEGER * The number of elements in DOTYPE. If it is zero, SCHKSB * does nothing. It must be at least zero. If it is MAXTYP+1 * and NSIZES is 1, then an additional type, MAXTYP+1 is * defined, which is to use whatever matrix is in A. This * is only useful if DOTYPE(1:MAXTYP) is .FALSE. and * DOTYPE(MAXTYP+1) is .TRUE. . * * DOTYPE (input) LOGICAL array, dimension (NTYPES) * If DOTYPE(j) is .TRUE., then for each size in NN a * matrix of that size and of type j will be generated. * If NTYPES is smaller than the maximum number of types * defined (PARAMETER MAXTYP), then types NTYPES+1 through * MAXTYP will not be generated. If NTYPES is larger * than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) * will be ignored. * * ISEED (input/output) INTEGER array, dimension (4) * On entry ISEED specifies the seed of the random number * generator. The array elements should be between 0 and 4095; * if not they will be reduced mod 4096. Also, ISEED(4) must * be odd. The random number generator uses a linear * congruential sequence limited to small integers, and so * should produce machine independent random numbers. The * values of ISEED are changed on exit, and can be used in the * next call to SCHKSB to continue the same random number * sequence. * * THRESH (input) REAL * 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. * * NOUNIT (input) INTEGER * The FORTRAN unit number for printing out error messages * (e.g., if a routine returns IINFO not equal to 0.) * * A (input/workspace) REAL array, dimension * (LDA, max(NN)) * Used to hold the matrix whose eigenvalues are to be * computed. * * LDA (input) INTEGER * The leading dimension of A. It must be at least 2 (not 1!) * and at least max( KK )+1. * * SD (workspace) REAL array, dimension (max(NN)) * Used to hold the diagonal of the tridiagonal matrix computed * by SSBTRD. * * SE (workspace) REAL array, dimension (max(NN)) * Used to hold the off-diagonal of the tridiagonal matrix * computed by SSBTRD. * * U (workspace) REAL array, dimension (LDU, max(NN)) * Used to hold the orthogonal matrix computed by SSBTRD. * * LDU (input) INTEGER * The leading dimension of U. It must be at least 1 * and at least max( NN ). * * WORK (workspace) REAL array, dimension (LWORK) * * LWORK (input) INTEGER * The number of entries in WORK. This must be at least * max( LDA+1, max(NN)+1 )*max(NN). * * RESULT (output) REAL array, dimension (4) * The values computed by the tests described above. * The values are currently limited to 1/ulp, to avoid * overflow. * * INFO (output) INTEGER * If 0, then everything ran OK. * *----------------------------------------------------------------------- * * Some Local Variables and Parameters: * ---- ----- --------- --- ---------- * ZERO, ONE Real 0 and 1. * MAXTYP The number of types defined. * NTEST The number of tests performed, or which can * be performed so far, for the current matrix. * NTESTT The total number of tests performed so far. * NMAX Largest value in NN. * NMATS The number of matrices generated so far. * NERRS The number of tests which have exceeded THRESH * so far. * COND, IMODE Values to be passed to the matrix generators. * ANORM Norm of A; passed to matrix generators. * * OVFL, UNFL Overflow and underflow thresholds. * ULP, ULPINV Finest relative precision and its inverse. * RTOVFL, RTUNFL Square roots of the previous 2 values. * The following four arrays decode JTYPE: * KTYPE(j) The general type (1-10) for type "j". * KMODE(j) The MODE value to be passed to the matrix * generator for type "j". * KMAGN(j) The order of magnitude ( O(1), * O(overflow^(1/2) ), O(underflow^(1/2) ) * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO, TEN PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0, \$ TEN = 10.0E0 ) REAL HALF PARAMETER ( HALF = ONE / TWO ) INTEGER MAXTYP PARAMETER ( MAXTYP = 15 ) * .. * .. Local Scalars .. LOGICAL BADNN, BADNNB INTEGER I, IINFO, IMODE, ITYPE, J, JC, JCOL, JR, JSIZE, \$ JTYPE, JWIDTH, K, KMAX, MTYPES, N, NERRS, \$ NMATS, NMAX, NTEST, NTESTT REAL ANINV, ANORM, COND, OVFL, RTOVFL, RTUNFL, \$ TEMP1, ULP, ULPINV, UNFL * .. * .. Local Arrays .. INTEGER IDUMMA( 1 ), IOLDSD( 4 ), KMAGN( MAXTYP ), \$ KMODE( MAXTYP ), KTYPE( MAXTYP ) * .. * .. External Functions .. REAL SLAMCH EXTERNAL SLAMCH * .. * .. External Subroutines .. EXTERNAL SLACPY, SLASUM, SLATMR, SLATMS, SLASET, SSBT21, \$ SSBTRD, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, REAL, SQRT * .. * .. Data statements .. DATA KTYPE / 1, 2, 5*4, 5*5, 3*8 / DATA KMAGN / 2*1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, \$ 2, 3 / DATA KMODE / 2*0, 4, 3, 1, 4, 4, 4, 3, 1, 4, 4, 0, \$ 0, 0 / * .. * .. Executable Statements .. * * Check for errors * NTESTT = 0 INFO = 0 * * Important constants * BADNN = .FALSE. NMAX = 1 DO 10 J = 1, NSIZES NMAX = MAX( NMAX, NN( J ) ) IF( NN( J ).LT.0 ) \$ BADNN = .TRUE. 10 CONTINUE * BADNNB = .FALSE. KMAX = 0 DO 20 J = 1, NSIZES KMAX = MAX( KMAX, KK( J ) ) IF( KK( J ).LT.0 ) \$ BADNNB = .TRUE. 20 CONTINUE KMAX = MIN( NMAX-1, KMAX ) * * Check for errors * IF( NSIZES.LT.0 ) THEN INFO = -1 ELSE IF( BADNN ) THEN INFO = -2 ELSE IF( NWDTHS.LT.0 ) THEN INFO = -3 ELSE IF( BADNNB ) THEN INFO = -4 ELSE IF( NTYPES.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.KMAX+1 ) THEN INFO = -11 ELSE IF( LDU.LT.NMAX ) THEN INFO = -15 ELSE IF( ( MAX( LDA, NMAX )+1 )*NMAX.GT.LWORK ) THEN INFO = -17 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SCHKSB', -INFO ) RETURN END IF * * Quick return if possible * IF( NSIZES.EQ.0 .OR. NTYPES.EQ.0 .OR. NWDTHS.EQ.0 ) \$ RETURN * * More Important constants * UNFL = SLAMCH( 'Safe minimum' ) OVFL = ONE / UNFL ULP = SLAMCH( 'Epsilon' )*SLAMCH( 'Base' ) ULPINV = ONE / ULP RTUNFL = SQRT( UNFL ) RTOVFL = SQRT( OVFL ) * * Loop over sizes, types * NERRS = 0 NMATS = 0 * DO 190 JSIZE = 1, NSIZES N = NN( JSIZE ) ANINV = ONE / REAL( MAX( 1, N ) ) * DO 180 JWIDTH = 1, NWDTHS K = KK( JWIDTH ) IF( K.GT.N ) \$ GO TO 180 K = MAX( 0, MIN( N-1, K ) ) * IF( NSIZES.NE.1 ) THEN MTYPES = MIN( MAXTYP, NTYPES ) ELSE MTYPES = MIN( MAXTYP+1, NTYPES ) END IF * DO 170 JTYPE = 1, MTYPES IF( .NOT.DOTYPE( JTYPE ) ) \$ GO TO 170 NMATS = NMATS + 1 NTEST = 0 * DO 30 J = 1, 4 IOLDSD( J ) = ISEED( J ) 30 CONTINUE * * Compute "A". * Store as "Upper"; later, we will copy to other format. * * Control parameters: * * KMAGN KMODE KTYPE * =1 O(1) clustered 1 zero * =2 large clustered 2 identity * =3 small exponential (none) * =4 arithmetic diagonal, (w/ eigenvalues) * =5 random log symmetric, w/ eigenvalues * =6 random (none) * =7 random diagonal * =8 random symmetric * =9 positive definite * =10 diagonally dominant tridiagonal * IF( MTYPES.GT.MAXTYP ) \$ GO TO 100 * ITYPE = KTYPE( JTYPE ) IMODE = KMODE( JTYPE ) * * Compute norm * GO TO ( 40, 50, 60 )KMAGN( JTYPE ) * 40 CONTINUE ANORM = ONE GO TO 70 * 50 CONTINUE ANORM = ( RTOVFL*ULP )*ANINV GO TO 70 * 60 CONTINUE ANORM = RTUNFL*N*ULPINV GO TO 70 * 70 CONTINUE * CALL SLASET( 'Full', LDA, N, ZERO, ZERO, A, LDA ) IINFO = 0 IF( JTYPE.LE.15 ) THEN COND = ULPINV ELSE COND = ULPINV*ANINV / TEN END IF * * Special Matrices -- Identity & Jordan block * * Zero * IF( ITYPE.EQ.1 ) THEN IINFO = 0 * ELSE IF( ITYPE.EQ.2 ) THEN * * Identity * DO 80 JCOL = 1, N A( K+1, JCOL ) = ANORM 80 CONTINUE * ELSE IF( ITYPE.EQ.4 ) THEN * * Diagonal Matrix, [Eigen]values Specified * CALL SLATMS( N, N, 'S', ISEED, 'S', WORK, IMODE, COND, \$ ANORM, 0, 0, 'Q', A( K+1, 1 ), LDA, \$ WORK( N+1 ), IINFO ) * ELSE IF( ITYPE.EQ.5 ) THEN * * Symmetric, eigenvalues specified * CALL SLATMS( N, N, 'S', ISEED, 'S', WORK, IMODE, COND, \$ ANORM, K, K, 'Q', A, LDA, WORK( N+1 ), \$ IINFO ) * ELSE IF( ITYPE.EQ.7 ) THEN * * Diagonal, random eigenvalues * CALL SLATMR( N, N, 'S', ISEED, 'S', WORK, 6, ONE, ONE, \$ 'T', 'N', WORK( N+1 ), 1, ONE, \$ WORK( 2*N+1 ), 1, ONE, 'N', IDUMMA, 0, 0, \$ ZERO, ANORM, 'Q', A( K+1, 1 ), LDA, \$ IDUMMA, IINFO ) * ELSE IF( ITYPE.EQ.8 ) THEN * * Symmetric, random eigenvalues * CALL SLATMR( N, N, 'S', ISEED, 'S', WORK, 6, ONE, ONE, \$ 'T', 'N', WORK( N+1 ), 1, ONE, \$ WORK( 2*N+1 ), 1, ONE, 'N', IDUMMA, K, K, \$ ZERO, ANORM, 'Q', A, LDA, IDUMMA, IINFO ) * ELSE IF( ITYPE.EQ.9 ) THEN * * Positive definite, eigenvalues specified. * CALL SLATMS( N, N, 'S', ISEED, 'P', WORK, IMODE, COND, \$ ANORM, K, K, 'Q', A, LDA, WORK( N+1 ), \$ IINFO ) * ELSE IF( ITYPE.EQ.10 ) THEN * * Positive definite tridiagonal, eigenvalues specified. * IF( N.GT.1 ) \$ K = MAX( 1, K ) CALL SLATMS( N, N, 'S', ISEED, 'P', WORK, IMODE, COND, \$ ANORM, 1, 1, 'Q', A( K, 1 ), LDA, \$ WORK( N+1 ), IINFO ) DO 90 I = 2, N TEMP1 = ABS( A( K, I ) ) / \$ SQRT( ABS( A( K+1, I-1 )*A( K+1, I ) ) ) IF( TEMP1.GT.HALF ) THEN A( K, I ) = HALF*SQRT( ABS( A( K+1, \$ I-1 )*A( K+1, I ) ) ) END IF 90 CONTINUE * ELSE * IINFO = 1 END IF * IF( IINFO.NE.0 ) THEN WRITE( NOUNIT, FMT = 9999 )'Generator', IINFO, N, \$ JTYPE, IOLDSD INFO = ABS( IINFO ) RETURN END IF * 100 CONTINUE * * Call SSBTRD to compute S and U from upper triangle. * CALL SLACPY( ' ', K+1, N, A, LDA, WORK, LDA ) * NTEST = 1 CALL SSBTRD( 'V', 'U', N, K, WORK, LDA, SD, SE, U, LDU, \$ WORK( LDA*N+1 ), IINFO ) * IF( IINFO.NE.0 ) THEN WRITE( NOUNIT, FMT = 9999 )'SSBTRD(U)', IINFO, N, \$ JTYPE, IOLDSD INFO = ABS( IINFO ) IF( IINFO.LT.0 ) THEN RETURN ELSE RESULT( 1 ) = ULPINV GO TO 150 END IF END IF * * Do tests 1 and 2 * CALL SSBT21( 'Upper', N, K, 1, A, LDA, SD, SE, U, LDU, \$ WORK, RESULT( 1 ) ) * * Convert A from Upper-Triangle-Only storage to * Lower-Triangle-Only storage. * DO 120 JC = 1, N DO 110 JR = 0, MIN( K, N-JC ) A( JR+1, JC ) = A( K+1-JR, JC+JR ) 110 CONTINUE 120 CONTINUE DO 140 JC = N + 1 - K, N DO 130 JR = MIN( K, N-JC ) + 1, K A( JR+1, JC ) = ZERO 130 CONTINUE 140 CONTINUE * * Call SSBTRD to compute S and U from lower triangle * CALL SLACPY( ' ', K+1, N, A, LDA, WORK, LDA ) * NTEST = 3 CALL SSBTRD( 'V', 'L', N, K, WORK, LDA, SD, SE, U, LDU, \$ WORK( LDA*N+1 ), IINFO ) * IF( IINFO.NE.0 ) THEN WRITE( NOUNIT, FMT = 9999 )'SSBTRD(L)', IINFO, N, \$ JTYPE, IOLDSD INFO = ABS( IINFO ) IF( IINFO.LT.0 ) THEN RETURN ELSE RESULT( 3 ) = ULPINV GO TO 150 END IF END IF NTEST = 4 * * Do tests 3 and 4 * CALL SSBT21( 'Lower', N, K, 1, A, LDA, SD, SE, U, LDU, \$ WORK, RESULT( 3 ) ) * * End of Loop -- Check for RESULT(j) > THRESH * 150 CONTINUE NTESTT = NTESTT + NTEST * * Print out tests which fail. * DO 160 JR = 1, NTEST IF( RESULT( JR ).GE.THRESH ) THEN * * If this is the first test to fail, * print a header to the data file. * IF( NERRS.EQ.0 ) THEN WRITE( NOUNIT, FMT = 9998 )'SSB' WRITE( NOUNIT, FMT = 9997 ) WRITE( NOUNIT, FMT = 9996 ) WRITE( NOUNIT, FMT = 9995 )'Symmetric' WRITE( NOUNIT, FMT = 9994 )'orthogonal', '''', \$ 'transpose', ( '''', J = 1, 4 ) END IF NERRS = NERRS + 1 WRITE( NOUNIT, FMT = 9993 )N, K, IOLDSD, JTYPE, \$ JR, RESULT( JR ) END IF 160 CONTINUE * 170 CONTINUE 180 CONTINUE 190 CONTINUE * * Summary * CALL SLASUM( 'SSB', NOUNIT, NERRS, NTESTT ) RETURN * 9999 FORMAT( ' SCHKSB: ', A, ' returned INFO=', I6, '.', / 9X, 'N=', \$ I6, ', JTYPE=', I6, ', ISEED=(', 3( I5, ',' ), I5, ')' ) * 9998 FORMAT( / 1X, A3, \$ ' -- Real Symmetric Banded Tridiagonal Reduction Routines' ) 9997 FORMAT( ' Matrix types (see SCHKSB for details): ' ) * 9996 FORMAT( / ' Special Matrices:', \$ / ' 1=Zero matrix. ', \$ ' 5=Diagonal: clustered entries.', \$ / ' 2=Identity matrix. ', \$ ' 6=Diagonal: large, evenly spaced.', \$ / ' 3=Diagonal: evenly spaced entries. ', \$ ' 7=Diagonal: small, evenly spaced.', \$ / ' 4=Diagonal: geometr. spaced entries.' ) 9995 FORMAT( ' Dense ', A, ' Banded Matrices:', \$ / ' 8=Evenly spaced eigenvals. ', \$ ' 12=Small, evenly spaced eigenvals.', \$ / ' 9=Geometrically spaced eigenvals. ', \$ ' 13=Matrix with random O(1) entries.', \$ / ' 10=Clustered eigenvalues. ', \$ ' 14=Matrix with large random entries.', \$ / ' 11=Large, evenly spaced eigenvals. ', \$ ' 15=Matrix with small random entries.' ) * 9994 FORMAT( / ' Tests performed: (S is Tridiag, U is ', A, ',', \$ / 20X, A, ' means ', A, '.', / ' UPLO=''U'':', \$ / ' 1= | A - U S U', A1, ' | / ( |A| n ulp ) ', \$ ' 2= | I - U U', A1, ' | / ( n ulp )', / ' UPLO=''L'':', \$ / ' 3= | A - U S U', A1, ' | / ( |A| n ulp ) ', \$ ' 4= | I - U U', A1, ' | / ( n ulp )' ) 9993 FORMAT( ' N=', I5, ', K=', I4, ', seed=', 4( I4, ',' ), ' type ', \$ I2, ', test(', I2, ')=', G10.3 ) * * End of SCHKSB * END