TESTING/EIG/cchkgg.f(3) Library Functions Manual TESTING/EIG/cchkgg.f(3) NAME TESTING/EIG/cchkgg.f SYNOPSIS Functions/Subroutines subroutine cchkgg (nsizes, nn, ntypes, dotype, iseed, thresh, tstdif, thrshn, nounit, a, lda, b, h, t, s1, s2, p1, p2, u, ldu, v, q, z, alpha1, beta1, alpha3, beta3, evectl, evectr, work, lwork, rwork, llwork, result, info) CCHKGG Function/Subroutine Documentation subroutine cchkgg (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, logical tstdif, real thrshn, integer nounit, complex, dimension( lda, * ) a, integer lda, complex, dimension( lda, * ) b, complex, dimension( lda, * ) h, complex, dimension( lda, * ) t, complex, dimension( lda, * ) s1, complex, dimension( lda, * ) s2, complex, dimension( lda, * ) p1, complex, dimension( lda, * ) p2, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldu, * ) v, complex, dimension( ldu, * ) q, complex, dimension( ldu, * ) z, complex, dimension( * ) alpha1, complex, dimension( * ) beta1, complex, dimension( * ) alpha3, complex, dimension( * ) beta3, complex, dimension( ldu, * ) evectl, complex, dimension( ldu, * ) evectr, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, logical, dimension( * ) llwork, real, dimension( 15 ) result, integer info) CCHKGG Purpose: CCHKGG checks the nonsymmetric generalized eigenvalue problem routines. H H H CGGHRD factors A and B as U H V and U T V , where means conjugate transpose, H is hessenberg, T is triangular and U and V are unitary. H H CHGEQZ factors H and T as Q S Z and Q P Z , where P and S are upper triangular and Q and Z are unitary. It also computes the generalized eigenvalues (alpha(1),beta(1)),...,(alpha(n),beta(n)), where alpha(j)=S(j,j) and beta(j)=P(j,j) -- thus, w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue problem det( A - w(j) B ) = 0 and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent problem det( m(j) A - B ) = 0 CTGEVC computes the matrix L of left eigenvectors and the matrix R of right eigenvectors for the matrix pair ( S, P ). In the description below, l and r are left and right eigenvectors corresponding to the generalized eigenvalues (alpha,beta). When CCHKGG is called, a number of matrix 'sizes' ('n's') and a number of matrix 'types' are specified. For each size ('n') and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 13 tests will be performed. The first twelve 'test ratios' should be small -- O(1). They will be compared with the threshold THRESH: H (1) | A - U H V | / ( |A| n ulp ) H (2) | B - U T V | / ( |B| n ulp ) H (3) | I - UU | / ( n ulp ) H (4) | I - VV | / ( n ulp ) H (5) | H - Q S Z | / ( |H| n ulp ) H (6) | T - Q P Z | / ( |T| n ulp ) H (7) | I - QQ | / ( n ulp ) H (8) | I - ZZ | / ( n ulp ) (9) max over all left eigenvalue/-vector pairs (beta/alpha,l) of H | (beta A - alpha B) l | / ( ulp max( |beta A|, |alpha B| ) ) (10) max over all left eigenvalue/-vector pairs (beta/alpha,l') of H | (beta H - alpha T) l' | / ( ulp max( |beta H|, |alpha T| ) ) where the eigenvectors l' are the result of passing Q to STGEVC and back transforming (JOB='B'). (11) max over all right eigenvalue/-vector pairs (beta/alpha,r) of | (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) ) (12) max over all right eigenvalue/-vector pairs (beta/alpha,r') of | (beta H - alpha T) r' | / ( ulp max( |beta H|, |alpha T| ) ) where the eigenvectors r' are the result of passing Z to STGEVC and back transforming (JOB='B'). The last three test ratios will usually be small, but there is no mathematical requirement that they be so. They are therefore compared with THRESH only if TSTDIF is .TRUE. (13) | S(Q,Z computed) - S(Q,Z not computed) | / ( |S| ulp ) (14) | P(Q,Z computed) - P(Q,Z not computed) | / ( |P| ulp ) (15) max( |alpha(Q,Z computed) - alpha(Q,Z not computed)|/|S| , |beta(Q,Z computed) - beta(Q,Z not computed)|/|P| ) / ulp In addition, the normalization of L and R are checked, and compared with the threshold THRSHN. Test Matrices ---- -------- The sizes of the test matrices 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) ( 0, 0 ) (a pair of zero matrices) (2) ( I, 0 ) (an identity and a zero matrix) (3) ( 0, I ) (an identity and a zero matrix) (4) ( I, I ) (a pair of identity matrices) t t (5) ( J , J ) (a pair of transposed Jordan blocks) t ( I 0 ) (6) ( X, Y ) where X = ( J 0 ) and Y = ( t ) ( 0 I ) ( 0 J ) and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N-1)/2 (7) ( D, I ) where D is P*D1, P is a random unitary diagonal matrix (i.e., with random magnitude 1 entries on the diagonal), and D1=diag( 0, 1,..., N-1 ) (i.e., a diagonal matrix with D1(1,1)=0, D1(2,2)=1, ..., D1(N,N)=N-1.) (8) ( I, D ) (9) ( big*D, small*I ) where 'big' is near overflow and small=1/big (10) ( small*D, big*I ) (11) ( big*I, small*D ) (12) ( small*I, big*D ) (13) ( big*D, big*I ) (14) ( small*D, small*I ) (15) ( D1, D2 ) where D1=P*diag( 0, 0, 1, ..., N-3, 0 ) and D2=Q*diag( 0, N-3, N-4,..., 1, 0, 0 ), and P and Q are random unitary diagonal matrices. t t (16) U ( J , J ) V where U and V are random unitary matrices. (17) U ( T1, T2 ) V where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = Q*( 0, N-3, N-4,..., 1, 0, 0 ) (18) U ( T1, T2 ) V diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) U ( T1, T2 ) V diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ) N-5 (20) U ( T1, T2 ) V diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) U ( T1, T2 ) V diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) where r1,..., r(N-4) are random. (22) U ( big*T1, small*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) U ( small*T1, big*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) U ( small*T1, small*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) U ( big*T1, big*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) U ( T1, T2 ) V where T1 and T2 are random upper-triangular matrices. Parameters NSIZES NSIZES is INTEGER The number of sizes of matrices to use. If it is zero, CCHKGG does nothing. It must be at least zero. NN NN is 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. NTYPES NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, CCHKGG 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 DOTYPE is 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 ISEED is 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 CCHKGG to continue the same random number sequence. THRESH THRESH is 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. TSTDIF TSTDIF is LOGICAL Specifies whether test ratios 13-15 will be computed and compared with THRESH. = .FALSE.: Only test ratios 1-12 will be computed and tested. Ratios 13-15 will be set to zero. = .TRUE.: All the test ratios 1-15 will be computed and tested. THRSHN THRSHN is REAL Threshold for reporting eigenvector normalization error. If the normalization of any eigenvector differs from 1 by more than THRSHN*ulp, then a special error message will be printed. (This is handled separately from the other tests, since only a compiler or programming error should cause an error message, at least if THRSHN is at least 5--10.) NOUNIT NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) A A is COMPLEX array, dimension (LDA, max(NN)) Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. LDA LDA is INTEGER The leading dimension of A, B, H, T, S1, P1, S2, and P2. It must be at least 1 and at least max( NN ). B B is COMPLEX array, dimension (LDA, max(NN)) Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. H H is COMPLEX array, dimension (LDA, max(NN)) The upper Hessenberg matrix computed from A by CGGHRD. T T is COMPLEX array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by CGGHRD. S1 S1 is COMPLEX array, dimension (LDA, max(NN)) The Schur (upper triangular) matrix computed from H by CHGEQZ when Q and Z are also computed. S2 S2 is COMPLEX array, dimension (LDA, max(NN)) The Schur (upper triangular) matrix computed from H by CHGEQZ when Q and Z are not computed. P1 P1 is COMPLEX array, dimension (LDA, max(NN)) The upper triangular matrix computed from T by CHGEQZ when Q and Z are also computed. P2 P2 is COMPLEX array, dimension (LDA, max(NN)) The upper triangular matrix computed from T by CHGEQZ when Q and Z are not computed. U U is COMPLEX array, dimension (LDU, max(NN)) The (left) unitary matrix computed by CGGHRD. LDU LDU is INTEGER The leading dimension of U, V, Q, Z, EVECTL, and EVECTR. It must be at least 1 and at least max( NN ). V V is COMPLEX array, dimension (LDU, max(NN)) The (right) unitary matrix computed by CGGHRD. Q Q is COMPLEX array, dimension (LDU, max(NN)) The (left) unitary matrix computed by CHGEQZ. Z Z is COMPLEX array, dimension (LDU, max(NN)) The (left) unitary matrix computed by CHGEQZ. ALPHA1 ALPHA1 is COMPLEX array, dimension (max(NN)) BETA1 BETA1 is COMPLEX array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by CHGEQZ when Q, Z, and the full Schur matrices are computed. ALPHA3 ALPHA3 is COMPLEX array, dimension (max(NN)) BETA3 BETA3 is COMPLEX array, dimension (max(NN)) The generalized eigenvalues of (A,B) computed by CHGEQZ when neither Q, Z, nor the Schur matrices are computed. EVECTL EVECTL is COMPLEX array, dimension (LDU, max(NN)) The (lower triangular) left eigenvector matrix for the matrices in S1 and P1. EVECTR EVECTR is COMPLEX array, dimension (LDU, max(NN)) The (upper triangular) right eigenvector matrix for the matrices in S1 and P1. WORK WORK is COMPLEX array, dimension (LWORK) LWORK LWORK is INTEGER The number of entries in WORK. This must be at least max( 4*N, 2 * N**2, 1 ), for all N=NN(j). RWORK RWORK is REAL array, dimension (2*max(NN)) LLWORK LLWORK is LOGICAL array, dimension (max(NN)) RESULT RESULT is REAL array, dimension (15) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. INFO INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 498 of file cchkgg.f. Author Generated automatically by Doxygen for LAPACK from the source code. LAPACK Version 3.12.0 TESTING/EIG/cchkgg.f(3)