TESTING/EIG/sdrgev.f(3) Library Functions Manual TESTING/EIG/sdrgev.f(3) NAME TESTING/EIG/sdrgev.f SYNOPSIS Functions/Subroutines subroutine sdrgev (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, b, s, t, q, ldq, z, qe, ldqe, alphar, alphai, beta, alphr1, alphi1, beta1, work, lwork, result, info) SDRGEV Function/Subroutine Documentation subroutine sdrgev (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, integer nounit, real, dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b, real, dimension( lda, * ) s, real, dimension( lda, * ) t, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldq, * ) z, real, dimension( ldqe, * ) qe, integer ldqe, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( * ) alphr1, real, dimension( * ) alphi1, real, dimension( * ) beta1, real, dimension( * ) work, integer lwork, real, dimension( * ) result, integer info) SDRGEV Purpose: SDRGEV checks the nonsymmetric generalized eigenvalue problem driver routine SGGEV. SGGEV computes for a pair of n-by-n nonsymmetric matrices (A,B) the generalized eigenvalues and, optionally, the left and right eigenvectors. A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is reasonable interpretation for beta=0, and even for both being zero. A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A - wB) * r = 0. A left generalized eigenvector is a vector l such that l**H * (A - wB) = 0, where l**H is the conjugate-transpose of l. When SDRGEV 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, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following tests will be performed and compared with the threshold THRESH. Results from SGGEV: (1) max over all left eigenvalue/-vector pairs (alpha/beta,l) of | VL**H * (beta A - alpha B) |/( ulp max(|beta A|, |alpha B|) ) where VL**H is the conjugate-transpose of VL. (2) | |VL(i)| - 1 | / ulp and whether largest component real VL(i) denotes the i-th column of VL. (3) max over all left eigenvalue/-vector pairs (alpha/beta,r) of | (beta A - alpha B) * VR | / ( ulp max(|beta A|, |alpha B|) ) (4) | |VR(i)| - 1 | / ulp and whether largest component real VR(i) denotes the i-th column of VR. (5) W(full) = W(partial) W(full) denotes the eigenvalues computed when both l and r are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and r, or only W and l are computed. (6) VL(full) = VL(partial) VL(full) denotes the left eigenvectors computed when both l and r are computed, and VL(partial) denotes the result when only l is computed. (7) VR(full) = VR(partial) VR(full) denotes the right eigenvectors computed when both l and r are also computed, and VR(partial) denotes the result when only l is computed. 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 diag( 0, 1,..., N-1 ) (a diagonal matrix with those diagonal entries.) (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 is diag( 0, 0, 1, ..., N-3, 0 ) and D2 is diag( 0, N-3, N-4,..., 1, 0, 0 ) t t (16) Q ( J , J ) Z where Q and Z are random orthogonal matrices. (17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, N-3, N-4,..., 1, 0, 0 ) (18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ) s = machine precision. (19) Q ( T1, T2 ) Z 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) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 ) diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ) (21) Q ( T1, T2 ) Z 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) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) diag(T2) = ( 0, 1, ..., 1, 0, 0 ) (26) Q ( T1, T2 ) Z 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, SDRGES does nothing. NSIZES >= 0. NN NN is INTEGER array, dimension (NSIZES) An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0. NTYPES NTYPES is INTEGER The number of elements in DOTYPE. If it is zero, SDRGES 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 SDRGES 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. NOUNIT NOUNIT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IERR not equal to 0.) A A is REAL 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, S, and T. It must be at least 1 and at least max( NN ). B B is REAL 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. S S is REAL array, dimension (LDA, max(NN)) The Schur form matrix computed from A by SGGES. On exit, S contains the Schur form matrix corresponding to the matrix in A. T T is REAL array, dimension (LDA, max(NN)) The upper triangular matrix computed from B by SGGES. Q Q is REAL array, dimension (LDQ, max(NN)) The (left) eigenvectors matrix computed by SGGEV. LDQ LDQ is INTEGER The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). Z Z is REAL array, dimension( LDQ, max(NN) ) The (right) orthogonal matrix computed by SGGES. QE QE is REAL array, dimension( LDQ, max(NN) ) QE holds the computed right or left eigenvectors. LDQE LDQE is INTEGER The leading dimension of QE. LDQE >= max(1,max(NN)). ALPHAR ALPHAR is REAL array, dimension (max(NN)) ALPHAI ALPHAI is REAL array, dimension (max(NN)) BETA BETA is REAL array, dimension (max(NN)) \verbatim The generalized eigenvalues of (A,B) computed by SGGEV. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. ALPHR1 ALPHR1 is REAL array, dimension (max(NN)) ALPHI1 ALPHI1 is REAL array, dimension (max(NN)) BETA1 BETA1 is REAL array, dimension (max(NN)) Like ALPHAR, ALPHAI, BETA, these arrays contain the eigenvalues of A and B, but those computed when SGGEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. WORK WORK is REAL array, dimension (LWORK) LWORK LWORK is INTEGER The number of entries in WORK. LWORK >= MAX( 8*N, N*(N+1) ). RESULT RESULT is REAL array, dimension (2) 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 404 of file sdrgev.f. Author Generated automatically by Doxygen for LAPACK from the source code. LAPACK Version 3.12.0 TESTING/EIG/sdrgev.f(3)