TESTING/EIG/zdrgsx.f(3) Library Functions Manual TESTING/EIG/zdrgsx.f(3) NAME TESTING/EIG/zdrgsx.f SYNOPSIS Functions/Subroutines subroutine zdrgsx (nsize, ncmax, thresh, nin, nout, a, lda, b, ai, bi, z, q, alpha, beta, c, ldc, s, work, lwork, rwork, iwork, liwork, bwork, info) ZDRGSX Function/Subroutine Documentation subroutine zdrgsx (integer nsize, integer ncmax, double precision thresh, integer nin, integer nout, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( lda, * ) b, complex*16, dimension( lda, * ) ai, complex*16, dimension( lda, * ) bi, complex*16, dimension( lda, * ) z, complex*16, dimension( lda, * ) q, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) s, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info) ZDRGSX Purpose: ZDRGSX checks the nonsymmetric generalized eigenvalue (Schur form) problem expert driver ZGGESX. ZGGES factors A and B as Q*S*Z' and Q*T*Z' , where ' means conjugate transpose, S and T are upper triangular (i.e., in generalized Schur form), and Q and Z are unitary. It also computes the generalized eigenvalues (alpha(j),beta(j)), j=1,...,n. Thus, w(j) = alpha(j)/beta(j) is a root of the characteristic equation det( A - w(j) B ) = 0 Optionally it also reorders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms; computes a reciprocal condition number for the average of the selected eigenvalues; and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues. When ZDRGSX is called with NSIZE > 0, five (5) types of built-in matrix pairs are used to test the routine ZGGESX. When ZDRGSX is called with NSIZE = 0, it reads in test matrix data to test ZGGESX. (need more details on what kind of read-in data are needed). For each matrix pair, the following tests will be performed and compared with the threshold THRESH except for the tests (7) and (9): (1) | A - Q S Z' | / ( |A| n ulp ) (2) | B - Q T Z' | / ( |B| n ulp ) (3) | I - QQ' | / ( n ulp ) (4) | I - ZZ' | / ( n ulp ) (5) if A is in Schur form (i.e. triangular form) (6) maximum over j of D(j) where: |alpha(j) - S(j,j)| |beta(j) - T(j,j)| D(j) = ------------------------ + ----------------------- max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|) (7) if sorting worked and SDIM is the number of eigenvalues which were selected. (8) the estimated value DIF does not differ from the true values of Difu and Difl more than a factor 10*THRESH. If the estimate DIF equals zero the corresponding true values of Difu and Difl should be less than EPS*norm(A, B). If the true value of Difu and Difl equal zero, the estimate DIF should be less than EPS*norm(A, B). (9) If INFO = N+3 is returned by ZGGESX, the reordering 'failed' and we check that DIF = PL = PR = 0 and that the true value of Difu and Difl is < EPS*norm(A, B). We count the events when INFO=N+3. For read-in test matrices, the same tests are run except that the exact value for DIF (and PL) is input data. Additionally, there is one more test run for read-in test matrices: (10) the estimated value PL does not differ from the true value of PLTRU more than a factor THRESH. If the estimate PL equals zero the corresponding true value of PLTRU should be less than EPS*norm(A, B). If the true value of PLTRU equal zero, the estimate PL should be less than EPS*norm(A, B). Note that for the built-in tests, a total of 10*NSIZE*(NSIZE-1) matrix pairs are generated and tested. NSIZE should be kept small. SVD (routine ZGESVD) is used for computing the true value of DIF_u and DIF_l when testing the built-in test problems. Built-in Test Matrices ====================== All built-in test matrices are the 2 by 2 block of triangular matrices A = [ A11 A12 ] and B = [ B11 B12 ] [ A22 ] [ B22 ] where for different type of A11 and A22 are given as the following. A12 and B12 are chosen so that the generalized Sylvester equation A11*R - L*A22 = -A12 B11*R - L*B22 = -B12 have prescribed solution R and L. Type 1: A11 = J_m(1,-1) and A_22 = J_k(1-a,1). B11 = I_m, B22 = I_k where J_k(a,b) is the k-by-k Jordan block with ``a'' on diagonal and ``b'' on superdiagonal. Type 2: A11 = (a_ij) = ( 2(.5-sin(i)) ) and B11 = (b_ij) = ( 2(.5-sin(ij)) ) for i=1,...,m, j=i,...,m A22 = (a_ij) = ( 2(.5-sin(i+j)) ) and B22 = (b_ij) = ( 2(.5-sin(ij)) ) for i=m+1,...,k, j=i,...,k Type 3: A11, A22 and B11, B22 are chosen as for Type 2, but each second diagonal block in A_11 and each third diagonal block in A_22 are made as 2 by 2 blocks. Type 4: A11 = ( 20(.5 - sin(ij)) ) and B22 = ( 2(.5 - sin(i+j)) ) for i=1,...,m, j=1,...,m and A22 = ( 20(.5 - sin(i+j)) ) and B22 = ( 2(.5 - sin(ij)) ) for i=m+1,...,k, j=m+1,...,k Type 5: (A,B) and have potentially close or common eigenvalues and very large departure from block diagonality A_11 is chosen as the m x m leading submatrix of A_1: | 1 b | | -b 1 | | 1+d b | | -b 1+d | A_1 = | d 1 | | -1 d | | -d 1 | | -1 -d | | 1 | and A_22 is chosen as the k x k leading submatrix of A_2: | -1 b | | -b -1 | | 1-d b | | -b 1-d | A_2 = | d 1+b | | -1-b d | | -d 1+b | | -1+b -d | | 1-d | and matrix B are chosen as identity matrices (see DLATM5). Parameters NSIZE NSIZE is INTEGER The maximum size of the matrices to use. NSIZE >= 0. If NSIZE = 0, no built-in tests matrices are used, but read-in test matrices are used to test DGGESX. NCMAX NCMAX is INTEGER Maximum allowable NMAX for generating Kroneker matrix in call to ZLAKF2 THRESH THRESH is 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. THRESH >= 0. NIN NIN is INTEGER The FORTRAN unit number for reading in the data file of problems to solve. NOUT NOUT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.) A A is COMPLEX*16 array, dimension (LDA, NSIZE) Used to store the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. LDA LDA is INTEGER The leading dimension of A, B, AI, BI, Z and Q, LDA >= max( 1, NSIZE ). For the read-in test, LDA >= max( 1, N ), N is the size of the test matrices. B B is COMPLEX*16 array, dimension (LDA, NSIZE) Used to store the matrix whose eigenvalues are to be computed. On exit, B contains the last matrix actually used. AI AI is COMPLEX*16 array, dimension (LDA, NSIZE) Copy of A, modified by ZGGESX. BI BI is COMPLEX*16 array, dimension (LDA, NSIZE) Copy of B, modified by ZGGESX. Z Z is COMPLEX*16 array, dimension (LDA, NSIZE) Z holds the left Schur vectors computed by ZGGESX. Q Q is COMPLEX*16 array, dimension (LDA, NSIZE) Q holds the right Schur vectors computed by ZGGESX. ALPHA ALPHA is COMPLEX*16 array, dimension (NSIZE) BETA BETA is COMPLEX*16 array, dimension (NSIZE) On exit, ALPHA/BETA are the eigenvalues. C C is COMPLEX*16 array, dimension (LDC, LDC) Store the matrix generated by subroutine ZLAKF2, this is the matrix formed by Kronecker products used for estimating DIF. LDC LDC is INTEGER The leading dimension of C. LDC >= max(1, LDA*LDA/2 ). S S is DOUBLE PRECISION array, dimension (LDC) Singular values of C WORK WORK is COMPLEX*16 array, dimension (LWORK) LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= 3*NSIZE*NSIZE/2 RWORK RWORK is DOUBLE PRECISION array, dimension (5*NSIZE*NSIZE/2 - 4) IWORK IWORK is INTEGER array, dimension (LIWORK) LIWORK LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= NSIZE + 2. BWORK BWORK is LOGICAL array, dimension (NSIZE) 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. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line 346 of file zdrgsx.f. Author Generated automatically by Doxygen for LAPACK from the source code. LAPACK Version 3.12.0 TESTING/EIG/zdrgsx.f(3)