# zdrgsx.f - Man Page

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.

## Referenced By

The man page zdrgsx(3) is an alias of zdrgsx.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK