# sdrgsx.f - Man Page

TESTING/EIG/sdrgsx.f

## Synopsis

### Functions/Subroutines

subroutine **sdrgsx** (nsize, ncmax, thresh, nin, nout, a, lda, b, ai, bi, z, q, alphar, alphai, beta, c, ldc, s, work, lwork, iwork, liwork, bwork, info)**SDRGSX**

## Function/Subroutine Documentation

### subroutine sdrgsx (integer nsize, integer ncmax, real thresh, integer nin, integer nout, real, dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b, real, dimension( lda, * ) ai, real, dimension( lda, * ) bi, real, dimension( lda, * ) z, real, dimension( lda, * ) q, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) s, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

**SDRGSX**

**Purpose:**

SDRGSX checks the nonsymmetric generalized eigenvalue (Schur form) problem expert driver SGGESX. SGGESX factors A and B as Q S Z' and Q T Z', where ' means transpose, T is upper triangular, S is in generalized Schur form (block upper triangular, with 1x1 and 2x2 blocks on the diagonal, the 2x2 blocks corresponding to complex conjugate pairs of generalized eigenvalues), and Q and Z are orthogonal. It also computes the generalized eigenvalues (alpha(1),beta(1)), ..., (alpha(n),beta(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 SDRGSX is called with NSIZE > 0, five (5) types of built-in matrix pairs are used to test the routine SGGESX. When SDRGSX is called with NSIZE = 0, it reads in test matrix data to test SGGESX. 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. quasi-triangular form) (6) maximum over j of D(j) where: if alpha(j) is real: |alpha(j) - S(j,j)| |beta(j) - T(j,j)| D(j) = ------------------------ + ----------------------- max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|) if alpha(j) is complex: | det( s S - w T ) | D(j) = --------------------------------------------------- ulp max( s norm(S), |w| norm(T) )*norm( s S - w T ) and S and T are here the 2 x 2 diagonal blocks of S and T corresponding to the j-th and j+1-th eigenvalues. (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 SGGESX, 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 above 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 SGESVD) 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 SLATM5).

**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 SGGESX.

*NCMAX*NCMAX is INTEGER Maximum allowable NMAX for generating Kroneker matrix in call to SLAKF2

*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. 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 IINFO not equal to 0.)

*A*A is REAL 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 REAL 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 REAL array, dimension (LDA, NSIZE) Copy of A, modified by SGGESX.

*BI*BI is REAL array, dimension (LDA, NSIZE) Copy of B, modified by SGGESX.

*Z*Z is REAL array, dimension (LDA, NSIZE) Z holds the left Schur vectors computed by SGGESX.

*Q*Q is REAL array, dimension (LDA, NSIZE) Q holds the right Schur vectors computed by SGGESX.

*ALPHAR*ALPHAR is REAL array, dimension (NSIZE)

*ALPHAI*ALPHAI is REAL array, dimension (NSIZE)

*BETA*BETA is REAL array, dimension (NSIZE) \verbatim On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues.

*C*C is REAL array, dimension (LDC, LDC) Store the matrix generated by subroutine SLAKF2, 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 REAL array, dimension (LDC) Singular values of C

*WORK*WORK is REAL array, dimension (LWORK)

*LWORK*LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX( 5*NSIZE*NSIZE/2 - 2, 10*(NSIZE+1) )

*IWORK*IWORK is INTEGER array, dimension (LIWORK)

*LIWORK*LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= NSIZE + 6.

*BWORK*BWORK is LOGICAL array, dimension (LDA)

*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 **356** of file **sdrgsx.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

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

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