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LAPACK
3.7.0
LAPACK: Linear Algebra PACKage
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subroutine cdrgsx | ( | integer | NSIZE, |
integer | NCMAX, | ||
real | THRESH, | ||
integer | NIN, | ||
integer | NOUT, | ||
complex, dimension( lda, * ) | A, | ||
integer | LDA, | ||
complex, dimension( lda, * ) | B, | ||
complex, dimension( lda, * ) | AI, | ||
complex, dimension( lda, * ) | BI, | ||
complex, dimension( lda, * ) | Z, | ||
complex, dimension( lda, * ) | Q, | ||
complex, dimension( * ) | ALPHA, | ||
complex, dimension( * ) | BETA, | ||
complex, dimension( ldc, * ) | C, | ||
integer | LDC, | ||
real, dimension( * ) | S, | ||
complex, dimension( * ) | WORK, | ||
integer | LWORK, | ||
real, dimension( * ) | RWORK, | ||
integer, dimension( * ) | IWORK, | ||
integer | LIWORK, | ||
logical, dimension( * ) | BWORK, | ||
integer | INFO | ||
) |
CDRGSX
CDRGSX checks the nonsymmetric generalized eigenvalue (Schur form) problem expert driver CGGESX. CGGES 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 CDRGSX is called with NSIZE > 0, five (5) types of built-in matrix pairs are used to test the routine CGGESX. When CDRGSX is called with NSIZE = 0, it reads in test matrix data to test CGGESX. (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 CGGESX, 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 CGESVD) 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).
[in] | 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. |
[in] | NCMAX | NCMAX is INTEGER Maximum allowable NMAX for generating Kroneker matrix in call to CLAKF2 |
[in] | 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. |
[in] | NIN | NIN is INTEGER The FORTRAN unit number for reading in the data file of problems to solve. |
[in] | NOUT | NOUT is INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.) |
[out] | A | A is COMPLEX array, dimension (LDA, NSIZE) Used to store the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. |
[in] | 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. |
[out] | B | B is COMPLEX array, dimension (LDA, NSIZE) Used to store the matrix whose eigenvalues are to be computed. On exit, B contains the last matrix actually used. |
[out] | AI | AI is COMPLEX array, dimension (LDA, NSIZE) Copy of A, modified by CGGESX. |
[out] | BI | BI is COMPLEX array, dimension (LDA, NSIZE) Copy of B, modified by CGGESX. |
[out] | Z | Z is COMPLEX array, dimension (LDA, NSIZE) Z holds the left Schur vectors computed by CGGESX. |
[out] | Q | Q is COMPLEX array, dimension (LDA, NSIZE) Q holds the right Schur vectors computed by CGGESX. |
[out] | ALPHA | ALPHA is COMPLEX array, dimension (NSIZE) |
[out] | BETA | BETA is COMPLEX array, dimension (NSIZE) On exit, ALPHA/BETA are the eigenvalues. |
[out] | C | C is COMPLEX array, dimension (LDC, LDC) Store the matrix generated by subroutine CLAKF2, this is the matrix formed by Kronecker products used for estimating DIF. |
[in] | LDC | LDC is INTEGER The leading dimension of C. LDC >= max(1, LDA*LDA/2 ). |
[out] | S | S is REAL array, dimension (LDC) Singular values of C |
[out] | WORK | WORK is COMPLEX array, dimension (LWORK) |
[in] | LWORK | LWORK is INTEGER The dimension of the array WORK. LWORK >= 3*NSIZE*NSIZE/2 |
[out] | RWORK | RWORK is REAL array, dimension (5*NSIZE*NSIZE/2 - 4) |
[out] | IWORK | IWORK is INTEGER array, dimension (LIWORK) |
[in] | LIWORK | LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= NSIZE + 2. |
[out] | BWORK | BWORK is LOGICAL array, dimension (NSIZE) |
[out] | 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. |
Definition at line 351 of file cdrgsx.f.