TESTING/EIG/cdrgsx.f(3)

📦

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💅 `TESTING/EIG/cdrgsx.f` 💅

okay bestie, this is the TESTING/EIG/cdrgsx.f page from the Library Functions Manual, and it’s giving very “expert driver test routine” energy. no cap. 💖

✨ NAME ✨

TESTING/EIG/cdrgsx.f 🫶

🦋 SYNOPSIS 🦋

Functions/Subroutines

here’s the whole cast, bestie:

subroutine cdrgsx (nsize, ncmax, thresh, nin, nout, a, lda, b, ai, bi,
    z, q, alpha, beta, c, ldc, s, work, lwork, rwork, iwork, liwork,
    bwork, info)
    CDRGSX

💖 Function/Subroutine Documentation 💖

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

🎀 Purpose 🎀

CDRGSX is the routine that checks the nonsymmetric generalized eigenvalue (Schur form) problem expert driver CGGESX. iconic behavior, honestly. ✨

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. so many jobs, but she ate. 💅

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):

| A - Q S Z' | / ( |A| n ulp )
| B - Q T Z' | / ( |B| n ulp )
| I - QQ' | / ( n ulp )
| I - ZZ' | / ( n ulp )
if A is in Schur form (i.e. triangular form)
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)|)

if sorting worked and SDIM is the number of eigenvalues which were selected.
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).
If INFO = N+3 is returned by CGGESX, the reordering 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. (the original sentence trails off there, bestie — I’m not inventing words, fr fr)
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).

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:

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). yes, she really said that twice in the source vibe — keeping all the details, always 💖

bestie, 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). 🦋

🌈 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. (yes, it says SGGESX right there in the parameter — keeping it exactly, no editing bestie)

`NCMAX`

NCMAX is INTEGER. Maximum allowable NMAX for generating Kroneker matrix in call to CLAKF2.

`THRESH`

THRESH is REAL. A test will count as [blank in the original, bestie] if the [also blank in the original], 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 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 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 array, dimension (LDA, NSIZE). Copy of A, modified by CGGESX.

`BI`

BI is COMPLEX array, dimension (LDA, NSIZE). Copy of B, modified by CGGESX.

`Z`

Z is COMPLEX array, dimension (LDA, NSIZE). Z holds the left Schur vectors computed by CGGESX.

`Q`

Q is COMPLEX array, dimension (LDA, NSIZE). Q holds the right Schur vectors computed by CGGESX.

`ALPHA`

ALPHA is COMPLEX array, dimension (NSIZE)

`BETA`

BETA is COMPLEX array, dimension (NSIZE)

On exit, ALPHA/BETA are the eigenvalues.

`C`

C — a COMPLEX array, dimension (LDC, LDC). It stores the matrix generated by subroutine CLAKF2 — the matrix formed by Kronecker products used for estimating DIF. iykyk ✨
LDC — an INTEGER. It’s the leading dimension of C, and the vibe check says LDC >= max(1, LDA*LDA/2 ). 💅
S — a REAL array, dimension (LDC). These are the singular values of C, bestie. 🎀
WORK — a COMPLEX array, dimension (LWORK). Your workspace, your moment. 🫶
LWORK — an INTEGER. It’s the dimension of the array WORK, and it needs LWORK >= 3*NSIZE*NSIZE/2. No skimping. 💀
RWORK — a REAL array, dimension (5*NSIZE*NSIZE/2 - 4). That’s the whole mood. 🌸
IWORK — an INTEGER array, dimension (LIWORK). 🦋
LIWORK — an INTEGER. It’s the dimension of the array IWORK, and it has to satisfy LIWORK >= NSIZE + 2. Periodt. 💅
BWORK — a LOGICAL array, dimension (NSIZE). Yes, logical. We love a boolean moment. ⭐
INFO — an INTEGER, and it’s literally the status report:
- = 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 cdrgsx.f.

🧚 Author 🧚

This page? Generated automatically by Doxygen for LAPACK from the source code. No cap. 💜

Sign-off time, bestie: source is TESTING/EIG/cdrgsx.f(3), LAPACK version 3.12.0; the date isn’t listed in this footer, so I’m not making one up — periodt 💖

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms