DDRGSX(3)

💅 `ddrgsx.f(3)` — Library Functions Manual

NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/ddrgsx.f

okay bestie, that’s the file this page is about — yes, the whole path, because LAPACK build paths are extra like that 💅

SYNOPSIS

Functions/Subroutines

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

DDRGSX

Function/Subroutine Documentation

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

DDRGSX

Purpose

okay so DDRGSX checks the nonsymmetric generalized eigenvalue (Schur form) problem expert driver DGGESX 😤✨

DGGESX 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 DDRGSX is called with NSIZE > 0, five (5) types of built-in matrix pairs are used to test the routine DGGESX.

When DDRGSX is called with NSIZE = 0, it reads in test matrix data to test DGGESX.

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. quasi-triangular form)
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.

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 DGGESX, 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.

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:

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 DGESVD) 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 DLAKF2.

`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 IINFO not equal to 0.)

`A`

A is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDA, NSIZE). Copy of A, modified by DGGESX.

`BI`

BI is DOUBLE PRECISION array, dimension (LDA, NSIZE). Copy of B, modified by DGGESX.

`Z`

okay bestie, parameter round two — same energy, zero facts dropped. 💅

Z — DOUBLE PRECISION array, dimension (LDA, NSIZE). Z holds the left Schur vectors computed by DGGESX. 🫶
Q — DOUBLE PRECISION array, dimension (LDA, NSIZE). Q holds the right Schur vectors computed by DGGESX. ✨
ALPHAR — DOUBLE PRECISION array, dimension (NSIZE). 🎀
ALPHAI — DOUBLE PRECISION array, dimension (NSIZE). 🎀
BETA — DOUBLE PRECISION array, dimension (NSIZE). On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues. no cap. 🔥
C — DOUBLE PRECISION array, dimension (LDC, LDC). Store the matrix generated by subroutine DLAKF2, this is the matrix formed by Kronecker products used for estimating DIF. 🧚
LDC — INTEGER. The leading dimension of C. LDC >= max(1, LDA*LDA/2 ). 💖
S — DOUBLE PRECISION array, dimension (LDC). Singular values of C. ⭐
WORK — DOUBLE PRECISION array, dimension (LWORK). 🌸
LWORK — INTEGER. The dimension of the array WORK. LWORK >= MAX( 5*NSIZE*NSIZE/2 - 2, 10*(NSIZE+1) ). 💜
IWORK — INTEGER array, dimension (LIWORK). 🦋
LIWORK — INTEGER. The dimension of the array IWORK. LIWORK >= NSIZE + 6. 🎀
BWORK — LOGICAL array, dimension (LDA). 🌈
INFO — 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 ddrgsx.f. 💅

Author ✨

Generated automatically by Doxygen for LAPACK from the source code. 🫶

and that’s the whole parameter family, bestie — every flag, bound, and INFO case accounted for. This cutie is from LAPACK Version 3.12.0, sourced from /home/abui...IG/ddrgsx.f(3) — date? not listed here, but the vibe is timeless. mwah 💖🎀

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