TESTING/EIG/sdrges3.f(3)

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💅 TESTING/EIG/sdrges3.f 💅

okay bestie, this is TESTING/EIG/sdrges3.f from the Library Functions Manual, section (3) — and yes, it's giving “Fortran testing routine checks the nonsymmetric generalized eigenvalue (Schur form) problem driver SGGES3” energy. This is part 1, so we're stopping exactly where the page stops. 🫶

🌸 NAME 🌸

TESTING/EIG/sdrges3.f

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

subroutine sdrges3 — bestie, here's the signature:

subroutine sdrges3 (nsizes, nn, ntypes, dotype, iseed, thresh, nounit,
    a, lda, b, s, t, q, ldq, z, alphar, alphai, beta, work, lwork,
    result, bwork, info)

SDRGES3

💖 Function/Subroutine Documentation 💖

🌟 subroutine `sdrges3` 🌟

subroutine sdrges3 (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    real thresh, integer nounit, real, dimension( lda, * ) a, integer lda,
    real, dimension( lda, * ) b, real, dimension( lda, * ) s, real,
    dimension( lda, * ) t, real, dimension( ldq, * ) q, integer ldq, real,
    dimension( ldq, * ) z, real, dimension( * ) alphar, real, dimension( * )
    alphai, real, dimension( * ) beta, real, dimension( * ) work, integer
    lwork, real, dimension( 13 ) result, logical, dimension( * ) bwork,
    integer info)

SDRGES3

🦋 Purpose 🦋

okay so SDRGES3 checks the nonsymmetric generalized eigenvalue (Schur form) problem driver SGGES3. no cap.

SGGES3 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(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 reorder the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms. (that's literally the page's wording, bestie — i'm not fixing the grammar 💅)

When SDRGES3 is called, a number of matrix () and a number of matrix are specified. For each size () and each TYPE of matrix, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following 13 tests will be performed and compared with the threshold THRESH except the tests (5), (11) and (13).

(1) — | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues)
(2) — | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues)
(3) — | I - QQ' | / ( n ulp ) (no sorting of eigenvalues)
(4) — | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues)
(5) — if A is in Schur form (i.e. quasi-triangular form) (no sorting of eigenvalues)
(6) — if eigenvalues = diagonal blocks of the Schur form (S, T), i.e., test the 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. (no sorting of eigenvalues)

(7) — | (A,B) - Q (S,T) Z' | / ( | (A,B) | n ulp ) (with sorting of eigenvalues).
(8) — | I - QQ' | / ( n ulp ) (with sorting of eigenvalues).
(9) — | I - ZZ' | / ( n ulp ) (with sorting of eigenvalues).
(10) — if A is in Schur form (i.e. quasi-triangular form) (with sorting of eigenvalues).
(11) — if eigenvalues = diagonal blocks of the Schur form (S, T), i.e. test the 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. (with sorting of eigenvalues).

(12) — if sorting worked and SDIM is the number of eigenvalues which were SELECTed.

🧪 Test Matrices 🧪

The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type will be generated. Currently, the list of possible types is:

(1)  ( 0, 0 )         (a pair of zero matrices)

(2)  ( I, 0 )         (an identity and a zero matrix)

(3)  ( 0, I )         (an identity and a zero matrix)

(4)  ( I, I )         (a pair of identity matrices)

        t   t
(5)  ( J , J  )       (a pair of transposed Jordan blocks)

                                    t                ( I   0  )
(6)  ( X, Y )         where  X = ( J   0  )  and Y = (      t )
                                 ( 0   I  )          ( 0   J  )
                      and I is a k x k identity and J a (k+1)x(k+1)
                      Jordan block; k=(N-1)/2

(7)  ( D, I )         where D is diag( 0, 1,..., N-1 ) (a diagonal
                      matrix with those diagonal entries.)
(8)  ( I, D )

(9)  ( big*D, small*I ) where  is near overflow and small=1/big

(10) ( small*D, big*I )

(11) ( big*I, small*D )

(12) ( small*I, big*D )

(13) ( big*D, big*I )

(14) ( small*D, small*I )

(15) ( D1, D2 )        where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
                       D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
          t   t
(16) Q ( J , J ) Z     where Q and Z are random orthogonal matrices.

(17) Q ( T1, T2 ) Z    where T1 and T2 are upper triangular matrices
                       with random O(1) entries above the diagonal
                       and diagonal entries diag(T1) =
                       ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
                       ( 0, N-3, N-4,..., 1, 0, 0 )

(18) Q ( T1, T2 ) Z    diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
                       diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
                       s = machine precision.

(19) Q ( T1, T2 ) Z    diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
                       diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )

                                                       N-5
(20) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
                       diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )

(21) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
                       diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
                       where r1,..., r(N-4) are random.

(22) Q ( big*T1, small*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(23) Q ( small*T1, big*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(24) Q ( small*T1, small*T2 ) Z  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(25) Q ( big*T1, big*T2 ) Z      diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(26) Q ( T1, T2 ) Z     where T1 and T2 are random upper-triangular
                        matrices.

🍓 Parameters 🍓

`NSIZES`

NSIZES is INTEGER — the number of sizes of matrices to use. If it is zero, SDRGES3 does nothing. NSIZES >= 0.

`NN`

NN is INTEGER array, dimension (NSIZES) — an array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.

`NTYPES`

NTYPES is INTEGER — the number of elements in DOTYPE. If it is zero, SDRGES3 does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A on input. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .

`DOTYPE`

DOTYPE is LOGICAL array, dimension (NTYPES) — if DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.

`ISEED`

ISEED — INTEGER array, dimension (4). On entry, ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not, they will be reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to SDRGES3 to continue the same random number sequence. 🎲
THRESH — REAL. A test will count as … if the …, computed as described above, exceeds THRESH (the source has blank spots there, bestie 😭). 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. 🔥
NOUNIT — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 🖨️
A — REAL array, dimension(LDA, max(NN)). Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 💅
LDA — INTEGER. The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). ✨
B — REAL array, dimension(LDA, max(NN)). Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 🎀
S — REAL array, dimension (LDA, max(NN)). The Schur form matrix computed from A by SGGES3. On exit, S contains the Schur form matrix corresponding to the matrix in A. 👑
T — REAL array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by SGGES3. 🌸
Q — REAL array, dimension (LDQ, max(NN)). The (left) orthogonal matrix computed by SGGES3. 🦋
LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). 💜
Z — REAL array, dimension( LDQ, max(NN) ). The (right) orthogonal matrix computed by SGGES3. 🌈
ALPHAR — REAL array, dimension (max(NN)). ⭐
ALPHAI — REAL array, dimension (max(NN)). 💫
BETA — REAL array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by SGGES3. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. 🍓
WORK — REAL array, dimension (LWORK). 🧚
LWORK — INTEGER. The dimension of the array WORK. LWORK >= MAX( 10*(N+1), 3*N*N ), where N is the largest matrix dimension. 🔥
RESULT — REAL array, dimension (15). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 😭
BWORK — LOGICAL array, dimension (N). 🌟
INFO — INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned. 💀

Author 👑

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 399 of file sdrges3.f.

Author (the Doxygen one) ✨

Generated automatically by Doxygen for LAPACK from the source code.

mwah 💖 — source/version vibes: LAPACK Version 3.12.0, page TESTING/EIG/sdrges3.f(3); the footer didn’t give a date, so no invented date, periodt. 🫶

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