/home/abui.../zdrges3.f(3)

💖 NAME 💖

okay so bestie, this is the manual page for /home/abui.../zdrges3.f(3) from the Library Functions Manual. The actual file path is:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/zdrges3.f

it's giving LAPACK testing energy, no cap 💅.

✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

subroutine ZDRGES3 (nsizes, nn, ntypes, dotype, iseed, thresh, nounit,
    a, lda, b, s, t, q, ldq, z, alpha, beta, work, lwork, rwork, result,
    bwork, info)
    ZDRGES3

🦋 Function/Subroutine Documentation 🦋

here's the full signature, character for character:

subroutine ZDRGES3 (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    double precision thresh, integer nounit, complex*16, dimension( lda, * )
    a, integer lda, complex*16, dimension( lda, * ) b, complex*16,
    dimension( lda, * ) s, complex*16, dimension( lda, * ) t, complex*16,
    dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldq, * ) z,
    complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta,
    complex*16, dimension( * ) work, integer lwork, double precision,
    dimension( * ) rwork, double precision, dimension( 13 ) result, logical,
    dimension( * ) bwork, integer info)
    ZDRGES3

🎯 Purpose 🎯

okay so ZDRGES3 is literally the vibe check for the nonsymmetric generalized eigenvalue (Schur form) problem driver ZGGES3.

ZGGES3 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 reorder the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms.

When ZDRGES3 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).

the source says 13 tests but the list goes to (12) — I'm not editing the source, bestie, I'm just reporting the news 💅

here's the test lineup, no cap:

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

(no sorting of eigenvalues)

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

(with sorting of eigenvalues).

Test (12): if sorting worked and SDIM is the number of eigenvalues which were CELECTed.

that's the test lineup — obsessed with the drama of D(j) ngl 😭.

🧪 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:

here's the raw matrix-type menu, copied with all its spacing because the formulas are doing the most:

(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.

periodt. that's the whole menu ✨.

🧚 Parameters 🧚

NSIZES

NSIZES is INTEGER.

The number of sizes of matrices to use. If it is zero, DDRGES3 does nothing. NSIZES >= 0.

yes, it says DDRGES3 there — the source said that, not me 🙈

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, DDRGES3 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 is 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 DDRGES3 to continue the same random number sequence.

THRESH

THRESH is DOUBLE PRECISION.

A test will count as if the , 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.

the source text has blanks there, bestie — not my fault 🙈

NOUNIT

NOUNIT is INTEGER.

The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)

A

A — this is a COMPLEX*16 array, dimension dimension(LDA, max(NN)). It holds the original A matrix. It’s used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. — iykyk, bestie 💅
LDA — an INTEGER. This is the leading dimension of A, B, S, and T. It has to be at least 1 and at least max( NN ). No skipping that vibe check 🌸
B — a COMPLEX*16 array, dimension dimension(LDA, max(NN)). It holds the original B matrix. Same deal: used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 💖
S — a COMPLEX*16 array, dimension dimension (LDA, max(NN)). This is the Schur form matrix computed from A by ZGGES3. On exit, S contains the Schur form matrix corresponding to the matrix in A. She’s the main character now ✨
T — a COMPLEX*16 array, dimension dimension (LDA, max(NN)). This is the upper triangular matrix computed from B by ZGGES3. Upper triangular and unbothered 🔺
Q — a COMPLEX*16 array, dimension dimension (LDQ, max(NN)). This is the (left) orthogonal matrix computed by ZGGES3. Left orthogonal, slaying as usual 👑
LDQ — an INTEGER. This is the leading dimension of Q and Z. It must be at least 1 and at least max( NN ). Respect the dimension bounds, fr fr 🫶
Z — a COMPLEX*16 array, dimension dimension( LDQ, max(NN) ). This is the (right) orthogonal matrix computed by ZGGES3. Right orthogonal, zero notes 🦋
ALPHA — a COMPLEX*16 array, dimension dimension (max(NN)). Bestie, that’s the whole note for this one 💜
BETA — a COMPLEX*16 array, dimension dimension (max(NN)). These are the generalized eigenvalues of (A,B) computed by ZGGES3. And yes: ALPHA(k) / BETA(k) is the k-th generalized eigenvalue of A and B. Math girlies rise 🎀
WORK — a COMPLEX*16 array, dimension dimension (LWORK). Workspace moment 💅
LWORK — an INTEGER. This is the dimension of the array WORK. The rule is LWORK >= 3*N*N. Do not test her. 🔥
RWORK — a DOUBLE PRECISION array, dimension dimension ( 8*N ). This is real workspace. Real ones only 🌈
RESULT — a DOUBLE PRECISION array, dimension dimension (15). These are the values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. Safety first, cutie ⭐
BWORK — a LOGICAL array, dimension dimension (N). Logic array, doing logic things 🧚
INFO — an INTEGER. The whole status report:
- = 0: successful exit. We love a clean finish 💖
- < 0: if INFO = -i, the i-th argument had an illegal value. Ruh-roh 🙈
- > 0: a routine returned an error code. INFO is the absolute value of the INFO value returned. Not iconic, but important 💀

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 378 of file zdrges3.f.

📝 Author

Generated automatically by Doxygen for LAPACK from the source code.

That’s the whole parameter lineup, bestie — no cap, every detail stayed intact. Catch you in the next subroutine 💋 — this one’s brought to you by LAPACK Version 3.12.0, source /home/abui.../zdrges3.f(3). Mwah 🫶✨

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