TESTING/EIG/zchkgg.f(3)

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🌸 TESTING/EIG/zchkgg.f 🌸

bestie, welcome to the zchkgg page — it's giving linear algebra realness and we're gonna get through it together 💅

✨ NAME ✨

TESTING/EIG/zchkgg.f

🌈 SYNOPSIS 🌈

Functions/Subroutines

subroutine zchkgg (nsizes, nn, ntypes, dotype, iseed, thresh, tstdif,
    thrshn, nounit, a, lda, b, h, t, s1, s2, p1, p2, u, ldu, v, q, z,
    alpha1, beta1, alpha3, beta3, evectl, evectr, work, lwork, rwork,
    llwork, result, info)

ZCHKGG

💖 Function/Subroutine Documentation 💖

`subroutine zchkgg`

okay so here's the full signature with all the types and dimensions — copy it exactly if you're calling this:

subroutine zchkgg (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    double precision thresh, logical tstdif, double precision thrshn,
    integer nounit, complex*16, dimension( lda, * ) a, integer lda,
    complex*16, dimension( lda, * ) b, complex*16, dimension( lda, * ) h,
    complex*16, dimension( lda, * ) t, complex*16, dimension( lda, * ) s1,
    complex*16, dimension( lda, * ) s2, complex*16, dimension( lda, * ) p1,
    complex*16, dimension( lda, * ) p2, complex*16, dimension( ldu, * ) u,
    integer ldu, complex*16, dimension( ldu, * ) v, complex*16, dimension(
    ldu, * ) q, complex*16, dimension( ldu, * ) z, complex*16, dimension( *
    ) alpha1, complex*16, dimension( * ) beta1, complex*16, dimension( * )
    alpha3, complex*16, dimension( * ) beta3, complex*16, dimension( ldu, *
    ) evectl, complex*16, dimension( ldu, * ) evectr, complex*16, dimension(
    * ) work, integer lwork, double precision, dimension( * ) rwork,
    logical, dimension( * ) llwork, double precision, dimension( 15 )
    result, integer info)

ZCHKGG

🎀 Purpose 🎀

so what does this even do? ZCHKGG checks the nonsymmetric generalized eigenvalue problem routines. periodt.

ZGGHRD factors A and B as U H V^H and U T V^H, where ^H means conjugate transpose, H is hessenberg, T is triangular, and U and V are unitary.

ZHGEQZ factors H and T as Q S Z^H and Q P Z^H, where P and S are upper triangular and Q and Z are unitary. It also computes the generalized eigenvalues (alpha(1),beta(1)),...,(alpha(n),beta(n)), where alpha(j)=S(j,j) and beta(j)=P(j,j) — thus, w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue problem

det( A - w(j) B ) = 0

and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent problem

det( m(j) A - B ) = 0

ZTGEVC computes the matrix L of left eigenvectors and the matrix R of right eigenvectors for the matrix pair ( S, P ). In the description below, l and r are left and right eigenvectors corresponding to the generalized eigenvalues (alpha,beta).

When ZCHKGG is called, a number of matrix sizes and a number of matrix types are specified. For each size and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 13 tests will be performed. The first twelve tests should be small — O(1). They will be compared with the threshold THRESH:

                  H
(1)   | A - U H V  | / ( |A| n ulp )

                  H
(2)   | B - U T V  | / ( |B| n ulp )

               H
(3)   | I - UU  | / ( n ulp )

               H
(4)   | I - VV  | / ( n ulp )

                  H
(5)   | H - Q S Z  | / ( |H| n ulp )

                  H
(6)   | T - Q P Z  | / ( |T| n ulp )

               H
(7)   | I - QQ  | / ( n ulp )

               H
(8)   | I - ZZ  | / ( n ulp )

(9)   max over all left eigenvalue/-vector pairs (beta/alpha,l) of
                          H
      | (beta A - alpha B) l | / ( ulp max( |beta A|, |alpha B| ) )

(10)  max over all left eigenvalue/-vector pairs (beta/alpha,l') of
                          H
      | (beta H - alpha T) l' | / ( ulp max( |beta H|, |alpha T| ) )

      where the eigenvectors l' are the result of passing Q to
      DTGEVC and back transforming (JOB='B').

(11)  max over all right eigenvalue/-vector pairs (beta/alpha,r) of

      | (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) )

(12)  max over all right eigenvalue/-vector pairs (beta/alpha,r') of

      | (beta H - alpha T) r' | / ( ulp max( |beta H|, |alpha T| ) )

      where the eigenvectors r' are the result of passing Z to
      DTGEVC and back transforming (JOB='B').

(13)  | S(Q,Z computed) - S(Q,Z not computed) | / ( |S| ulp )

(14)  | P(Q,Z computed) - P(Q,Z not computed) | / ( |P| ulp )

(15)  max( |alpha(Q,Z computed) - alpha(Q,Z not computed)|/|S| ,
           |beta(Q,Z computed) - beta(Q,Z not computed)|/|P| ) / ulp

the last three test ratios will usually be small, but there is no mathematical requirement that they be so. they are therefore compared with THRESH only if TSTDIF is .TRUE.

In addition, the normalization of L and R are checked, and compared with the threshold THRSHN.

🧪 Test Matrices 🧪

okay so 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 types are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type j will be generated. currently, the list of possible types is:

Type 1: ( 0, 0 ) — a pair of zero matrices
Type 2: ( I, 0 ) — an identity and a zero matrix
Type 3: ( 0, I ) — an identity and a zero matrix
Type 4: ( I, I ) — a pair of identity matrices
Type 5:

     t   t
( J , J  )

— a pair of transposed Jordan blocks

Type 6:

                                 t                ( I   0  )
( 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

Type 7: ( D, I ) — where D is P*D1, P is a random unitary diagonal matrix (i.e., with random magnitude 1 entries on the diagonal), and D1=diag( 0, 1,..., N-1 ) (i.e., a diagonal matrix with D1(1,1)=0, D1(2,2)=1, ..., D1(N,N)=N-1.)
Type 8: ( I, D )
Type 9: ( big*D, small*I ) — where big is near overflow and small=1/big
Type 10: ( small*D, big*I )
Type 11: ( big*I, small*D )
Type 12: ( small*I, big*D )
Type 13: ( big*D, big*I )
Type 14: ( small*D, small*I )
Type 15: ( D1, D2 ) — where D1=P*diag( 0, 0, 1, ..., N-3, 0 ) and D2=Q*diag( 0, N-3, N-4,..., 1, 0, 0 ), and P and Q are random unitary diagonal matrices.
Type 16:

      t   t
U ( J , J ) V     where U and V are random unitary matrices.

Type 17:

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

Type 18:

U ( T1, T2 ) V    diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
                  diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
                  s = machine precision.

Type 19:

U ( T1, T2 ) V    diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
                  diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )

Type 20:

                                                      N-5
U ( T1, T2 ) V    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
                  diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )

Type 21:

U ( T1, T2 ) V    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.

Type 22:

U ( big*T1, small*T2 ) V   diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
                           diag(T2) = ( 0, 1, ..., 1, 0, 0 )

Type 23:

U ( small*T1, big*T2 ) V   diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
                           diag(T2) = ( 0, 1, ..., 1, 0, 0 )

Type 24:

U ( small*T1, small*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
                           diag(T2) = ( 0, 1, ..., 1, 0, 0 )

Type 25:

U ( big*T1, big*T2 ) V     diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
                           diag(T2) = ( 0, 1, ..., 1, 0, 0 )

Type 26:

U ( T1, T2 ) V     where T1 and T2 are random upper-triangular
                   matrices.

that's the whole test matrix menu, bestie — it's giving variety ✨

🌟 Parameters 🌟

`NSIZES`

NSIZES is INTEGER — the number of sizes of matrices to use. if it is zero, ZCHKGG does nothing. it must be at least zero.

`NN`

NN is INTEGER array, dimension (NSIZES) — an array containing the sizes to be used for the matrices. zero values will be skipped. the values must be at least zero.

`NTYPES`

NTYPES is INTEGER — the number of elements in DOTYPE. if it is zero, ZCHKGG 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. this is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .

`DOTYPE`

DOTYPE — 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. And if NTYPES is larger than MAXTYP, then DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) just get ignored, bestie. Nothing happens. Crickets. 💀
ISEED — INTEGER array, dimension (4) On entry, ISEED specifies the seed of the random number generator. Okay so the array elements should be between 0 and 4095 — if they're not, they get reduced mod 4096. Also, ISEED(4) must be odd (no exceptions, it's giving strict). The random number generator uses a linear congruential sequence limited to small integers, so it should produce machine independent random numbers ✨. The values of ISEED are changed on exit, and can be used in the next call to ZCHKGG to continue the same random number sequence. So like, it's a little continuity thread you can pick back up 🧵
THRESH — DOUBLE PRECISION A test will count as [blank] if the [blank], computed as described above, exceeds THRESH. Ngl the source text literally has gaps there — we're not gonna invent words for it, that's not the vibe 🙈. 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. It must be at least zero.
TSTDIF — LOGICAL Specifies whether test ratios 13-15 will be computed and compared with THRESH. 👀
- = .FALSE.: Only test ratios 1-12 will be computed and tested. Ratios 13-15 will be set to zero.
- = .TRUE.: All the test ratios 1-15 will be computed and tested. Full send 🔥
THRSHN — DOUBLE PRECISION Threshold for reporting eigenvector normalization error. If the normalization of any eigenvector differs from 1 by more than THRSHN*ulp, then a special error message will be printed. (This is handled separately from the other tests, since only a compiler or programming error should cause an error message, at least if THRSHN is at least 5--10.) So like, if this one screams at you, something is genuinely cursed 🫠
NOUNIT — INTEGER The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
A — COMPLEX16 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, H, T, S1, P1, S2, and P2. It must be at least 1 and at least max( NN ).
B — COMPLEX16 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.
H — COMPLEX16 array, dimension (LDA, max(NN))* The upper Hessenberg matrix computed from A by ZGGHRD.
T — COMPLEX16 array, dimension (LDA, max(NN))* The upper triangular matrix computed from B by ZGGHRD.
S1 — COMPLEX16 array, dimension (LDA, max(NN))* The Schur (upper triangular) matrix computed from H by ZHGEQZ when Q and Z are also computed.
S2 — COMPLEX16 array, dimension (LDA, max(NN))* The Schur (upper triangular) matrix computed from H by ZHGEQZ when Q and Z are not computed.
P1 — COMPLEX16 array, dimension (LDA, max(NN))* The upper triangular matrix computed from T by ZHGEQZ when Q and Z are also computed.
P2 — COMPLEX16 array, dimension (LDA, max(NN))* The upper triangular matrix computed from T by ZHGEQZ when Q and Z are not computed.
U — COMPLEX16 array, dimension (LDU, max(NN))* The (left) unitary matrix computed by ZGGHRD.
LDU — INTEGER The leading dimension of U, V, Q, Z, EVECTL, and EVEZTR. It must be at least 1 and at least max( NN ).
V — COMPLEX16 array, dimension (LDU, max(NN))* The (right) unitary matrix computed by ZGGHRD.
Q — COMPLEX16 array, dimension (LDU, max(NN))* The (left) unitary matrix computed by ZHGEQZ.
Z — COMPLEX16 array, dimension (LDU, max(NN))* The (left) unitary matrix computed by ZHGEQZ.
ALPHA1 — COMPLEX16 array, dimension (max(NN))* (yes bestie, that's the whole description — it's giving mysterious main character energy 🎀)
BETA1 — COMPLEX16 array, dimension (max(NN))* The generalized eigenvalues of (A,B) computed by ZHGEQZ when Q, Z, and the full Schur matrices are computed.
ALPHA3 — COMPLEX16 array, dimension (max(NN))* (also just vibes, no description. iconic honestly)
BETA3 — COMPLEX16 array, dimension (max(NN))* The generalized eigenvalues of (A,B) computed by ZHGEQZ when neither Q, Z, nor the Schur matrices are computed.
EVECTL — COMPLEX16 array, dimension (LDU, max(NN))* The (lower triangular) left eigenvector matrix for the matrices in S1 and P1.
EVECTR — COMPLEX16 array, dimension (LDU, max(NN))* The (upper triangular) right eigenvector matrix for the matrices in S1 and P1.
WORK — COMPLEX16 array, dimension (LWORK)* (just the workspace, doing workspace things ✨)
LWORK — INTEGER The number of entries in WORK. This must be at least max( 4*N, 2 * N**2, 1 ), for all N=NN(j). Do not skimp, she needs room 💅
RWORK — DOUBLE PRECISION array, dimension (2max(NN))* (the real-arithmetic bestie of WORK)
LLWORK — LOGICAL array, dimension (max(NN)) (a whole array of little .TRUE./.FALSE. guys 🦋)
RESULT — DOUBLE PRECISION array, dimension (15) The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. So like, they're capped — no runaway numbers allowed 😭
INFO — INTEGER
- = 0: successful exit. slay 🎉
- < 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 498 of file zchkgg.f.

🖨️ Author

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okay that's the whole page, bestie, you made it to the end 🫶 this absolute icon of a manual page is LAPACK Version 3.12.0, living its truth in TESTING/EIG/zchkgg.f(3), and the little footer stamp — version and date and all — is right there at the bottom of the source, so go peek at it if you need the exact date. periodt. 🌈👑

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