CCHKGG(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 CCHKGG(3) — Library Functions Manual 💅

okay so bestie, this is the CCHKGG(3) page from the Library Functions Manual, and it is lowkey a whole vibe. It lives at this exact path:

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

🌸 NAME 🌸

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

Bestie, that's literally the file this page is about. No cap. 🫶

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine CCHKGG is the main character here:

subroutine CCHKGG (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)

CCHKGG

👑 Function/Subroutine Documentation 👑

💖 Purpose 💖

CCHKGG checks the nonsymmetric generalized eigenvalue problem routines. Slay. ✨

CGGHRD 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.

CHGEQZ 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

CTGEVC 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 CCHKGG 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 should be small — O(1). They will be compared with the threshold THRESH:

The page says 13 tests and then lists fifteen ratios. I'm just reporting the facts, bestie — both things are in the original. 😭
                  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
      STGEVC 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
      STGEVC 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 🌈

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) — a pair of transposed Jordan blocks:
        t   t
(5)  ( J , J  )       (a pair of transposed Jordan blocks)
  • Type (6) — ( X, Y ) where X = ( J^t 0 ; 0 I ) and Y = ( I 0 ; 0 J^t ), and I is a k x k identity and J a (k+1)x(k+1) Jordan block; k=(N-1)/2.
                                     t                ( I   0  )
(6)  ( X, Y )         where  X = ( J   0  )  and Y = (      t )
                                  ( 0   I  )          ( 0   J  )
  • 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) — U ( J^t, J^t ) V where U and V are random unitary matrices.
           t   t
(16) 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 with diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ) and diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ), where s = machine precision.
  • Type (19) — U ( T1, T2 ) V with diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 ) and diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
  • Type (20) — U ( T1, T2 ) V with diag(T1)=( 0, 0, 1, 1, a, ..., a^(N-5)=s, 0 ) and diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
                                                        N-5
(20) 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 with diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 ) and diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ), where r1,..., r(N-4) are random.
  • Type (22) — U ( big*T1, small*T2 ) V with diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • Type (23) — U ( small*T1, big*T2 ) V with diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • Type (24) — U ( small*T1, small*T2 ) V with diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • Type (25) — U ( big*T1, big*T2 ) V with diag(T1) = P*( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • Type (26) — U ( T1, T2 ) V where T1 and T2 are random upper-triangular matrices.

🎀 Parameters 🎀

NSIZES

NSIZES is INTEGER. It's the number of sizes of matrices to use. If it is zero, CCHKGG does nothing. It must be at least zero.

NN

NN is INTEGER array, dimension (NSIZES). It's 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. It's the number of elements in DOTYPE. If it is zero, CCHKGG 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 (continued)

DOTYPE is a 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. lowkey that last bit is just "we're not reading those, bestie" 💅

🎲 ISEED

ISEED is an 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. like, mandatory odd, no exceptions ✨ 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 CCHKGG to continue the same random number sequence. iykyk — it's a little seed you pass along 🍓

📏 THRESH

THRESH is REAL 💜

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. It must be at least zero. at least zero, fr fr, that's not a vibe suggestion it's a requirement 🫶

🔍 TSTDIF

TSTDIF is 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. 💀 (they're just... not invited to the party)
  • = .TRUE.: All the test ratios 1-15 will be computed and tested. 👑

🧚 THRSHN

THRSHN is REAL 🌸

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.) it's giving "if this fires, something is deeply wrong and it's not your math" 😭

🖨️ 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 is COMPLEX 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. those three conditions together are the secret handshake ✨

📐 LDA

LDA is 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

B is COMPLEX 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. (same secret handshake as A, no cap 🫶)

🏔️ H

H is COMPLEX array, dimension (LDA, max(NN)) ⭐

The upper Hessenberg matrix computed from A by CGGHRD.

🔺 T

T is COMPLEX array, dimension (LDA, max(NN)) 🌟

The upper triangular matrix computed from B by CGGHRD.

💎 S1

S1 is COMPLEX array, dimension (LDA, max(NN)) 🌸

The Schur (upper triangular) matrix computed from H by CHGEQZ when Q and Z are also computed.

💎 S2

S2 is COMPLEX array, dimension (LDA, max(NN)) 🌸

The Schur (upper triangular) matrix computed from H by CHGEQZ when Q and Z are not computed. see the difference? that's the whole point of having two 💅

🧩 P1

P1 is COMPLEX array, dimension (LDA, max(NN)) 🎀

The upper triangular matrix computed from T by CHGEQZ when Q and Z are also computed.

🧩 P2

P2 is COMPLEX array, dimension (LDA, max(NN)) 🎀

The upper triangular matrix computed from T by CHGEQZ when Q and Z are not computed.

🌙 U

U is COMPLEX array, dimension (LDU, max(NN)) 💜

The (left) unitary matrix computed by CGGHRD.

📏 LDU

LDU is INTEGER 🌈

The leading dimension of U, V, Q, Z, EVECTL, and EVECTR. It must be at least 1 and at least max( NN ).

🌙 V

V is COMPLEX array, dimension (LDU, max(NN)) 💜

The (right) unitary matrix computed by CGGHRD.

🌙 Q

Q is COMPLEX array, dimension (LDU, max(NN)) 💜

The (left) unitary matrix computed by CHGEQZ.

🌙 Z

Z is COMPLEX array, dimension (LDU, max(NN)) 💜

The (left) unitary matrix computed by CHGEQZ. yes, it says left again — we're reporting it exactly as written, bestie, don't come for me 🙈

🍒 ALPHA1

ALPHA1 is COMPLEX array, dimension (max(NN)) ✨

🍒 BETA1

BETA1 is COMPLEX array, dimension (max(NN)) ✨

The generalized eigenvalues of (A,B) computed by CHGEQZ when Q, Z, and the full Schur matrices are computed.

🍒 ALPHA3

ALPHA3 is COMPLEX array, dimension (max(NN)) ✨

🍒 BETA3

BETA3 is COMPLEX array, dimension (max(NN)) ✨

The generalized eigenvalues of (A,B) computed by CHGEQZ when neither Q, Z, nor the Schur matrices are computed.

🦋 EVECTL

EVECTL is COMPLEX array, dimension (LDU, max(NN)) 🌸

The (lower triangular) left eigenvector matrix for the matrices in S1 and P1.

🦋 EVECTR

EVECTR is COMPLEX array, dimension (LDU, max(NN)) 🌸

The (upper triangular) right eigenvector matrix for the matrices in S1 and P1.

⚙️ WORK

WORK is COMPLEX array, dimension (LWORK) 🎀

📏 LWORK

LWORK is INTEGER 💅

The number of entries in WORK. This must be at least max( 4*N, 2 * N**2, 1 ), for all N=NN(j). for ALL of them, not just your favourite one ✨

⚙️ RWORK

RWORK is REAL array, dimension (2*max(NN)) 🍓

⚙️ LLWORK

LLWORK is LOGICAL array, dimension (max(NN)) 🍓

📊 RESULT

RESULT is REAL array, dimension (15) 🔥

The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. capped so nothing goes feral and overflows on you 💀

ℹ️ INFO

INFO is 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. so it's the error, but make it positive 💅

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 498 of file cchkgg.f. 📍

📜 Author 📜

Generated automatically by Doxygen for LAPACK from the source code. 🤖✨

and that's a wrap on CCHKGG(3), bestie!! 🎀 You just read the whole thing and honestly? iconic behavior. This page came from /home/abui...IG/cchkgg.f(3) and it's LAPACK Version 3.12.0 — go drink some water, you've earned it 🫶💖

📦 where this page comes from