TESTING/EIG/sdrgev.f(3)

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🎀 TESTING/EIG/sdrgev.f 🎀

okay so this is a page from the Library Functions Manual for TESTING/EIG/sdrgev.f — and bestie, it's a chonky one. Buckle in 💖

✨ NAME ✨

TESTING/EIG/sdrgev.f — the file itself, main character energy, no notes 💅

🌸 SYNOPSIS 🌸

Functions/Subroutines

  • subroutine sdrgev (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, b, s, t, q, ldq, z, qe, ldqe, alphar, alphai, beta, alphr1, alphi1, beta1, work, lwork, result, info)
  • SDRGEV

💜 Function/Subroutine Documentation 💜

⭐ subroutine sdrgev ⭐

Here's the full signature, and yes every single argument is listed, we do not skip, we do not summarize, we are thorough in this house 🫶

subroutine sdrgev (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( ldqe, * ) qe, integer ldqe,
    real, dimension( * ) alphar, real, dimension( * ) alphai, real,
    dimension( * ) beta, real, dimension( * ) alphr1, real, dimension( * )
    alphi1, real, dimension( * ) beta1, real, dimension( * ) work, integer
    lwork, real, dimension( * ) result, integer info)

SDRGEV — that's the name, say it with your chest ✨

🦋 Purpose 🦋

!> !> SDRGEV checks the nonsymmetric generalized eigenvalue problem driver !> routine SGGEV. !>

So basically: SDRGEV is the vibe checker for SGGEV 🔥

SGGEV computes, for a pair of n-by-n nonsymmetric matrices (A,B), the generalized eigenvalues and — if you want them — the left and right eigenvectors. Iconic behavior.

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It's usually represented as the pair (alpha,beta), because there's a reasonable interpretation for beta=0, and honestly even for both of them being zero. slay.

A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that

(A - wB) * r = 0.

A left generalized eigenvector is a vector l such that

l**H * (A - wB) = 0,

where l**H is the conjugate-transpose of l. periodt 💅

When SDRGEV is called, a number of matrix sizes and a number of matrix types 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 tests will be performed and compared with the threshold THRESH. 👑

🌈 Results from SGGEV 🌈

(1) max over all left eigenvalue/-vector pairs (alpha/beta,l) of

| VL**H * (beta A - alpha B) |/( ulp max(|beta A|, |alpha B|) )

where VL**H is the conjugate-transpose of VL.

(2) | |VL(i)| - 1 | / ulp and whether largest component real

VL(i) denotes the i-th column of VL.

(3) max over all left eigenvalue/-vector pairs (alpha/beta,r) of

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

(4) | |VR(i)| - 1 | / ulp and whether largest component real

VR(i) denotes the i-th column of VR.

(5) W(full) = W(partial)

W(full) denotes the eigenvalues computed when both l and r are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and r, or only W and l are computed.

(6) VL(full) = VL(partial)

VL(full) denotes the left eigenvectors computed when both l and r are computed, and VL(partial) denotes the result when only l is computed.

(7) VR(full) = VR(partial)

VR(full) denotes the right eigenvectors computed when both l and r are also computed, and VR(partial) denotes the result when only l is computed.

🧚 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: 👇

(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.

twenty-six whole matrix types, we're so back 😭🔥

👑 Parameters 👑

🌟 NSIZES

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

Translation: how many sizes of matrices you wanna use. If it's zero, SDRGES does nothing (it's giving "no thoughts, head empty" 💀). NSIZES >= 0 — non-negotiable.

🌟 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. !>

The actual sizes, in an array, dimension (NSIZES). Zero values get skipped — we don't have time for them. NN >= 0 fr fr.

🌟 NTYPES

!> NTYPES is INTEGER !> The number of elements in DOTYPE. If it is zero, SDRGES !> 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. . !>

So how many elements are in DOTYPE. If it's zero, SDRGES does nothing. Gotta be at least zero. ✨ AND — special case — if it's MAXTYP+1 and NSIZES is 1, then there's an extra type, MAXTYP+1, defined, which means "just use whatever matrix is already in A". That's only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE.. iykyk 🫶

🌟 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. !>

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 gets generated. If NTYPES is smaller than the max 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) get ignored. rude but ok 💅

🌟 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 SDRGES to continue the same random number !> sequence. !>

The rng seed, 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 get reduced mod 4096. Also ISEED(4) must be odd. The 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 reused in the next call to SDRGES to continue the same random number sequence. it's giving continuity 🧚

🌟 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. !>

A test will count as a fail if the error, computed as described above, exceeds THRESH. ⚠️ Important: 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. no cap.

🌟 NOUNIT

!> NOUNIT is INTEGER !> The FORTRAN unit number for printing out error messages !> (e.g., if a routine returns IERR not equal to 0.) !>

The FORTRAN unit number for printing out error messages (like, if a routine returns IERR not equal to 0). your little error megaphone 📢

🌟 A

!> A is 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. !>

A REAL array, dimension(LDA, max(NN)). Holds the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.. otherwise it's just chilling 🦋

🌟 LDA

  • LDA — INTEGER, bestie. It's the leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). no cap 💅
  • 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. — that's the exact vibe check.
  • S — REAL array, dimension (LDA, max(NN)). The Schur form matrix computed from A by SGGES. On exit, S contains the Schur form matrix corresponding to the matrix in A. iconic ✨
  • T — REAL array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by SGGES. periodt.
  • Q — REAL array, dimension (LDQ, max(NN)). The (left) eigenvectors matrix computed by SGGEV. obsessed 🫶
  • LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). don't skip this one fr fr.
  • Z — REAL array, dimension( LDQ, max(NN) ). The (right) orthogonal matrix computed by SGGES. main character energy 👑
  • QE — REAL array, dimension( LDQ, max(NN) ). QE holds the computed right or left eigenvectors. cute.
  • LDQE — INTEGER. The leading dimension of QE. LDQE >= max(1,max(NN)). noted 💅
  • ALPHAR — REAL array, dimension (max(NN)). yes bestie, that's the whole entry.
  • ALPHAI — REAL array, dimension (max(NN)). also exactly that.
  • BETA — REAL array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by SGGEV. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. math slay 🔥
  • ALPHR1 — REAL array, dimension (max(NN)). yup.
  • ALPHI1 — REAL array, dimension (max(NN)). yup again.
  • BETA1 — REAL array, dimension (max(NN)). Like ALPHAR, ALPHAI, BETA, these arrays contain the eigenvalues of A and B, but those computed when SGGEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. iykyk 🌸
  • WORK — REAL array, dimension (LWORK). workspace bestie.
  • LWORK — INTEGER. The number of entries in WORK. LWORK >= MAX( 8*N, N*(N+1) ). write that down 📝
  • RESULT — REAL array, dimension (2). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. safety first 🎀
  • 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. that's the whole status update 💀

👑 Author

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

📍 Definition at line 404 of file sdrgev.f.

🧚 Author

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💖 that's a wrap, bestie! this page is from LAPACK Version 3.12.0, file TESTING/EIG/sdrgev.f(3), with all the source/version/date footer energy included and not copy-pasted. periodt. 🎀

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