tgsen(3)

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🌸 tgsen(3) — Library Functions Manual 🌸

okay so bestie, welcome to tgsen(3) from the Library Functions Manual. This is part 1 of the page, and it's giving generalized Schur reorder energy — we are literally about to move a selected cluster of eigenvalues into the leading diagonal blocks. 💅✨

🎀 NAME 🎀

tgsen — reorder generalized Schur form. That's the whole vibe: reorder the generalized Schur form, bestie. 🦋

✨ SYNOPSIS ✨

💖 Functions 💖

  • ctgsen — the complex single-precision one, and yes it gets its own whole signature:
subroutine ctgsen (ijob, wantq, wantz, select, n, a, lda, b, ldb, alpha,
           beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork,
           info)
           CTGSEN
  • dtgsen — the double-precision real one with alphar and alphai in the mix:
subroutine dtgsen (ijob, wantq, wantz, select, n, a, lda, b, ldb,
           alphar, alphai, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork,
           iwork, liwork, info)
           DTGSEN
  • stgsen — same real double-precision energy but single precision, slay:
subroutine stgsen (ijob, wantq, wantz, select, n, a, lda, b, ldb,
           alphar, alphai, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork,
           iwork, liwork, info)
           STGSEN
  • ztgsen — the complex double-precision icon:
subroutine ztgsen (ijob, wantq, wantz, select, n, a, lda, b, ldb, alpha,
           beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork,
           info)
           ZTGSEN

🌈 Detailed Description 🌈

This is the big-picture section before we get into the function docs, bestie. 🫶

🦋 Function Documentation 🦋

💅 ctgsen — CTGSEN 💅

subroutine ctgsen (integer ijob, logical wantq, logical wantz, logical,
     dimension( * ) select, integer n, complex, dimension( lda, * ) a,
     integer lda, complex, dimension( ldb, * ) b, integer ldb, complex,
     dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension(
     ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz,
     integer m, real pl, real pr, real, dimension( * ) dif, complex,
     dimension( * ) work, integer lwork, integer, dimension( * ) iwork,
     integer liwork, integer info)
     CTGSEN

✨ Purpose ✨

CTGSEN reorders the generalized Schur decomposition of a complex matrix pair (A, B) — in terms of an unitary equivalence transformation Q**H * (A, B) * Z — so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the pair (A,B). The leading columns of Q and Z form unitary bases of the corresponding left and right eigenspaces (deflating subspaces). (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular. 🦋

CTGSEN also computes the generalized eigenvalues

w(j)= ALPHA(j) / BETA(j)

of the reordered matrix pair (A, B). ✨

Optionally, the routine computes estimates of reciprocal condition numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) between the matrix pairs (A11, B11) and (A22,B22) that correspond to the selected cluster and the eigenvalues outside the cluster, resp., and norms of onto left and right eigenspaces w.r.t. the selected cluster in the (1,1)-block. 💜

🌸 Parameters 🌸

  • IJOB — IJOB is INTEGER. Specifies whether condition numbers are required for the cluster of eigenvalues (PL and PR) or the deflating subspaces (Difu and Difl):
    • =0: Only reorder w.r.t. SELECT. No extras.
    • =1: Reciprocal of norms of onto left and right eigenspaces w.r.t. the selected cluster (PL and PR).
    • =2: Upper bounds on Difu and Difl. F-norm-based estimate (DIF(1:2)).
    • =3: Estimate of Difu and Difl. 1-norm-based estimate (DIF(1:2)). About 5 times as expensive as IJOB = 2.
    • =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic version to get it all.
    • =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
  • WANTQ — WANTQ is LOGICAL. .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q.
  • WANTZ — WANTZ is LOGICAL. .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z.
  • SELECT — SELECT is LOGICAL array, dimension (N). SELECT specifies the eigenvalues in the selected cluster. To select an eigenvalue w(j), SELECT(j) must be set to .TRUE..
  • N — N is INTEGER. The order of the matrices A and B. N >= 0.
  • A — A is COMPLEX array, dimension(LDA,N). On entry, the upper triangular matrix A, in generalized Schur canonical form. On exit, A is overwritten by the reordered matrix A.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is COMPLEX array, dimension(LDB,N). On entry, the upper triangular matrix B, in generalized Schur canonical form. On exit, B is overwritten by the reordered matrix B.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • ALPHA — ALPHA is COMPLEX array, dimension (N).
  • BETA — BETA is COMPLEX array, dimension (N). The diagonal elements of A and B, respectively, when the pair (A,B) has been reduced to generalized Schur form. ALPHA(i)/BETA(i) i=1,...,N are the generalized eigenvalues.
  • Q — Q is COMPLEX array, dimension (LDQ,N). On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. On exit, Q has been postmultiplied by the left unitary transformation matrix which reorder (A, B); The leading M columns of Q form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). If WANTQ = .FALSE., Q is not referenced.
  • LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N.
  • Z — Z is COMPLEX array, dimension (LDZ,N). On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. On exit, Z has been postmultiplied by the left unitary transformation matrix which reorder (A, B); The leading M columns of Z form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). If WANTZ = .FALSE., Z is not referenced.
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N.
  • M — M is INTEGER. The dimension of the specified pair of left and right eigenspaces, (deflating subspaces) 0 <= M <= N.
  • PL — PL is REAL.
  • PR — PR is REAL. If IJOB = 1, 4 or 5, PL, PR are lower bounds on the reciprocal of the norm of onto left and right eigenspace with respect to the selected cluster. 0 < PL, PR <= 1. If M = 0 or M = N, PL = PR = 1. If IJOB = 0, 2 or 3 PL, PR are not referenced.
  • DIF — DIF is REAL array, dimension (2). If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based estimates of Difu and Difl, computed using reversed communication with CLACN2. If M = 0 or N, DIF(1:2) = F-norm([A, B]). If IJOB = 0 or 1, DIF is not referenced.
  • WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= 1. If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M). If IJOB = 3 or 5, LWORK >= 4*M*(N-M). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
  • LIWORK — LIWORK is INTEGER. The dimension of the array IWORK. LIWORK >= 1. If IJOB = 1, 2 or 4, LIWORK >= N+2; If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M)); If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA.
  • INFO — INFO is INTEGER. =0: Successful exit. <0: If INFO = -i, the i-th argument had an illegal value. =1: Reordering of (A, B) failed because the transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is very ill-conditioned. (A, B) may have been partially reordered. If requested, 0 is returned in DIF(*), PL and PR.

👑 Author 👑

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

🦋 Further Details: 🦋

okay so back to the math, bestie: CTGSEN first collects the selected eigenvalues by computing unitary U and W that move them to the top left corner of (A, B). In other words, the selected eigenvalues are the eigenvalues of (A11, B11) in

U**H*(A, B)*W = (A11 A12) (B11 B12) n1
                              ( 0  A22),( 0  B22) n2
                                n1  n2    n1  n2

where N = n1+n2 and U**H means the conjugate transpose of U. The first n1 columns of U and W span the specified pair of left and right eigenspaces (deflating subspaces) of (A, B). 💅

If (A, B) has been obtained from the generalized real Schur decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the reordered generalized Schur form of (C, D) is given by

(C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H,

and the first n1 columns of Q*U and Z*W span the corresponding deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). 🌸

⚠️ Note that if the selected eigenvalue is sufficiently ill-conditioned, then its value may differ significantly from its value before reordering. no cap, that's just how it is.

The reciprocal condition numbers of the left and right eigenspaces spanned by the first n1 columns of U and W (or Q*U and Z*W) may be returned in DIF(1:2), corresponding to Difu and Difl, resp. ✨

The Difu and Difl are defined as:

Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )

and

Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

where sigma-min(Zu) is the smallest singular value of the (2*n1*n2)-by-(2*n1*n2) matrix

Zu = [ kron(In2, A11)  -kron(A22**H, In1) ]
     [ kron(In2, B11)  -kron(B22**H, In1) ].

Here, Inx is the identity matrix of size nx and A22**H is the conjugate transpose of A22. kron(X, Y) is the Kronecker product between the matrices X and Y. 🧚

When DIF(2) is small, small changes in (A, B) can cause large changes in the deflating subspace. An approximate (asymptotic) bound on the maximum angular error in the computed deflating subspaces is

EPS * norm((A, B)) / DIF(2),

where EPS is the machine precision. 😭

The reciprocal norm of the projectors on the left and right eigenspaces associated with (A11, B11) may be returned in PL and PR. They are computed as follows. First we compute L and R so that P*(A, B)*Q is block diagonal, where

P = ( I -L ) n1           Q = ( I R ) n1
    ( 0  I ) n2    and        ( 0 I ) n2
      n1 n2                    n1 n2

and (L, R) is the solution to the generalized Sylvester equation

A11*R - L*A22 = -A12
B11*R - L*B22 = -B12

Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is

EPS * norm((A, B)) / PL.

There are also global error bounds which valid for perturbations up to a certain restriction: A lower bound (x) on the smallest F-norm(E,F) for which an eigenvalue of (A11, B11) may move and coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), (i.e. (A + E, B + F), is

x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

An approximate bound on x can be computed from DIF(1:2), PL and PR. 💖

If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed (L', R') and unperturbed (L, R) left and right deflating subspaces associated with the selected cluster in the (1,1)-blocks can be bounded as

max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

See LAPACK User's Guide section 4.11 or the following references for more information. 🦋

Note that if the default method for computing the Frobenius-norm-based estimate DIF is not wanted (see CLATDF), then the parameter IDIFJB (see below) should be changed from 3 to 4 (routine CLATDF (IJOB = 2 will be used)). See CTGSYL for more details. 💀

💖 Contributors 💖

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

📚 References 📚

  • [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
  • [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
  • [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

Definition at line 430 of file ctgsen.f.

subroutine dtgsen (integer ijob, logical wantq, logical wantz, logical,
    dimension( * ) select, integer n, double precision, dimension( lda, * )
    a, integer lda, double precision, dimension( ldb, * ) b, integer ldb,
    double precision, dimension( * ) alphar, double precision, dimension( *
    ) alphai, double precision, dimension( * ) beta, double precision,
    dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, *
    ) z, integer ldz, integer m, double precision pl, double precision pr,
    double precision, dimension( * ) dif, double precision, dimension( * )
    work, integer lwork, integer, dimension( * ) iwork, integer liwork,
    integer info)

🦋 DTGSEN 🦋

🌟 Purpose 🌟

okay so DTGSEN reorders the generalized real Schur decomposition of a real matrix pair (A, B) (in terms of an orthonormal equivalence transformation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix A and the upper triangular B. The leading columns of Q and Z form orthonormal bases of the corresponding left and right eigenspaces (deflating subspaces). (A, B) must be in generalized real Schur canonical form (as returned by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular. 🔥

DTGSEN also computes the generalized eigenvalues

w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)

of the reordered matrix pair (A, B). 👑

Optionally, DTGSEN computes the estimates of reciprocal condition numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) between the matrix pairs (A11, B11) and (A22,B22) that correspond to the selected cluster and the eigenvalues outside the cluster, resp., and norms of onto left and right eigenspaces w.r.t. the selected cluster in the (1,1)-block. 🫶

✨ Parameters ✨

  • IJOB — IJOB is INTEGER. Specifies whether condition numbers are required for the cluster of eigenvalues (PL and PR) or the deflating subspaces (Difu and Difl):
    • =0: Only reorder w.r.t. SELECT. No extras.
    • =1: Reciprocal of norms of onto left and right eigenspaces w.r.t. the selected cluster (PL and PR).
    • =2: Upper bounds on Difu and Difl. F-norm-based estimate (DIF(1:2)).
    • =3: Estimate of Difu and Difl. 1-norm-based estimate (DIF(1:2)). About 5 times as expensive as IJOB = 2.
    • =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic version to get it all.
    • =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
  • WANTQ — WANTQ is LOGICAL. .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q.
  • WANTZ — WANTZ is LOGICAL. .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z.
  • SELECT — SELECT is LOGICAL array, dimension (N). SELECT specifies the eigenvalues in the selected cluster. To select a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select a complex conjugate pair of eigenvalues w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, either SELECT(j) or SELECT(j+1) or both must be set to .TRUE.; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded. 💅
  • N — N is INTEGER. The order of the matrices A and B. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the upper quasi-triangular matrix A, with (A, B) in generalized real Schur canonical form. On exit, A is overwritten by the reordered matrix A.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is DOUBLE PRECISION array, dimension (LDB,N). On entry, the upper triangular matrix B, with (A, B) in generalized real Schur canonical form. On exit, B is overwritten by the reordered matrix B.
  • LDB —
  • LDB — INTEGER. It's the leading dimension of array B, bestie. Rule: LDB >= max(1,N).
  • ALPHAR — DOUBLE PRECISION array, dimension (N).
  • ALPHAI — DOUBLE PRECISION array, dimension (N).
  • BETA — DOUBLE PRECISION array, dimension (N).

Okay so on exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), for j=1,...,N, gives you the generalized eigenvalues. Slay. And ALPHAR(j) + ALPHAI(j)*i together with BETA(j), j=1,...,N, are the diagonals of the complex Schur form (S,T) — the one you'd get if those 2-by-2 diagonal blocks of the real generalized Schur form of (A,B) got further reduced to triangular form using complex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real 🌸. If it's positive, then the j-th and (j+1)-st eigenvalues form a complex conjugate pair, with ALPHAI(j+1) negative. Iconic behaviour tbh.

  • Q — DOUBLE PRECISION array, dimension (LDQ,N).

On entry: if WANTQ = .TRUE., Q is an N-by-N matrix. On exit, Q has been postmultiplied by the left orthogonal transformation matrix which reorder (A, B) — and yes, that's exactly how the original words it 😭. The leading M columns of Q form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). If WANTQ = .FALSE., Q is simply not referenced. Leave her alone 💅.

  • LDQ — INTEGER. The leading dimension of the array Q. LDQ >= 1; and if WANTQ = .TRUE., then LDQ >= N.
  • Z — DOUBLE PRECISION array, dimension (LDZ,N).

On entry: if WANTZ = .TRUE., Z is an N-by-N matrix. On exit, Z has been postmultiplied by the left orthogonal transformation matrix which reorder (A, B); the leading M columns of Z form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). If WANTZ = .FALSE., Z is not referenced, no thoughts, head empty 🫶.

  • LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1; if WANTZ = .TRUE., then LDZ >= N.
  • M — INTEGER. The dimension of the specified pair of left and right eigenspaces (deflating subspaces). Range: 0 <= M <= N.
  • PL — DOUBLE PRECISION.
  • PR — DOUBLE PRECISION.

If IJOB = 1, 4 or 5, then PL and PR are lower bounds on the reciprocal of the norm of onto left and right eigenspaces with respect to the selected cluster. And 0 < PL, PR <= 1. If M = 0 or M = N, then PL = PR = 1. If IJOB = 0, 2 or 3, then PL and PR are just… not referenced ✨.

  • DIF — DOUBLE PRECISION array, dimension (2).

If IJOB >= 2, then DIF(1:2) store the estimates of Difu and Difl. If IJOB = 2 or 4, then DIF(1:2) are F-norm-based upper bounds on Difu and Difl. If IJOB = 3 or 5, then DIF(1:2) are 1-norm-based estimates of Difu and Difl. And if M = 0 or N, then DIF(1:2) = F-norm([A, B]). If IJOB = 0 or 1, DIF is not referenced. Periodt.

  • WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)).

On exit, if INFO = 0, then WORK(1) returns the optimal LWORK. Chef's kiss 👑

  • LWORK — INTEGER. The dimension of the array WORK. Constraint: LWORK >= 4*N+16. If IJOB = 1, 2 or 4, then LWORK >= MAX(4*N+16, 2*M*(N-M)). If IJOB = 3 or 5, then LWORK >= MAX(4*N+16, 4*M*(N-M)).

And check this: if LWORK = -1, then a workspace query is assumed 🧚 — the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. Free info, no scolding.

  • IWORK — INTEGER array, dimension (MAX(1,LIWORK)).

On exit, if INFO = 0, then IWORK(1) returns the optimal LIWORK.

  • LIWORK — INTEGER. The dimension of the array IWORK. Constraint: LIWORK >= 1. If IJOB = 1, 2 or 4, then LIWORK >= N+6. If IJOB = 3 or 5, then LIWORK >= MAX(2*M*(N-M), N+6).

Same deal as before: if LIWORK = -1, then a workspace query is assumed — the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. Very demure, very mindful 🙈.

  • INFO — INTEGER. The receipts:
    • =0: successful exit, we ate 💅
    • <0: if INFO = -i, the i-th argument had an illegal value
    • =1: reordering of (A, B) failed because the transformed matrix pair (A, B) would be too far from generalized Schur form — the problem is very ill-conditioned. Heads up: (A, B) may have been partially reordered. And if requested, 0 is returned in DIF(*), PL and PR. Not a full slay, sorry bestie 😭

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌈 Further Details:

ok this is the part where it gets really good, grab a snack 🍓

DTGSEN first collects the selected eigenvalues by computing orthogonal U and W that move them to the top left corner of (A, B). In other words, the selected eigenvalues are the eigenvalues of (A11, B11) in:

    U**T*(A, B)*W = (A11 A12) (B11 B12) n1
                    ( 0  A22),( 0  B22) n2
                      n1  n2    n1  n2

where N = n1+n2 and U**T means the transpose of U. The first n1 columns of U and W span the specified pair of left and right eigenspaces (deflating subspaces) of (A, B) ✨

If (A, B) has been obtained from the generalized real Schur decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the reordered generalized real Schur form of (C, D) is given by:

    (C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,

and the first n1 columns of Q*U and Z*W span the corresponding deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). That's the whole vibe 🤝

⚠️ Note that if the selected eigenvalue is sufficiently ill-conditioned, then its value may differ significantly from its value before reordering. So like… don't be surprised. It's giving chaos.

The reciprocal condition numbers of the left and right eigenspaces spanned by the first n1 columns of U and W (or Q*U and Z*W) may be returned in DIF(1:2), corresponding to Difu and Difl, resp.

The Difu and Difl are defined as:

    Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )

and

    Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

where sigma-min(Zu) is the smallest singular value of the (2*n1*n2)-by-(2*n1*n2) matrix

    Zu = [ kron(In2, A11)  -kron(A22**T, In1) ]
         [ kron(In2, B11)  -kron(B22**T, In1) ].

Here, Inx is the identity matrix of size nx and A22**T is the transpose of A22. And kron(X, Y) is the Kronecker product between the matrices X and Y. Big math energy, we love to see it 💜

When DIF(2) is small, small changes in (A, B) can cause large changes in the deflating subspace. An approximate (asymptotic) bound on the maximum angular error in the computed deflating subspaces is

    EPS * norm((A, B)) / DIF(2),

where EPS is the machine precision. Tiny DIF(2) = big oof, basically 🧚

The reciprocal norm of the projectors on the left and right eigenspaces associated with (A11, B11) may be returned in PL and PR. They are computed as follows. First we compute L and R so that P*(A, B)*Q is block diagonal, where

    P = ( I -L ) n1           Q = ( I R ) n1
        ( 0  I ) n2    and        ( 0 I ) n2
          n1 n2                    n1 n2

and (L, R) is the solution to the generalized Sylvester equation

    A11*R - L*A22 = -A12
    B11*R - L*B22 = -B12

Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is

    EPS * norm((A, B)) / PL.

There are also global error bounds which valid for perturbations up to a certain restriction: a lower bound (x) on the smallest F-norm(E,F) for which an eigenvalue of (A11, B11) may move and coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), (i.e. (A + E, B + F), is

 x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

An approximate bound on x can be computed from DIF(1:2), PL and PR. Bookkeeping queen behaviour fr fr 📚

If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed (L', R') and unperturbed (L, R) left and right deflating subspaces associated with the selected cluster in the (1,1)-blocks can be bounded as

 max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
 max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

See LAPACK User's Guide section 4.11 or the following references for more information 🎀

⭐ Note that if the default method for computing the Frobenius-norm-based estimate DIF is not wanted (see DLATDF), then the parameter IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF (IJOB = 2 will be used)). See DTGSYL for more details. Iykyk.

🌸 Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

🦋 References:

okay so the reference receipts for dtgsen first, because citations are lowkey the moment 📚✨:

  • [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
  • [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
  • [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

and the exact coordinate: Definition at line 448 of file dtgsen.f.

now bestie, spot the single-precision twin — STGSEN is here and it's giving “same energy, different precision” 💅:

   subroutine stgsen (integer ijob, logical wantq, logical wantz, logical,
       dimension( * ) select, integer n, real, dimension( lda, * ) a, integer
       lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * )
       alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real,
       dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer
       ldz, integer m, real pl, real pr, real, dimension( * ) dif, real,
       dimension( * ) work, integer lwork, integer, dimension( * ) iwork,
       integer liwork, integer info)
       STGSEN

🌸 STGSEN 🌸

✨ Purpose ✨

okay so STGSEN reorders the generalized real Schur decomposition of a real matrix pair (A, B) — and yes, it does that through an orthonormal equivalence transformation Q**T * (A, B) * Z — so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix A and the upper triangular B. the leading columns of Q and Z form orthonormal bases of the corresponding left and right eigenspaces (deflating subspaces). and listen: (A, B) must be in generalized real Schur canonical form (as returned by SGGES), meaning A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks, and B is upper triangular. no cap. 💜

STGSEN also computes the generalized eigenvalues

w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)

of the reordered matrix pair (A, B).

optionally, STGSEN computes the estimates of reciprocal condition numbers for eigenvalues and eigenspaces. these are Difu[(A11,B11), (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) between the matrix pairs (A11, B11) and (A22,B22) that correspond to the selected cluster and the eigenvalues outside the cluster, resp., and norms of onto left and right eigenspaces w.r.t. the selected cluster in the (1,1)-block.

🎀 Parameters 🎀

bestie, here's every single arg — no skips, no “we'll come back to it,” we're doing this properly 🫶:

  • IJOB — IJOB is INTEGER. specifies whether condition numbers are required for the cluster of eigenvalues (PL and PR) or the deflating subspaces (Difu and Difl):
    • =0: Only reorder w.r.t. SELECT. No extras.
    • =1: Reciprocal of norms of onto left and right eigenspaces w.r.t. the selected cluster (PL and PR).
    • =2: Upper bounds on Difu and Difl. F-norm-based estimate (DIF(1:2)).
    • =3: Estimate of Difu and Difl. 1-norm-based estimate (DIF(1:2)). About 5 times as expensive as IJOB = 2.
    • =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic version to get it all.
    • =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
  • WANTQ — WANTQ is LOGICAL. .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q.
  • WANTZ — WANTZ is LOGICAL. .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z.
  • SELECT — SELECT is LOGICAL array, dimension (N). SELECT specifies the eigenvalues in the selected cluster. to select a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. to select a complex conjugate pair of eigenvalues w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, either SELECT(j) or SELECT(j+1) or both must be set to .TRUE.; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.
  • N — N is INTEGER. the order of the matrices A and B. N >= 0.
  • A — A is REAL array, dimension(LDA,N). on entry, the upper quasi-triangular matrix A, with (A, B) in generalized real Schur canonical form. on exit, A is overwritten by the reordered matrix A.
  • LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1,N).
  • B — B is REAL array, dimension(LDB,N). on entry, the upper triangular matrix B, with (A, B) in generalized real Schur canonical form. on exit, B is overwritten by the reordered matrix B.
  • LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= max(1,N).
  • ALPHAR — ALPHAR is REAL array, dimension (N).
  • ALPHAI — ALPHAI is REAL array, dimension (N).
  • BETA — BETA is REAL array, dimension (N). on exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i and BETA(j),j=1,...,N are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real generalized Schur form of (A,B) were further reduced to triangular form using complex unitary transformations. if ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative.
  • Q — Q is REAL array, dimension (LDQ,N). on entry, if WANTQ = .TRUE., Q is an N-by-N matrix. on exit, Q has been postmultiplied by the left orthogonal transformation matrix which reorder (A, B); The leading M columns of Q form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). if WANTQ = .FALSE., Q is not referenced.
  • LDQ — LDQ is INTEGER. the leading dimension of the array Q. LDQ >= 1; and if WANTQ = .TRUE., LDQ >= N.
  • Z — Z is REAL array, dimension (LDZ,N). on entry, if WANTZ = .TRUE., Z is an N-by-N matrix. on exit, Z has been postmultiplied by the left orthogonal transformation matrix which reorder (A, B); The leading M columns of Z form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). if WANTZ = .FALSE., Z is not referenced.
  • LDZ — LDZ is INTEGER. the leading dimension of the array Z. LDZ >= 1; If WANTZ = .TRUE., LDZ >= N.
  • M — M is INTEGER. the dimension of the specified pair of left and right eigenspaces (deflating subspaces). 0 <= M <= N.
  • PL — PL is REAL.
  • PR — PR is REAL. if IJOB = 1, 4 or 5, PL, PR are lower bounds on the reciprocal of the norm of onto left and right eigenspaces with respect to the selected cluster. 0 < PL, PR <= 1. if M = 0 or M = N, PL = PR = 1. if IJOB = 0, 2 or 3, PL and PR are not referenced.
  • DIF — DIF is REAL array, dimension (2). if IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. if IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on Difu and Difl. if IJOB = 3 or 5, DIF(1:2) are 1-norm-based estimates of Difu and Difl. if M = 0 or N, DIF(1:2) = F-norm([A, B]). if IJOB = 0 or 1, DIF is not referenced.
  • WORK — WORK is REAL array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — LWORK is INTEGER. the dimension of the array WORK. LWORK >= 4*N+16. if IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). if IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)). if LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
  • LIWORK

It's an INTEGER 💜 — the dimension of the array IWORK. So the rules are:

  • LIWORK >= 1.
  • If IJOB = 1, 2 or 4, then LIWORK >= N+6.
  • If IJOB = 3 or 5, then LIWORK >= MAX(2*M*(N-M), N+6).
🫶 Special case, bestie, listen up: if LIWORK = -1, that means a workspace query is happening. The routine only calculates the optimal size of the IWORK array, returns that value as the first entry of the IWORK array, and no error message related to LIWORK gets issued by XERBLA.
  • INFO is an INTEGER 🩺 — the vibe report card:
    • =0: Successful exit. slay.
    • <0: If INFO = -i, the i-th argument had an illegal value.
    • =1: Reordering of (A, B) failed because the transformed matrix pair (A, B) would be too far from generalized Schur form — the problem is very ill-conditioned. 😭 (A, B) may have been partially reordered. If requested, 0 is returned in DIF(*), PL and PR.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🧮 Further Details

okay so STGSEN first collects the selected eigenvalues by computing orthogonal U and W that move them to the top left corner of (A, B). In other words, the selected eigenvalues are the eigenvalues of (A11, B11) in:

              U**T*(A, B)*W = (A11 A12) (B11 B12) n1
                              ( 0  A22),( 0  B22) n2
                                n1  n2    n1  n2

where N = n1+n2 and U**T means the transpose of U. The first n1 columns of U and W span the specified pair of left and right eigenspaces (deflating subspaces) of (A, B).

If (A, B) has been obtained from the generalized real Schur decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the reordered generalized real Schur form of (C, D) is given by

           (C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,

and the first n1 columns of Q*U and Z*W span the corresponding deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).

⚠️ Note that if the selected eigenvalue is sufficiently ill-conditioned, then its value may differ significantly from its value before reordering. 🫠

The reciprocal condition numbers of the left and right eigenspaces spanned by the first n1 columns of U and W (or Q*U and Z*W) may be returned in DIF(1:2), corresponding to Difu and Difl, resp.

The Difu and Difl are defined as:

       Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
  and
       Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

where sigma-min(Zu) is the smallest singular value of the (2*n1*n2)-by-(2*n1*n2) matrix

       Zu = [ kron(In2, A11)  -kron(A22**T, In1) ]
            [ kron(In2, B11)  -kron(B22**T, In1) ].

Here, Inx is the identity matrix of size nx and A22**T is the transpose of A22. kron(X, Y) is the Kronecker product between the matrices X and Y.

When DIF(2) is small, small changes in (A, B) can cause large changes in the deflating subspace. An approximate (asymptotic) bound on the maximum angular error in the computed deflating subspaces is

       EPS * norm((A, B)) / DIF(2),

where EPS is the machine precision.

The reciprocal norm of the projectors on the left and right eigenspaces associated with (A11, B11) may be returned in PL and PR. They are computed as follows. First we compute L and R so that P*(A, B)*Q is block diagonal, where

       P = ( I -L ) n1           Q = ( I R ) n1
           ( 0  I ) n2    and        ( 0 I ) n2
             n1 n2                    n1 n2

and (L, R) is the solution to the generalized Sylvester equation

       A11*R - L*A22 = -A12
       B11*R - L*B22 = -B12

Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is

       EPS * norm((A, B)) / PL.

There are also global error bounds which valid for perturbations up to a certain restriction: A lower bound (x) on the smallest F-norm(E,F) for which an eigenvalue of (A11, B11) may move and coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), (i.e. (A + E, B + F), is

   x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

An approximate bound on x can be computed from DIF(1:2), PL and PR.

If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed (L', R') and unperturbed (L, R) left and right deflating subspaces associated with the selected cluster in the (1,1)-blocks can be bounded as

   max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
   max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

See LAPACK User's Guide section 4.11 or the following references for more information.

🌸 Heads up: if the default method for computing the Frobenius-norm-based estimate DIF is not wanted (see SLATDF), then the parameter IDIFJB (see below) should be changed from 3 to 4 (routine SLATDF (IJOB = 2 will be used)). See STGSYL for more details.

🌸 Contributors

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

📚 References

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software,
      Report UMINF - 94.04, Department of Computing Science, Umea
      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
      Note 87. To appear in Numerical Algorithms, 1996.

  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,
      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
      1996.

Definition at line 448 of file stgsen.f.

okay and now the complex twin enters the chat 💅

   subroutine ztgsen (integer ijob, logical wantq, logical wantz, logical,
       dimension( * ) select, integer n, complex*16, dimension( lda, * ) a,
       integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16,
       dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16,
       dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z,
       integer ldz, integer m, double precision pl, double precision pr, double
       precision, dimension( * ) dif, complex*16, dimension( * ) work, integer
       lwork, integer, dimension( * ) iwork, integer liwork, integer info)
       ZTGSEN

💅 ZTGSEN

Purpose

okay so ZTGSEN reorders the generalized Schur decomposition of a complex matrix pair (A, B) (in terms of an unitary equivalence transformation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the pair (A,B). The leading columns of Q and Z form unitary bases of the corresponding left and right eigenspaces (deflating subspaces). (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular. 👑

ZTGSEN also computes the generalized eigenvalues

          w(j)= ALPHA(j) / BETA(j)

of the reordered matrix pair (A, B).

Optionally, the routine computes estimates of reciprocal condition numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) between the matrix pairs (A11, B11) and (A22,B22) that correspond to the selected cluster and the eigenvalues outside the cluster, resp., and norms of onto left and right eigenspaces w.r.t. the selected cluster in the (1,1)-block.

Parameters

  • IJOB is an INTEGER 🎛️ — specifies whether condition numbers are required for the cluster of eigenvalues (PL and PR) or the deflating subspaces (Difu and Difl):
    • =0: Only reorder w.r.t. SELECT. No extras.
    • =1: Reciprocal of norms of onto left and right eigenspaces w.r.t. the selected cluster (PL and PR).
    • =2: Upper bounds on Difu and Difl. F-norm-based estimate (DIF(1:2)).
    • =3: Estimate of Difu and Difl. 1-norm-based estimate (DIF(1:2)). About 5 times as expensive as IJOB = 2.
    • =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic version to get it all. 🛍️
    • =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
  • WANTQ is a LOGICAL 🦋
    • .TRUE. : update the left transformation matrix Q;
    • .FALSE.: do not update Q.
  • WANTZ is a LOGICAL 🌈
    • .TRUE. : update the right transformation matrix Z;
    • .FALSE.: do not update Z.
  • SELECT is a LOGICAL array, dimension (N) 🎯 — SELECT specifies the eigenvalues in the selected cluster. To select an eigenvalue w(j), SELECT(j) must be set to .TRUE..
  • N —
  • N — N is INTEGER. The order of the matrices A and B. N >= 0. 🌸
  • A — A is COMPLEX*16 array, dimension(LDA,N). On entry, the upper triangular matrix A, in generalized Schur canonical form. On exit, A is overwritten by the reordered matrix A. ✨
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💅
  • B — B is COMPLEX*16 array, dimension(LDB,N). On entry, the upper triangular matrix B, in generalized Schur canonical form. On exit, B is overwritten by the reordered matrix B. 🦋
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N). 💖
  • ALPHA — ALPHA is COMPLEX*16 array, dimension (N). 🎀
  • BETA — BETA is COMPLEX*16 array, dimension (N). The diagonal elements of A and B, respectively, when the pair (A,B) has been reduced to generalized Schur form. ALPHA(i)/BETA(i) i=1,...,N are the generalized eigenvalues. 👑
  • Q — Q is COMPLEX*16 array, dimension (LDQ,N). On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. On exit, Q has been postmultiplied by the left unitary transformation matrix which reorder (A, B); The leading M columns of Q form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). If WANTQ = .FALSE., Q is not referenced. 🌈
  • LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N. 💜
  • Z — Z is COMPLEX*16 array, dimension (LDZ,N). On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. On exit, Z has been postmultiplied by the left unitary transformation matrix which reorder (A, B); The leading M columns of Z form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). If WANTZ = .FALSE., Z is not referenced. ⭐
  • LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N. 💫
  • M — M is INTEGER. The dimension of the specified pair of left and right eigenspaces, (deflating subspaces) 0 <= M <= N. 🫶
  • PL — PL is DOUBLE PRECISION. 🍓
  • PR — PR is DOUBLE PRECISION. If IJOB = 1, 4 or 5, PL, PR are lower bounds on the reciprocal of the norm of onto left and right eigenspace with respect to the selected cluster. 0 < PL, PR <= 1. If M = 0 or M = N, PL = PR = 1. If IJOB = 0, 2 or 3 PL, PR are not referenced. 🧚
  • DIF — DIF is DOUBLE PRECISION array, dimension (2). If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based estimates of Difu and Difl, computed using reversed communication with ZLACN2. If M = 0 or N, DIF(1:2) = F-norm([A, B]). If IJOB = 0 or 1, DIF is not referenced. 🔥
  • WORK — WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🌟
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= 1 If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M) If IJOB = 3 or 5, LWORK >= 4*M*(N-M) If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 💀
  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 🙈
  • LIWORK — LIWORK is INTEGER. The dimension of the array IWORK. LIWORK >= 1. If IJOB = 1, 2 or 4, LIWORK >= N+2; If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M)); If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. 🎀
  • INFO — INFO is INTEGER
    • =0: Successful exit. 🎉
    • <0: If INFO = -i, the i-th argument had an illegal value. 😭
    • =1: Reordering of (A, B) failed because the transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is very ill-conditioned. (A, B) may have been partially reordered. If requested, 0 is returned in DIF(*), PL and PR. 💀

👑 Author

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

🔎 Further Details:

Okay so here's the lore, bestie: ZTGSEN first collects the selected eigenvalues by computing unitary U and W that move them to the top left corner of (A, B). In other words, the selected eigenvalues are the eigenvalues of (A11, B11) in:

U**H*(A, B)*W = (A11 A12) (B11 B12) n1
                ( 0  A22),( 0  B22) n2
                  n1  n2    n1  n2

where N = n1+n2 and U**H means the conjugate transpose of U. The first n1 columns of U and W span the specified pair of left and right eigenspaces (deflating subspaces) of (A, B). ✨

If (A, B) has been obtained from the generalized real Schur decomposition of a matrix pair (C, D) = Q*(A, B)*Z**H, then the reordered generalized Schur form of (C, D) is given by

(C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H,

and the first n1 columns of Q*U and Z*W span the corresponding deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). 💖

⚠️ Note that if the selected eigenvalue is sufficiently ill-conditioned, then its value may differ significantly from its value before reordering. That's not a vibe, that's a warning, bestie.

The reciprocal condition numbers of the left and right eigenspaces spanned by the first n1 columns of U and W (or Q*U and Z*W) may be returned in DIF(1:2), corresponding to Difu and Difl, resp. 🌸

The Difu and Difl are defined as:

Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )

and

Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

where sigma-min(Zu) is the smallest singular value of the (2*n1*n2)-by-(2*n1*n2) matrix

Zu = [ kron(In2, A11)  -kron(A22**H, In1) ]
     [ kron(In2, B11)  -kron(B22**H, In1) ].

Here, Inx is the identity matrix of size nx and A22**H is the conjugate transpose of A22. kron(X, Y) is the Kronecker product between the matrices X and Y. 🦋

When DIF(2) is small, small changes in (A, B) can cause large changes in the deflating subspace. An approximate (asymptotic) bound on the maximum angular error in the computed deflating subspaces is

EPS * norm((A, B)) / DIF(2),

where EPS is the machine precision. 💅

The reciprocal norm of the projectors on the left and right eigenspaces associated with (A11, B11) may be returned in PL and PR. They are computed as follows. First we compute L and R so that P*(A, B)*Q is block diagonal, where

P = ( I -L ) n1           Q = ( I R ) n1
    ( 0  I ) n2    and        ( 0 I ) n2
      n1 n2                    n1 n2

and (L, R) is the solution to the generalized Sylvester equation

A11*R - L*A22 = -A12
B11*R - L*B22 = -B12

Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is

EPS * norm((A, B)) / PL.

There are also global error bounds which valid for perturbations up to a certain restriction: A lower bound (x) on the smallest F-norm(E,F) for which an eigenvalue of (A11, B11) may move and coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), (i.e. (A + E, B + F), is

x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

An approximate bound on x can be computed from DIF(1:2), PL and PR. 🔥

If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed (L', R') and unperturbed (L, R) left and right deflating subspaces associated with the selected cluster in the (1,1)-blocks can be bounded as

max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

See LAPACK User's Guide section 4.11 or the following references for more information. 📖

Note that if the default method for computing the Frobenius-norm-based estimate DIF is not wanted (see ZLATDF), then the parameter IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF (IJOB = 2 will be used)). See ZTGSYL for more details. 🫶

💖 Contributors 💖

Okay so the literal legends credited here are:

  • Bo Kagstrom
  • Peter Poromaa

Both are from the Department of Computing Science, Umea University, S-901 87 Umea, Sweden. Iconic behavior, no cap. 🎀

📚 References 📚

These are the receipts, bestie — all the citation details, kept exactly as printed:

  • [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
  • [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
  • [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

And the source pin is right here: Definition at line 430 of file ztgsen.f. 🫶

✨ Author ✨

Generated automatically by Doxygen for LAPACK from the source code. That’s the whole author situation — Doxygen did its little autogenerated thing, periodt. 💅

And that’s the whole page, bestie! 🌈 The footer credits LAPACK, Version 3.12.0, for tgsen(3) — that’s the source + version, and there’s no separate date printed on that footer line, so I’m not gonna invent one. 💀🎀

📦 where this page comes from