ZTRSEN(3)

🌸 NAME 🌸

Okay bestie, this is the ztrsen.f(3) page from the Library Functions Manual, and the full source path is literally:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ztrsen.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

   subroutine ZTRSEN (job, compq, select, n, t, ldt, q, ldq, w, m, s, sep,
           work, lwork, info)
           ZTRSEN

🦋 Function/Subroutine Documentation 🦋

   subroutine ZTRSEN (character job, character compq, logical, dimension( * )
       select, integer n, complex*16, dimension( ldt, * ) t, integer ldt,
       complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( *
       ) w, integer m, double precision s, double precision sep, complex*16,
       dimension( * ) work, integer lwork, integer info)
       ZTRSEN

🎯 Purpose 🎯

So ZTRSEN reorders the Schur factorization of a complex matrix

A = Q*T*Q**H

so that a selected cluster of eigenvalues appears in the leading positions on the diagonal of the upper triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace.

Optionally, the routine computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace. Math girl summer, fr fr. 💅

🧸 Parameters 🧸

• JOB — JOB is CHARACTER*1. Specifies whether condition numbers are required for the cluster of eigenvalues (S) or the invariant subspace (SEP):
  • = 'N': none;
  • = 'E': for eigenvalues only (S);
  • = 'V': for invariant subspace only (SEP);
  • = 'B': for both eigenvalues and invariant subspace (S and SEP).
• COMPQ — COMPQ is CHARACTER*1:
  • = 'V': update the matrix Q of Schur vectors;
  • = 'N': do not update Q.
• SELECT — SELECT is LOGICAL array, dimension (N). SELECT specifies the eigenvalues in the selected cluster. To select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
• N — N is INTEGER. The order of the matrix T. N >= 0.
• T — T is COMPLEX*16 array, dimension (LDT,N). On entry, the upper triangular matrix T. On exit, T is overwritten by the reordered matrix T, with the selected eigenvalues as the leading diagonal elements.
• LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= max(1,N).
• Q — Q is COMPLEX*16 array, dimension (LDQ,N). On entry, if COMPQ = 'V', the matrix Q of Schur vectors. On exit, if COMPQ = 'V', Q has been postmultiplied by the unitary transformation matrix which reorders T; the leading M columns of Q form an orthonormal basis for the specified invariant subspace. If COMPQ = 'N', Q is not referenced.
• LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
• W — W is COMPLEX*16 array, dimension (N). The reordered eigenvalues of T, in the same order as they appear on the diagonal of T.
• M — M is INTEGER. The dimension of the specified invariant subspace. 0 <= M <= N.
• S — S is DOUBLE PRECISION. If JOB = 'E' or 'B', S is a lower bound on the reciprocal condition number for the selected cluster of eigenvalues. S cannot underestimate the true reciprocal condition number by more than a factor of sqrt(N). If M = 0 or N, S = 1. If JOB = 'N' or 'V', S is not referenced.
• SEP — SEP is DOUBLE PRECISION. If JOB = 'V' or 'B', SEP is the estimated reciprocal condition number of the specified invariant subspace. If M = 0 or N, SEP = norm(T). If JOB = 'N' or 'E', SEP 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. If JOB = 'N', LWORK >= 1; if JOB = 'E', LWORK = max(1,M*(N-M)); if JOB = 'V' or 'B', LWORK >= max(1,2*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.

• INFO — INFO is INTEGER:
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value

👑 Author 👑

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

🌈 Further Details 🌈

Okay, this is the part where it gets extra, and we love that for us. 💖

ZTRSEN first collects the selected eigenvalues by computing a unitary transformation Z to move them to the top left corner of T. In other words, the selected eigenvalues are the eigenvalues of T11 in:

         Z**H * T * Z = ( T11 T12 ) n1
                        (  0  T22 ) n2
                           n1  n2

where N = n1+n2. The first n1 columns of Z span the specified invariant subspace of T.

If T has been obtained from the Schur factorization of a matrix

A = Q*T*Q**H

then the reordered Schur factorization of A is given by

A = (Q*Z)*(Z**H*T*Z)*(Q*Z)**H

and the first n1 columns of Q*Z span the corresponding invariant subspace of A.

The reciprocal condition number of the average of the eigenvalues of T11 may be returned in S. S lies between 0 (very badly conditioned) and 1 (very well conditioned). It is computed as follows. First we compute R so that

                        P = ( I  R ) n1
                            ( 0  0 ) n2
                              n1 n2

is the projector on the invariant subspace associated with T11. R is the solution of the Sylvester equation:

                       T11*R - R*T22 = T12.

Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote the two-norm of M. Then S is computed as the lower bound

                     (1 + F-norm(R)**2)**(-1/2)

on the reciprocal of 2-norm(P), the true reciprocal condition number. S cannot underestimate 1 / 2-norm(P) by more than a factor of sqrt(N).

An approximate error bound for the computed average of the eigenvalues of T11 is

                        EPS * norm(T) / S

where EPS is the machine precision.

The reciprocal condition number of the right invariant subspace spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. SEP is defined as the separation of T11 and T22:

                    sep( T11, T22 ) = sigma-min( C )

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

    C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

I(m) is an m by m identity matrix, and kprod denotes the Kronecker product. We estimate sigma-min(C) by the reciprocal of an estimate of the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

When SEP is small, small changes in T can cause large changes in the invariant subspace. An approximate bound on the maximum angular error in the computed right invariant subspace is

                     EPS * norm(T) / SEP

Definition at line 262 of file ztrsen.f.

🤖 Author 🤖

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That’s the whole page, bestie. Source/version sign-off: LAPACK Version 3.12.0, and the footer path is /home/abui...RC/ztrsen.f(3). Bye 💋✨