trsen(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

, in arch rolling

💖 trsen(3) — Library Functions Manual ✨

🌸 NAME 🌸

trsen — trsen: reorder Schur form

✨ SYNOPSIS ✨

Functions:

subroutine ctrsen (job, compq, select, n, t, ldt, q, ldq, w, m, s, sep,
    work, lwork, info)
    CTRSEN
subroutine dtrsen (job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s,
    sep, work, lwork, iwork, liwork, info)
    DTRSEN
subroutine strsen (job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s,
    sep, work, lwork, iwork, liwork, info)
    STRSEN
subroutine ztrsen (job, compq, select, n, t, ldt, q, ldq, w, m, s, sep,
    work, lwork, info)
    ZTRSEN

🫶 Detailed Description 🫶

okay so this is where the actual function tea lives — the function docs are right below, and they are giving very “reorder the Schur form” main character energy. 💅

🧚 Function Documentation 🧚

💅 CTRSEN 💅

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

CTRSEN — bestie, this is the complex one, and it is lowkey iconic. 🦋

Purpose:

CTRSEN 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.

Parameters:

JOB — JOB is CHARACTER*1. It 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 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 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 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 REAL. 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 REAL. 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 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:

CTRSEN 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: text 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 text 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: text 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 text (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 text 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: text sep( T11, T22 ) = sigma-min( C ) where sigma-min(C) is the smallest singular value of the n1*n2-by-n1*n2 matrix text 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 text EPS * norm(T) / SEP

Definition: at line 262 of file ctrsen.f.

🎀 DTRSEN 🎀

subroutine dtrsen (character job, character compq, logical, dimension( * )
    select, integer n, double precision, dimension( ldt, * ) t, integer ldt,
    double precision, dimension( ldq, * ) q, integer ldq, double precision,
    dimension( * ) wr, double precision, dimension( * ) wi, integer m,
    double precision s, double precision sep, double precision, dimension( *
    ) work, integer lwork, integer, dimension( * ) iwork, integer liwork,
    integer info)

DTRSEN — real matrix bestie, we are reordering the real Schur factorization and it is highkey slaying. 🌈

Purpose:

DTRSEN reorders the real Schur factorization of a real matrix A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-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. T must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign.

Parameters:

JOB — JOB is CHARACTER*1. It 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 a LOGICAL array, dimension (N). It’s the VIP list for the selected cluster — the eigenvalues that actually make the cut. To select a real eigenvalue w(j), set SELECT(j) to .TRUE.. To select a complex conjugate pair of eigenvalues w(j) and w(j+1) that live in a 2-by-2 diagonal block, set either SELECT(j) or SELECT(j+1) or both to .TRUE.. But bestie, a complex conjugate pair must be either both included in the cluster or both excluded. No splitting the duo. 💅

`N`

N is INTEGER. It’s the order of the matrix T. N >= 0.

`T`

T is DOUBLE PRECISION array, dimension (LDT,N).

On entry, it’s the upper quasi-triangular matrix T, in Schur canonical form.

On exit, T is overwritten by the reordered matrix T, again in Schur canonical form, with the selected eigenvalues in the leading diagonal blocks. It’s giving a whole new arrangement, but mathematically legal. ✨

`LDT`

LDT is INTEGER. It’s the leading dimension of the array T. LDT >= max(1,N).

`Q`

Q is DOUBLE PRECISION array, dimension (LDQ,N).

On entry, if COMPQ = 'V', this is the matrix Q of Schur vectors.

On exit, if COMPQ = 'V', Q has been postmultiplied by the orthogonal 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. Not me checking Q when COMPQ = 'N' — it’s literally not invited. 🙈

`LDQ`

LDQ is INTEGER. It’s the leading dimension of the array Q.

LDQ >= 1; and if COMPQ = 'V', LDQ >= N.

`WR`

WR is DOUBLE PRECISION array, dimension (N).

`WI`

WI is DOUBLE PRECISION array, dimension (N).

WR and WI hold the real and imaginary parts, respectively, of the reordered eigenvalues of T. The eigenvalues are stored in the same order as on the diagonal of T, with WR(i) = T(i,i) and, if T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and WI(i+1) = -WI(i).

Note that if a complex eigenvalue is sufficiently ill-conditioned, then its value may differ significantly from its value before reordering. That’s the chaotic part. 😭

`M`

M is INTEGER. It’s 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK)).

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

`LWORK`

LWORK is INTEGER. It’s the dimension of the array WORK.

If JOB = 'N', LWORK >= max(1,N); if JOB = 'E', LWORK >= max(1,M*(N-M)); if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).

`IWORK`

IWORK is INTEGER array, dimension (MAX(1,LIWORK)).

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

`LIWORK`

LIWORK is INTEGER. It’s the dimension of the array IWORK.

If JOB = 'N' or 'E', LIWORK >= 1; if JOB = 'V' or 'B', LIWORK >= max(1,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 T failed because some eigenvalues are too close to separate (the problem is very ill-conditioned); T may have been partially reordered, and WR and WI contain the eigenvalues in the same order as in T; S and SEP (if requested) are set to zero

👑 Author

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

🔍 Further Details

DTRSEN first collects the selected eigenvalues by computing an orthogonal 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**T * T * Z = ( T11 T12 ) n1
               (  0  T22 ) n2
                  n1  n2

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

If T has been obtained from the real Schur factorization of a matrix A = Q*T*Q**T, then the reordered real Schur factorization of A is given by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, 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 311 of file dtrsen.f.

subroutine strsen (character job, character compq, logical, dimension( * )
    select, integer n, real, dimension( ldt, * ) t, integer ldt, real,
    dimension( ldq, * ) q, integer ldq, real, dimension( * ) wr, real,
    dimension( * ) wi, integer m, real s, real sep, real, dimension( * )
    work, integer lwork, integer, dimension( * ) iwork, integer liwork,
    integer info)

🌈 `STRSEN`

Purpose

STRSEN reorders the real Schur factorization of a real matrix A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-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.

T must be in Schur canonical form (as returned by SHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. This is the structure check, bestie — don’t skip it. 🫶

Parameters

JOB is CHARACTER*1. It 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 is CHARACTER*1.

= 'V': update the matrix Q of Schur vectors;
= 'N': do not update Q.

`SELECT`

Like, to select a complex conjugate pair of eigenvalues w(j) and w(j+1) — that's a 2-by-2 diagonal block, bestie, keep up 🫶 — then either SELECT(j) or SELECT(j+1) or literally both have to be set to .TRUE.. And a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded. No splitting them up, they're a package deal, periodt ✨

N — N is INTEGER. The order of the matrix T. N >= 0. (yes, zero is allowed, she's flexible 💅)
T — T is REAL array, dimension (LDT,N). On entry: the upper quasi-triangular matrix T, in Schur canonical form 🌸. On exit: T gets overwritten by the reordered matrix T, again in Schur canonical form, with the selected eigenvalues chilling in the leading diagonal blocks 👑
LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= max(1,N).
Q — Q is REAL array, dimension (LDQ,N). On entry, if COMPQ = 'V', this is the matrix Q of Schur vectors. On exit, if COMPQ = 'V', Q has been postmultiplied by the orthogonal transformation matrix which reorders T, and the leading M columns of Q form an orthonormal basis for the specified invariant subspace 🦋. If COMPQ = 'N', Q is not referenced — she simply doesn't exist to this routine, ghosted fr fr.
LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
WR — WR is REAL array, dimension (N)
WI — WI is REAL array, dimension (N)

Okay so WR and WI together are the real and imaginary parts, respectively, of the reordered eigenvalues of T 💖. The eigenvalues are stored in the same order as on the diagonal of T, with WR(i) = T(i,i) and, if T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and WI(i+1) = -WI(i). Heads up though: if a complex eigenvalue is sufficiently ill-conditioned, its value may differ significantly from its value before reordering 😭 that's just how it is.

M — M is INTEGER. The dimension of the specified invariant subspace. 0 < = M <= N.
S — S is REAL. 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 REAL. 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 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. If JOB = 'N', LWORK >= max(1,N); 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.

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. If JOB = 'N' or 'E', LIWORK >= 1; if JOB = 'V' or 'B', LIWORK >= max(1,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 T failed because some eigenvalues are too close to separate (the problem is very ill-conditioned); T may have been partially reordered, and WR and WI contain the eigenvalues in the same order as in T; S and SEP (if requested) are set to zero 😭

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔥 Further Details:

  STRSEN first collects the selected eigenvalues by computing an
  orthogonal 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**T * T * Z = ( T11 T12 ) n1
                         (  0  T22 ) n2
                            n1  n2

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

So basically: if T came from the real Schur factorization of a matrix A = Q*T*Q**T, then the reordered real Schur factorization of A is given by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span the corresponding invariant subspace of A 🦋. Gorgeous. Iconic. No notes.

                        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 😭 lowkey scary but that's the vibe.

An approximate bound on the maximum angular error in the computed right invariant subspace is

                     EPS * norm(T) / SEP

Definition at line 312 of file strsen.f.

🌈 subroutine ztrsen

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:

!>
!> 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.
!>

🎀 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. 💅 (simpler than the real version, no pair drama, we love a low-maintenance queen)
N — N is INTEGER. The order of the matrix T. N >= 0.
T

T 🦋

T is a COMPLEX*16 array, dimension (LDT,N). On entry, it's the upper triangular matrix T. On exit, T is overwritten by the reordered matrix T, with the selected eigenvalues as the leading diagonal elements.

Overwritten means exactly that, bestie — the old T is not sticking around. 💀

LDT

LDT is INTEGER. It's the leading dimension of the array T, and LDT >= max(1,N). Respect the leading dimension, periodt. 💅

Q

Q is a COMPLEX*16 array, dimension (LDQ,N).

On entry, if COMPQ = 'V', this is 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. Not even a little bit. 🫶

LDQ

LDQ is INTEGER. It's the leading dimension of the array Q.

LDQ >= 1; and if COMPQ = 'V', LDQ >= N. Simple as that. ✨

W

W is a COMPLEX*16 array, dimension (N).

These are the reordered eigenvalues of T, in the same order as they appear on the diagonal of T. Order matters, iykyk. 💜

M

M is INTEGER. It's the dimension of the specified invariant subspace.

0 <= M <= N.

S

S is DOUBLE PRECISION.

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 a COMPLEX*16 array, dimension (MAX(1,LWORK)).

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. Free real estate in the first slot, but make it useful. 🌿

LWORK

LWORK is INTEGER. It's 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)).

Workspace query mode: zero actual work, maximum info. Very main character. 👑

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: 🌈

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.

                      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) )

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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trsen(3)

💖 trsen(3) — Library Functions Manual ✨

🌸 NAME 🌸

✨ SYNOPSIS ✨

🫶 Detailed Description 🫶

🧚 Function Documentation 🧚

💅 CTRSEN 💅

🎀 DTRSEN 🎀

SELECT

N

T

LDT

Q

LDQ

WR

WI

M

S

SEP

WORK

LWORK

IWORK

LIWORK

INFO

👑 Author

🔍 Further Details

🌈 STRSEN

Purpose

Parameters

SELECT

👑 Author

🔥 Further Details:

🌈 subroutine ztrsen

✨ Purpose:

🎀 Parameters

T 🦋

LDT

Q

LDQ

W

M

S

SEP

WORK

LWORK

INFO

Author 🌸

Further Details: 🌈

Author 💖

`SELECT`

`N`

`T`

`LDT`

`Q`

`LDQ`

`WR`

`WI`

`M`

`S`

`SEP`

`WORK`

`LWORK`

`IWORK`

`LIWORK`

`INFO`

🌈 `STRSEN`

`SELECT`