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---
:name: ctrsen
:md5sum: d2b5291e456ad12fb0adea66a05d53de
:category: :subroutine
:arguments:
- job:
:type: char
:intent: input
- compq:
:type: char
:intent: input
- select:
:type: logical
:intent: input
:dims:
- n
- n:
:type: integer
:intent: input
- t:
:type: complex
:intent: input/output
:dims:
- ldt
- n
- ldt:
:type: integer
:intent: input
- q:
:type: complex
:intent: input/output
:dims:
- ldq
- n
- ldq:
:type: integer
:intent: input
- w:
:type: complex
:intent: output
:dims:
- n
- m:
:type: integer
:intent: output
- s:
:type: real
:intent: output
- sep:
:type: real
:intent: output
- work:
:type: complex
:intent: output
:dims:
- MAX(1,lwork)
- lwork:
:type: integer
:intent: input
:option: true
:default: "lsame_(&job,\"N\") ? n : lsame_(&job,\"E\") ? m*(n-m) : (lsame_(&job,\"V\")||lsame_(&job,\"B\")) ? 2*m*(n-m) : 0"
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE CTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* CTRSEN reorders the Schur factorization of a complex matrix\n\
* A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in\n\
* the leading positions on the diagonal of the upper triangular matrix\n\
* T, and the leading columns of Q form an orthonormal basis of the\n\
* corresponding right invariant subspace.\n\
*\n\
* Optionally the routine computes the reciprocal condition numbers of\n\
* the cluster of eigenvalues and/or the invariant subspace.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* JOB (input) CHARACTER*1\n\
* Specifies whether condition numbers are required for the\n\
* cluster of eigenvalues (S) or the invariant subspace (SEP):\n\
* = 'N': none;\n\
* = 'E': for eigenvalues only (S);\n\
* = 'V': for invariant subspace only (SEP);\n\
* = 'B': for both eigenvalues and invariant subspace (S and\n\
* SEP).\n\
*\n\
* COMPQ (input) CHARACTER*1\n\
* = 'V': update the matrix Q of Schur vectors;\n\
* = 'N': do not update Q.\n\
*\n\
* SELECT (input) LOGICAL array, dimension (N)\n\
* SELECT specifies the eigenvalues in the selected cluster. To\n\
* select the j-th eigenvalue, SELECT(j) must be set to .TRUE..\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix T. N >= 0.\n\
*\n\
* T (input/output) COMPLEX array, dimension (LDT,N)\n\
* On entry, the upper triangular matrix T.\n\
* On exit, T is overwritten by the reordered matrix T, with the\n\
* selected eigenvalues as the leading diagonal elements.\n\
*\n\
* LDT (input) INTEGER\n\
* The leading dimension of the array T. LDT >= max(1,N).\n\
*\n\
* Q (input/output) COMPLEX array, dimension (LDQ,N)\n\
* On entry, if COMPQ = 'V', the matrix Q of Schur vectors.\n\
* On exit, if COMPQ = 'V', Q has been postmultiplied by the\n\
* unitary transformation matrix which reorders T; the leading M\n\
* columns of Q form an orthonormal basis for the specified\n\
* invariant subspace.\n\
* If COMPQ = 'N', Q is not referenced.\n\
*\n\
* LDQ (input) INTEGER\n\
* The leading dimension of the array Q.\n\
* LDQ >= 1; and if COMPQ = 'V', LDQ >= N.\n\
*\n\
* W (output) COMPLEX array, dimension (N)\n\
* The reordered eigenvalues of T, in the same order as they\n\
* appear on the diagonal of T.\n\
*\n\
* M (output) INTEGER\n\
* The dimension of the specified invariant subspace.\n\
* 0 <= M <= N.\n\
*\n\
* S (output) REAL\n\
* If JOB = 'E' or 'B', S is a lower bound on the reciprocal\n\
* condition number for the selected cluster of eigenvalues.\n\
* S cannot underestimate the true reciprocal condition number\n\
* by more than a factor of sqrt(N). If M = 0 or N, S = 1.\n\
* If JOB = 'N' or 'V', S is not referenced.\n\
*\n\
* SEP (output) REAL\n\
* If JOB = 'V' or 'B', SEP is the estimated reciprocal\n\
* condition number of the specified invariant subspace. If\n\
* M = 0 or N, SEP = norm(T).\n\
* If JOB = 'N' or 'E', SEP is not referenced.\n\
*\n\
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))\n\
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.\n\
*\n\
* LWORK (input) INTEGER\n\
* The dimension of the array WORK.\n\
* If JOB = 'N', LWORK >= 1;\n\
* if JOB = 'E', LWORK = max(1,M*(N-M));\n\
* if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).\n\
*\n\
* If LWORK = -1, then a workspace query is assumed; the routine\n\
* only calculates the optimal size of the WORK array, returns\n\
* this value as the first entry of the WORK array, and no error\n\
* message related to LWORK is issued by XERBLA.\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* < 0: if INFO = -i, the i-th argument had an illegal value\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* CTRSEN first collects the selected eigenvalues by computing a unitary\n\
* transformation Z to move them to the top left corner of T. In other\n\
* words, the selected eigenvalues are the eigenvalues of T11 in:\n\
*\n\
* Z'*T*Z = ( T11 T12 ) n1\n\
* ( 0 T22 ) n2\n\
* n1 n2\n\
*\n\
* where N = n1+n2 and Z' means the conjugate transpose of Z. The first\n\
* n1 columns of Z span the specified invariant subspace of T.\n\
*\n\
* If T has been obtained from the Schur factorization of a matrix\n\
* A = Q*T*Q', then the reordered Schur factorization of A is given by\n\
* A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span the\n\
* corresponding invariant subspace of A.\n\
*\n\
* The reciprocal condition number of the average of the eigenvalues of\n\
* T11 may be returned in S. S lies between 0 (very badly conditioned)\n\
* and 1 (very well conditioned). It is computed as follows. First we\n\
* compute R so that\n\
*\n\
* P = ( I R ) n1\n\
* ( 0 0 ) n2\n\
* n1 n2\n\
*\n\
* is the projector on the invariant subspace associated with T11.\n\
* R is the solution of the Sylvester equation:\n\
*\n\
* T11*R - R*T22 = T12.\n\
*\n\
* Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote\n\
* the two-norm of M. Then S is computed as the lower bound\n\
*\n\
* (1 + F-norm(R)**2)**(-1/2)\n\
*\n\
* on the reciprocal of 2-norm(P), the true reciprocal condition number.\n\
* S cannot underestimate 1 / 2-norm(P) by more than a factor of\n\
* sqrt(N).\n\
*\n\
* An approximate error bound for the computed average of the\n\
* eigenvalues of T11 is\n\
*\n\
* EPS * norm(T) / S\n\
*\n\
* where EPS is the machine precision.\n\
*\n\
* The reciprocal condition number of the right invariant subspace\n\
* spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.\n\
* SEP is defined as the separation of T11 and T22:\n\
*\n\
* sep( T11, T22 ) = sigma-min( C )\n\
*\n\
* where sigma-min(C) is the smallest singular value of the\n\
* n1*n2-by-n1*n2 matrix\n\
*\n\
* C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )\n\
*\n\
* I(m) is an m by m identity matrix, and kprod denotes the Kronecker\n\
* product. We estimate sigma-min(C) by the reciprocal of an estimate of\n\
* the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)\n\
* cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).\n\
*\n\
* When SEP is small, small changes in T can cause large changes in\n\
* the invariant subspace. An approximate bound on the maximum angular\n\
* error in the computed right invariant subspace is\n\
*\n\
* EPS * norm(T) / SEP\n\
*\n\
* =====================================================================\n\
*\n"
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