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---
:name: ctrsna
:md5sum: e91cc890d0ea7dbba10a9adc422212aa
:category: :subroutine
:arguments:
- job:
:type: char
:intent: input
- howmny:
:type: char
:intent: input
- select:
:type: logical
:intent: input
:dims:
- n
- n:
:type: integer
:intent: input
- t:
:type: complex
:intent: input
:dims:
- ldt
- n
- ldt:
:type: integer
:intent: input
- vl:
:type: complex
:intent: input
:dims:
- ldvl
- m
- ldvl:
:type: integer
:intent: input
- vr:
:type: complex
:intent: input
:dims:
- ldvr
- m
- ldvr:
:type: integer
:intent: input
- s:
:type: real
:intent: output
:dims:
- mm
- sep:
:type: real
:intent: output
:dims:
- mm
- mm:
:type: integer
:intent: input
- m:
:type: integer
:intent: output
- work:
:type: complex
:intent: workspace
:dims:
- "lsame_(&job,\"E\") ? 0 : ldwork"
- "lsame_(&job,\"E\") ? 0 : n+6"
- ldwork:
:type: integer
:intent: input
- rwork:
:type: real
:intent: workspace
:dims:
- "lsame_(&job,\"E\") ? 0 : n"
- info:
:type: integer
:intent: output
:substitutions:
ldwork: "((lsame_(&job,\"V\")) || (lsame_(&job,\"B\"))) ? n : 1"
mm: m
:fortran_help: " SUBROUTINE CTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* CTRSNA estimates reciprocal condition numbers for specified\n\
* eigenvalues and/or right eigenvectors of a complex upper triangular\n\
* matrix T (or of any matrix Q*T*Q**H with Q unitary).\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* JOB (input) CHARACTER*1\n\
* Specifies whether condition numbers are required for\n\
* eigenvalues (S) or eigenvectors (SEP):\n\
* = 'E': for eigenvalues only (S);\n\
* = 'V': for eigenvectors only (SEP);\n\
* = 'B': for both eigenvalues and eigenvectors (S and SEP).\n\
*\n\
* HOWMNY (input) CHARACTER*1\n\
* = 'A': compute condition numbers for all eigenpairs;\n\
* = 'S': compute condition numbers for selected eigenpairs\n\
* specified by the array SELECT.\n\
*\n\
* SELECT (input) LOGICAL array, dimension (N)\n\
* If HOWMNY = 'S', SELECT specifies the eigenpairs for which\n\
* condition numbers are required. To select condition numbers\n\
* for the j-th eigenpair, SELECT(j) must be set to .TRUE..\n\
* If HOWMNY = 'A', SELECT is not referenced.\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix T. N >= 0.\n\
*\n\
* T (input) COMPLEX array, dimension (LDT,N)\n\
* The upper triangular matrix T.\n\
*\n\
* LDT (input) INTEGER\n\
* The leading dimension of the array T. LDT >= max(1,N).\n\
*\n\
* VL (input) COMPLEX array, dimension (LDVL,M)\n\
* If JOB = 'E' or 'B', VL must contain left eigenvectors of T\n\
* (or of any Q*T*Q**H with Q unitary), corresponding to the\n\
* eigenpairs specified by HOWMNY and SELECT. The eigenvectors\n\
* must be stored in consecutive columns of VL, as returned by\n\
* CHSEIN or CTREVC.\n\
* If JOB = 'V', VL is not referenced.\n\
*\n\
* LDVL (input) INTEGER\n\
* The leading dimension of the array VL.\n\
* LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.\n\
*\n\
* VR (input) COMPLEX array, dimension (LDVR,M)\n\
* If JOB = 'E' or 'B', VR must contain right eigenvectors of T\n\
* (or of any Q*T*Q**H with Q unitary), corresponding to the\n\
* eigenpairs specified by HOWMNY and SELECT. The eigenvectors\n\
* must be stored in consecutive columns of VR, as returned by\n\
* CHSEIN or CTREVC.\n\
* If JOB = 'V', VR is not referenced.\n\
*\n\
* LDVR (input) INTEGER\n\
* The leading dimension of the array VR.\n\
* LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.\n\
*\n\
* S (output) REAL array, dimension (MM)\n\
* If JOB = 'E' or 'B', the reciprocal condition numbers of the\n\
* selected eigenvalues, stored in consecutive elements of the\n\
* array. Thus S(j), SEP(j), and the j-th columns of VL and VR\n\
* all correspond to the same eigenpair (but not in general the\n\
* j-th eigenpair, unless all eigenpairs are selected).\n\
* If JOB = 'V', S is not referenced.\n\
*\n\
* SEP (output) REAL array, dimension (MM)\n\
* If JOB = 'V' or 'B', the estimated reciprocal condition\n\
* numbers of the selected eigenvectors, stored in consecutive\n\
* elements of the array.\n\
* If JOB = 'E', SEP is not referenced.\n\
*\n\
* MM (input) INTEGER\n\
* The number of elements in the arrays S (if JOB = 'E' or 'B')\n\
* and/or SEP (if JOB = 'V' or 'B'). MM >= M.\n\
*\n\
* M (output) INTEGER\n\
* The number of elements of the arrays S and/or SEP actually\n\
* used to store the estimated condition numbers.\n\
* If HOWMNY = 'A', M is set to N.\n\
*\n\
* WORK (workspace) COMPLEX array, dimension (LDWORK,N+6)\n\
* If JOB = 'E', WORK is not referenced.\n\
*\n\
* LDWORK (input) INTEGER\n\
* The leading dimension of the array WORK.\n\
* LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.\n\
*\n\
* RWORK (workspace) REAL array, dimension (N)\n\
* If JOB = 'E', RWORK is not referenced.\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\
* The reciprocal of the condition number of an eigenvalue lambda is\n\
* defined as\n\
*\n\
* S(lambda) = |v'*u| / (norm(u)*norm(v))\n\
*\n\
* where u and v are the right and left eigenvectors of T corresponding\n\
* to lambda; v' denotes the conjugate transpose of v, and norm(u)\n\
* denotes the Euclidean norm. These reciprocal condition numbers always\n\
* lie between zero (very badly conditioned) and one (very well\n\
* conditioned). If n = 1, S(lambda) is defined to be 1.\n\
*\n\
* An approximate error bound for a computed eigenvalue W(i) is given by\n\
*\n\
* EPS * norm(T) / S(i)\n\
*\n\
* where EPS is the machine precision.\n\
*\n\
* The reciprocal of the condition number of the right eigenvector u\n\
* corresponding to lambda is defined as follows. Suppose\n\
*\n\
* T = ( lambda c )\n\
* ( 0 T22 )\n\
*\n\
* Then the reciprocal condition number is\n\
*\n\
* SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )\n\
*\n\
* where sigma-min denotes the smallest singular value. We approximate\n\
* the smallest singular value by the reciprocal of an estimate of the\n\
* one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is\n\
* defined to be abs(T(1,1)).\n\
*\n\
* An approximate error bound for a computed right eigenvector VR(i)\n\
* is given by\n\
*\n\
* EPS * norm(T) / SEP(i)\n\
*\n\
* =====================================================================\n\
*\n"
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