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LinearAlgebra

 EigenConditionNumbers
 compute condition numbers of eigenvalues or eigenvectors of a numeric eigenvalue conditioning problem

 Calling Sequence EigenConditionNumbers(A, C, bal, out, options, outopts)

Parameters

 A - Matrix; numeric Matrix whose eigenvalues' or eigenvectors' condition numbers are required C - (optional) Matrix; numeric Matrix for the generalized numeric eigenvalue conditioning problem bal - (optional) identical('balance')=symbol; specifies whether to allow scaling or permutation, where the symbol is any of 'both', 'permute', 'scale', or 'none' out - (optional) identical('output')={name,list}; specifies whether to return condition numbers of the eigenvalues and eigenvectors or the eigenvalues and eigenvectors options - (optional); constructor options for the result object(s) outopts - (optional) equation(s) of the form outputoptions[o]=list where o is one of 'conditionvalues', 'conditionvectors', 'values', 'vectors', 'alpha', or 'beta'; constructor options for the specified result object

Description

 • The EigenConditionNumbers(A) command solves the simple eigenvalue conditioning problem by returning an expression sequence.  This expression sequence consists of two components.  The first is a Vector V1 of the floating-point condition numbers of the eigenvalues of A, and the second is a Vector V2 of the floating-point condition numbers of the eigenvectors of A.  Optionally, the floating-point eigenvalues or the floating-point eigenvectors are returned.
 In the simple eigenvalue problem, if A is real-symmetric or complex hermitian, the returned eigenvalues are real.
 • The EigenConditionNumbers(..) function solves the simple eigenvalue conditioning problem $A·x=\mathrm{\lambda }·x$ and the generalized eigenvalue conditioning problem $A·x=\mathrm{\lambda }·C·x$.  The eigenvalues V[i] and the nonzero eigenvectors E[1 .. -1, i] are such that for the simple eigenvalue problem $M\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{E}_{1..-1,i}={V}_{i}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{E}_{1..-1,i}$, and for the generalized problem $M\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{E}_{1..-1,i}={V}_{i}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}C\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{E}_{1..-1,i}$ or ${\mathrm{\beta }}_{i}·M·{E}_{1..-1,i}={\mathrm{\alpha }}_{i}·C·{E}_{1..-1,i}$.
 • The EigenConditionNumbers(A, C) command solves the generalized numeric eigenvalue conditioning problem.
 In the generalized eigenvalue conditioning problem, if A and C are real-symmetric or complex hermitian and C also has the positive_definite attribute, the eigenvalues are real.
 In the generalized eigenvalue conditioning problem, ${V}_{i}=\frac{{\mathrm{\beta }}_{i}}{{\mathrm{\alpha }}_{i}}$ for each finite eigenvalue ${V}_{i}$.
 • The out option specifies what the command should return.  It can be one of vectors, values, conditionvalues, conditionvectors or a list of these.
 • The options option provides additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix or Vector constructor that builds the result. These options may also be provided in the form outputoptions[o]=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions[o] option, the latter takes precedence (regardless of the order).
 • For extensive details of the relationships between condition numbers and error bounds, see
 E. Anderson et al. LAPACK users' guide. 3d ed. SIAM, 1999.

Examples

 > $\mathrm{with}\left(\mathrm{LinearAlgebra}\right):$
 > $A≔\mathrm{Matrix}\left(3,3,\left[\left[-1,-3,-6\right],\left[3,5,6\right],\left[-3,-3,-4\right]\right],'\mathrm{datatype}'='\mathrm{float}'\right)$
 ${A}{≔}\left[\begin{array}{ccc}{-}{1.}& {-}{3.}& {-}{6.}\\ {3.}& {5.}& {6.}\\ {-}{3.}& {-}{3.}& {-}{4.}\end{array}\right]$ (1)
 > $\mathrm{EigenConditionNumbers}\left(A\right)$
 $\left[\begin{array}{c}{0.471404520791032}\\ {0.436298377504987}\\ {0.745873405008495}\end{array}\right]{,}\left[\begin{array}{c}{6.}\\ {1.55431223447522}{}{{10}}^{{-15}}\\ {1.33226762955019}{}{{10}}^{{-15}}\end{array}\right]$ (2)
 > $v,e≔\mathrm{EigenConditionNumbers}\left(A,'\mathrm{output}'=\left['\mathrm{values}','\mathrm{vectors}'\right]\right)$
 ${v}{,}{e}{≔}\left[\begin{array}{c}{-}{4.00000000000000}{+}{0.}{}{I}\\ {2.00000000000000}{+}{0.}{}{I}\\ {2.}{+}{0.}{}{I}\end{array}\right]{,}\left[\begin{array}{ccc}{-}{0.577350269189626}{+}{0.}{}{I}& {-}{0.0866100166393946}{+}{0.}{}{I}& {-}{0.808190323654766}{+}{0.}{}{I}\\ {0.577350269189626}{+}{0.}{}{I}& {0.907714476800681}{+}{0.}{}{I}& {-}{0.254240841587667}{+}{0.}{}{I}\\ {-}{0.577350269189626}{+}{0.}{}{I}& {-}{0.410552230080643}{+}{0.}{}{I}& {0.531215582621217}{+}{0.}{}{I}\end{array}\right]$ (3)
 > $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{e}_{1..-1,2}={v}_{2}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{e}_{1..-1,2}$
 $\left[\begin{array}{c}{-}{0.173220033278790}{+}{0.}{}{I}\\ {1.81542895360136}{+}{0.}{}{I}\\ {-}{0.821104460161287}{+}{0.}{}{I}\end{array}\right]{=}\left[\begin{array}{c}{-}{0.173220033278789}{+}{0.}{}{I}\\ {1.81542895360136}{+}{0.}{}{I}\\ {-}{0.821104460161286}{+}{0.}{}{I}\end{array}\right]$ (4)
 > $M≔\mathrm{Matrix}\left(\left[\left[6.,8.,5.\right],\left[8.,8.,9.\right],\left[5.,9.,6.\right]\right],'\mathrm{datatype}'='\mathrm{float}'\right):$
 > $N≔\mathrm{Matrix}\left(\left[\left[6.,3.,3.\right],\left[3.,8.,8.\right],\left[3.,8.,9.\right]\right],'\mathrm{datatype}'='\mathrm{float}'\right):$
 > $\mathrm{EigenConditionNumbers}\left(M,N,\mathrm{output}=\left['\mathrm{vectors}','\mathrm{values}','\mathrm{conditionvalues}','\mathrm{conditionvectors}'\right]\right)$
 $\left[\begin{array}{ccc}{-}{0.139566030823037}{+}{0.}{}{I}& {-}{0.940687706820450}{+}{0.}{}{I}& {-}{1.}{+}{0.}{}{I}\\ {1.}{+}{0.}{}{I}& {-}{1.}{+}{0.}{}{I}& {0.0726922335169216}{+}{0.}{}{I}\\ {-}{0.911459942195583}{+}{0.}{}{I}& {0.399204400339096}{+}{0.}{}{I}& {0.953350738117875}{+}{0.}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{-}{4.55651173111532}{+}{0.}{}{I}\\ {1.56423606281299}{+}{0.}{}{I}\\ {0.223044899071558}{+}{0.}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{2.36042287591761}\\ {10.6687589561888}\\ {4.91085988501676}\end{array}\right]{,}\left[\begin{array}{c}{3.45594343706379}\\ {2.29879484036260}\\ {3.02673623742847}\end{array}\right]$ (5)
 > $\mathrm{EigenConditionNumbers}\left(M,N,\mathrm{output}='\mathrm{values}'\right)$
 $\left[\begin{array}{c}{-}{4.55651173111532}{+}{0.}{}{I}\\ {1.56423606281299}{+}{0.}{}{I}\\ {0.223044899071558}{+}{0.}{}{I}\end{array}\right]$ (6)
 > $\mathrm{EigenConditionNumbers}\left(M,N,\mathrm{output}=\left['\mathrm{α}','\mathrm{β}'\right]\right)$
 $\left[\begin{array}{c}{-}{2.99144878241616}{+}{0.}{}{I}\\ {14.9822766669746}{+}{0.}{}{I}\\ {1.38335073553634}{+}{0.}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0.656521690043783}\\ {9.57801512390126}\\ {6.20211778567742}\end{array}\right]$ (7)