Re: Quadratic Form Contours

*To*: mathgroup at smc.vnet.net*Subject*: [mg57756] Re: Quadratic Form Contours*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Tue, 7 Jun 2005 05:59:45 -0400 (EDT)*Organization*: The University of Western Australia*References*: <d7pa71$sqt$1@smc.vnet.net> <d83dqf$o1e$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <d83dqf$o1e$1 at smc.vnet.net>, dh <dh at metrohm.ch> wrote: > To use "Ellipsoid" we need the semi-length and directions of the axes. > The direction we obtain from: Eigenvectors[A] > the half-length from: Sqrt[ b/Eigenvalues[A]] > Therefore, an ellipsoid would be specified by: > > Ellipsoid[{x,y}, Sqrt[ b/Eigenvalues[A]], Eigenvectors[A] ] Not quite. If A is an exact matrix then Eigenvectors[A] will not be normalised. Also, you need to transpose the Eigenvectors. The following code does the job: << Statistics`MultiDescriptiveStatistics` MatrixToEllipsoid[m_, b_, origin_:{0,0}] := Module[{a = Transpose[Eigenvectors[m]], r = Sqrt[b/Eigenvalues[m]]}, Ellipsoid[origin, r, a/(Norm /@ a)]] For example, with A = {{5, -2}, {-2, 8}}; b = 36; the ellipsoid equation is obtained via eqn = ExpandAll[{x, y} . A . {x, y} == b] We can plot the contours of this as follows: ContourPlot[First[eqn], {x, -4, 4}, {y, -4, 4}, Contours -> {b}, ContourShading -> False]; Alternatively, we can visualize the ellipse using Show[Graphics[MatrixToEllipsoid[A,b]], AspectRatio -> Automatic, Axes -> True] The two pictures agree. Cheers, Paul > The erason for this is: > If you turn the coordinate system in direction of the Eigenvectors, the > quadratic form looks like: > > x1^2/ew1 + x2^2/ew2 == b > > where ew means eigenvalue. > sincerely, Daniel > > Joerg Schaber wrote: > > Hi, > > > > does anybody know a simple way to calculate 2-D ellipsoids in x={x1,x2} > > of a quadatric form solving xAx=b for a graphical output, i.e. contour > > lines of the quadrtic form expressed as the Graphics primitive Ellipsoid? > > I suppose that the option ParameterConfidenceRegion of NonlinearRegress > > does something like that, but how? > > > > best, > > > > joerg > > -- Paul Abbott Phone: +61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul http://InternationalMathematicaSymposium.org/IMS2005/