Re: Translating this algorithm into mathematica code
- To: mathgroup at smc.vnet.net
- Subject: [mg107410] Re: [mg107380] Translating this algorithm into mathematica code
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Fri, 12 Feb 2010 04:40:02 -0500 (EST)
- References: <201002111018.FAA28089@smc.vnet.net>
Kamil wrote: > Dear all, > > I am kind of new to this forum and mathematica but I have found the > post here quite interesting and helpful. I am having problem > translating this algorithm to mathematica code and I hope I can be > helped. The algorith is as given below. Thanks to you all. > > Suppose that for a given matrix A ( n x n ) that is a function of say > x the non-trivial solution is desired. > The following algorithm is used to compute the value of x that makes > the determinant of A zero. > > INPUT > (i) The matrix A and find matrix B, where B = - dA/dx (i.e derivative > of A wrt x) > (ii) The initial estimate of x > (iii) A tolerance for convergence say delta =EF=80 (a small positive number) > is selected. > ITERATION > (a) Choose the initial guess x=EF=80 (0) and start the iteration. > (b) Compute the eigenvalues of the matrix (A - B) based upon the > initial guess x=EF=80 (0) . > (c) Evaluate the minimum eigenvalue of found in step b and assign this > value to epsilon > (d) Compute the new estimate x=EF=80 (1)=x(0) + epsilon > (e) Compute the matrices A(x) and B(x) by substituting x= x(1) > > (f) Repeat steps (b)-(e) for kth iteration until the condition > Abs(epsilon.... in step c) < delta in step (iii) =EF=80 is satisfied. > (g) Stop the iteration. > (h) Store the value of x(k) > (i) Repeat steps (a)-(h) to evaluate another x(k) =EF=80 for different > starting value x=EF=80 (0) . If you are not tied to the particular method you indicate above, you could just use FindRoot on a suitably defined numeric function. numericalDet[mat:{{_?NumericQ..}..} /; MatrixQ[mat] && Length[mat]==Length[mat[[1]]]] := Det[mat] Here is an example. matrix[x_] := { {Cos[x], Sin[x], x}, {x-1, x+3/7, Cos[x]^2}, {-x^2+1/3, Sin[x]*Cos[x], 1}} In[29]:= detroot = FindRoot[numericalDet[matrix[x]], {x,-2}] Out[29]= {x -> -1.47929} Check the resulting determinant: In[30]:= numericalDet[matrix[x]/.detroot] // InputForm Out[30]//InputForm= -1.628393168080853*^-15 Daniel Lichtblau Wolfram Research
- References:
- Translating this algorithm into mathematica code
- From: Kamil <meetkamil@gmail.com>
- Translating this algorithm into mathematica code