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Translating this algorithm into mathematica code

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.

(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.
(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) .

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