MATLAB: How to get the inverse of a symmetric matrix and ensure accurancy

inverse of matrixMATLABsymmetrix matrix

Hi,

I have a flexibility matrix (20*20), F, which is symmetric and positively defined. I need to reverse it to get the stiffness matrix, K=F_inv, and then to obtain eigenvalues using K. Theoretically speaking, K should also be symmetric and positively defined, which also results in positive real eigenvalues. However, in Matlab, the inverse of F does not yield to a symmetric matrix anymore by using K=inv(F) probability due to numerical errors. Moreover, this phenomenon leads to a few complex eigenvalues due to the non-symmetric K.

I tested norm(K*F) very close to 1. So how can I deal with this situation? I also tried to use Cholesky decomposition to get the inverse matrix instead of build-in inv. This approach can definitely provides symmetric inverse matrix of F, however, the accurancy is reduced as well. norm(F_inv*F) using Cholesky is around 1.2, and F_inv*F is close to the identity matrix, but not accurate enough. So how can get the inverse of F to be symmetrix and accurate as well? Anyone has suggestion on this? Thank you very much!

Best Answer

Hi,

you could consider to do the calculation symbolic. Here is a small example:

A = [9 3 -6 12; 3 26 -7 -11; -6 -7 9 7; 12 -11 7 65]
B = inv(A)
C = A*B
% do the same symbolic
D = sym(A)
E = inv(D)
F = D*E

Note that maybe things are getting ugly, when using this approach on a 20x20 matrix.

Best regards

Stephan

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1. Why is computing the matrix exponential of A slow?

In the script, you defined matrix A in term of sym (each sym being applied to a MATLAB double). The only 'non-trivial' (i.e. exact irrational) input value is log(2).

Each double is approximated by a rational number. Thus, the matrix ending up in the symbolic engine consists of exact symbolic data (rational numbers and the occasional log(2)).

Note that the sparsity of matrix A does not help: for exact input data, the matrix has to be diagonalized symbolically. The resulting matrix of eigenvectors is not sparse. It contains many rational numbers with huge numerators and denominators.

The runtime is mainly determined by a final inversion of the (dense) eigenvector matrix (this is needed algorithmically for "expm"), and therefore causing the slow performance.

1. About using symbolic variables to get high-precision result:

Numerical exponential of a matrix is a very stable process. Notice that the input data consist of symbolic rational approximations of MATLAB doubles,and therefore contains numerical input errors even with the symbolic computation.

2. The workaround:

Thus, you should get a much faster (and same precision) computation when you use "vpa" to the input of "expm" when all input values are of purely numerical nature. (The resulting numerical vpa algorithm in "expm" is completely different from the "sym" algorithm and numerically stable, so you do not have to worry about numerical errors)

>> A = vpa(A);
>> E = expm(A);

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for x = -10:1:10
  for y = -10:1:10
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        [x y]
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