**for**

**Jacobi’s
Method****
**Jacobi’s method is an easily
understood algorithm for finding all eigenpairs for a symmetric
matrix. It is a reliable method that produces uniformly accurate
answers for the results. For matrices of order up to 10×10, the
algorithm is competitive with more sophisticated ones. If speed is
not a major consideration, it is quite acceptable for matrices up to
order 20×20. A solution is guaranteed for all real symmetric
matrices when Jacobi’s method is used. This limitation is not
severe since many practical problems of applied mathematics and
engineering involve symmetric matrices. From a theoretical viewpoint,
the method embodies techniques that are found in more sophisticated
algorithms. For instructive purposes, it is worthwhile to investigate
the details of Jacobi’s method.

**Jacobi Series of
Transformations****
**Start with the real symmetric
matrix .
Then construct the sequence of orthogonal
matrices as
follows:

and

for j = 1, 2, ... .

It is possible to construct the sequence so that

.

.

**Proof ****Jacobi's
Method** **Jacobi's
Method**

**Remark.** Current
research by James W. Demmel and Kresimir Veselic (1992) indicate that
Jacobi's method is more accurate than QR. You can check
out their research by following the link in the list of internet
resources. The abstract for their research follows
below.

**Abstract.** We show
that Jacobi's method (with a proper stopping criterion) computes
small eigenvalues of symmetric positive definite matrices with a
uniformly better relative accuracy bound than QR, divide and conquer,
traditional bisection, or any algorithm which first involves
tridiagonalizing the matrix. In fact, modulo an assumption based on
extensive numerical tests, we show that Jacobi's method is optimally
accurate in the following sense: if the matrix is such that small
relative errors in its entries cause small relative errors in its
eigenvalues, Jacobi will compute them with nearly this accuracy. In
other words, as long as the initial matrix has small relative errors
in each component, even using infinite precision will not improve on
Jacobi (modulo factors of dimensionality). ...

**Computer Programs ** **Jacobi's
Method** **Jacobi's
Method**

**Mathematica Subroutine
****(****Jacobi
iteration for eigenvectors).** To compute the
full set of eigen-pairs of
the n by n real symmetric
matrix **A**. Jacobi iteration is used to
find all eigenvalue-eigenvector pairs.

**Example 1.** Use the
classic Jacobi method to find all the eigenvalues and eigenvectors of
the symmetric matrix .

**Solution
1.**

**Example 2.** Use the
cyclic Jacobi method to find all the eigenvalues and eigenvectors of
the symmetric matrix .

**Solution
2.**

**Example
3.** Determine which method is faster: the
original Jacobi method or the cyclic Jacobi method.

The statement in the text says that the search for the maximum
element becomes time consuming.

However, it might be the computation of for
each step which is time consuming.

**Solution
3.**

**Example 4.** How fast
are our subroutines compared to *Mathematica'*s ?

**Solution
4.**

**Example 5.** Since
the cyclic Jacobi method seems faster, we should mention that calling
the subroutine costs time.

We could put it all together in one part and it will be
faster.

However, *Mathematica*, Maple, Matlab are all in some sense
interpreted languages such as BASIC.

Because the instruction code we write is
not compiled. Complied
languages such as FORTRAN and C will
always be the fast ones.

**Solution
5.**

**Example 6.** An
eigenvalue movie. Cleve Moler, inventor of the software
used graphs to illustrate convergence of eigenvalues. In
Matlab, eigmovie(A), shows a "movie" that depicts the computation of
the eigenvalues of a symmetric matrix. (Cleve
Moler is chairman and cofounder of The
MathWorks.) The following *Mathematica*
animation has the "spirit" of Cleve "eigmovie."

**Solution
6.**

**Research Experience for
Undergraduates**

**Jacobi's
Method** **Jacobi's
Method** Internet hyperlinks to web sites and
a bibliography of articles.

**Download this
Mathematica Notebook****
****Jacobi
Iteration for Eigenvectors****
**

**Return
to Numerical Methods - Numerical Analysis**

(c) John H. Mathews 2004