Symmetric Matrix Eigenvalue Problems - Numerical Methods in Engineering with MATLAB - page 331

However, there is a major snag—the transformation that annihilates an off-diagonal
term also undoes some of the previously created zeroes.Fortunately, itturnsoutthat
the off-diagonaltermsthat reappear will besmaller than before. ThusJacobi method
is an iterative procedurethat repeatedly applies Jacobi rotations until the off-diagonal
termshave virtually vanished. The finaltransformationmatrix P is the accumulation
of individual rotations R i :

P = R 1 · R 2 · R 3 ··· (9.14)

The columnsof P finish up being the eigenvectorsof A and the diagonal elements of
A ∗ = P T AP become the eigenvectors.

Let us now look at the details of a Jacobi rotation. FromEq. (9.13) we see that
A k =

0 if

( c 2 s 2 ) A k + − cs ( A kk − A ) = 0 (a)

Using the trigonometric identities c 2 s 2 − = cos 2 θ and cs = (1 / 2)sin 2 θ , weobtain

fromEq. (a)

2 A k
A kk −

tan 2 θ =− (b)

A

which couldbe solved for θ , followedbycomputation of c = cos θ and s = sin θ . How-

ever, the procedure describedbelow leadstoabetteralgorithm. 20
Introducing the notation

A kk − A

2 A k

φ = cot 2 θ =− (9.15)

and utilizing the trigonometric identity

2 t

tan 2 θ =

(1 − t 2 )

where t = tan θ , wecan write Eq. (b) as

t 2

+ φ − =

2 t 1 0

which has the roots

2

t =− φ ± φ + 1

| | ≤ | θ | ≤

45 ◦ ,leads to the

It has been found that the root t 1, which correspondsto

morestable transformation. Therefore, we choose the plussign if φ> 0 and the minus

20

The procedure is adapted fromW.H. Press et al ., Numerical Recipes in Fortran , 2nd ed. (1992),
CambridgeUniversity Press.