Symmetric Matrix Eigenvalue Problems - Numerical Methods in Engineering with MATLAB - page 332

φ ≤

sign if 0, which isequivalenttousing

) 1 φ − | φ | +

2

t = sgn( φ +

To f o restal l exces sive roundoff errorif φ islarge, wemultiplyboth sides of the equation

2

by | φ | + φ + 1 and solvefor t , which yields

sgn( φ )

t =

φ

(9.16a)

| φ | +

2

+ 1

φ

In the case of very large , we shouldreplace Eq. (9.16a) by the approximation

1
2

t = (9.16b)

φ

2 . Ha ving comp uted t , wecan use the

to preventoverflowinthe computation of φ

θ = √ 1 trigonometric relationship tan θ = sin θ/ cos − cos 2 θ/ cos θ to obtain

1

c = √ 1 s = tc (9.17)

+ t 2

We now improve the computational properties of the transformation formulas in
Eqs. (9.13).Solving Eq. (a)for A , weobtain

c 2 s 2 −

A = A kk + A k (c)

cs

Replacing all occurrences of A by Eq. (c) and simplifying, wecan write the transfor-
mation formulas in Eqs. (9.13) as

A kk =

A kk − tA k

A ∗ = A + tA k

A k = A ∗ k =

0 (9.18)

A ki A ik = = A ki − s ( A i + τ A ki ), i = k , i =

A ∗ i A i =

= A i + s ( A ki − τ A i ), i = k , i =

where

s

τ = (9.19)

1 + c

The introduction of τ allowedustoexpress each formulainthe form (original

value) (change), which is helpful in reducing the roundoff error.

At the startofJacobi's diagonalizationprocess the transformationmatrix P is
initialized to the identitymatrix. Each Jacobi rotation changes this matrix from P
to P ∗ =

+

PR . The corresponding changes in the elements of P can be shown to be