Numerical Methods Contents - SAM

(diagonal) entries of D are strictly positive and we can define
D = diag(λ 1 ,...,λ n ), λ i > 0 ⇒ D 1/2 := diag( √ λ 1 , . .., √ λ n ) .
This is generalized to
B 1/2 := Q T D 1/2 Q ,
and one easily verifies, using Q T = Q −1 , that (B 1/2 ) 2 = B and that B 1/2 is s.p.d. In fact, these two
requirements already determine B 1/2 uniquely.
However, if one formally applies Algorithm 4.2.1 to the transformed system from (4.3.1), it becomes
apparent that, after suitable transformation of the iteration variables p j and r j , B 1/2 and B −1/2 invariably
occur in products B −1/2 B −1/2 = B −1 and B 1/2 B −1/2 = I. Thus, thanks to this intrinsic
transformation square roots of B are not required for the implementation!
Algorithm 4.3.1 (Preconditioned CG method (PCG)).
Input: initial guess x ∈ R n ˆ= x (0) ∈ R n , tolerance τ > 0
Output: approximate solution x ˆ= x (l)
Notion 4.3.1 (Preconditioner).
A s.p.d. matrix B ∈ R n,n is called a preconditioner (ger.: Vorkonditionierer) for the s.p.d. matrix
A ∈ R n,n , if
1. κ(B −1/2 AB −1/2 ) is “small” and
2. the evaluation of B −1 x is about as expensive (in terms of elementary operations) as the
matrix×vector multiplication Ax, x ∈ R n .
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p := r := b − Ax; p := B −1 r; q := p; τ 0 := p T r;
for l = 1 to l max do {
β := r T q; h := Ap; α := β
p T h ;
x := x + αp;
r := r − αh;
q := B −1 r; β := rT q
β ;
if |q T r| ≤ τ · τ 0 then stop;
p := q + βp;
}
(4.3.2)
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There are several equivalent ways to express that κ(B −1/2 AB −1/2 ) is “small”:
Computational effort per step: 1 evaluation A×vector, 1 evaluation B −1 ×vector, 3 dot products,
3 AXPY-operations
• κ(B −1 A) is “small”,
because spectra agree σ(B −1 A) = σ(B −1/2 AB −1/2 ) due to similarity (→ Lemma 5.1.4)
• ∃0 < γ < Γ, Γ/γ "‘small”: γ (x T Bx) ≤ x T Ax ≤ Γ (x T Bx) ∀x ∈ R n ,
where equivalence is seen by transforming y := B −1/2 x and appealing to the min-max Theorem
5.3.5.
“Reader’s digest” version of notion 4.3.1:
Remark 4.3.2 (Convergence theory for PCG).
Assertions of Thm. 4.2.7 remain valid with κ(A) replaced with κ(B −1 A) and energy norm based on
à instead of A.
△
Example 4.3.3 (Simple preconditioners).
✗
✖
S.p.d. B preconditioner :⇔ B −1 = cheap approximate inverse of A
✔
✕
B = easily invertible “part” of A
Problem: B 1/2 , which occurs prominently in (4.3.1) is usually not available with acceptable computational
costs.
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B =diag(A): Jacobi preconditioner (diagonal scaling)
{
(B) ij =
(A) ij , if |i − j| ≤ k ,
0 else,
for some k ≪ n.
Symmetric Gauss-Seidel preconditioner
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