(0)

p(A) x = .

A A

i=1,...,m

j=1

(5.68)

Relations (5.66), (5.68) imply the following theorem:

Theorem 5.15 For the CG method and any p ∈ Pk satisfying p(0) = 1,

we have

x(k) ’ x ¤ max |p(»i )| x(0) ’ x ,

A A

i=1,...,m

with the eigenvalues »1 , . . . , »m of A.

If the eigenvalues of A are not known, but their location is, i.e., if one

knows a, b ∈ R such that

a ¤ »1 , . . . , »m ¤ b , (5.69)

then only the following weaker estimate can be used:

x(k) ’ x ¤ max |p(»)| x(0) ’ x . (5.70)

A A

»∈[a,b]

Therefore, we have to ¬nd p ∈ Pm with p(0) = 1 that minimizes

max |p(»)| » ∈ [a, b] .

5.2. Gradient and Conjugate Gradient Methods 225

This approximation problem in the maximum norm appeared already in

(5.43), because there is a bijection between the sets p ∈ Pk p(1) = 1

and p ∈ Pk p(0) = 1 through

p’p , p(t) := p(1 ’ t) .

˜ ˜ (5.71)

Its solution can represented by using the Chebyshev polynomials of the

¬rst kind (see, for example, [38, p. 302]). They are recursively de¬ned by

Tk+1 (x) := 2xTk (x) ’ Tk’1 (x) for x ∈ R

T0 (x) := 1 , T1 (x) := x ,

and have the representation

Tk (x) = cos(k arccos(x))

for |x| ¤ 1. This immediately implies

|Tk (x)| ¤ 1 for |x| ¤ 1 .

A further representation, valid for x ∈ R, is

1/2 k 1/2 k

1

x + x2 ’ 1 + x ’ x2 ’ 1

Tk (x) = . (5.72)

2

The optimal polynomial in (5.70) is then de¬ned by

Tk ((b + a ’ 2z)/(b ’ a))

for z ∈ R .

p(z) :=

Tk ((b + a)/(b ’ a))

This implies the following result:

Theorem 5.16 Let κ be the spectral condition number of A and assume

κ > 1. Then

k

κ1/2 ’ 1

1

’x ¤ ’x ¤2 x(0) ’ x

(k) (0)

x x . (5.73)

A A A

κ1/2 + 1

κ+1

Tk κ’1

Proof: Choose a as the smallest eigenvalue »min and b as the largest one

»max .

The ¬rst inequality follows immediately from (5.70) and κ = b/a. For

the second inequality note that due to (κ + 1)/(κ ’ 1) = 1 + 2/(κ ’ 1) =:

1 + 2· ≥ 1, (5.72) implies

1/2 k

κ+1 1

≥ 1 + 2· + (1 + 2·)2 ’ 1

Tk

κ’1 2

k

1 1/2

= 1 + 2· + 2 (·(· + 1)) .

2

Finally,

(· + 1)1/2 + · 1/2

2

1/2 1/2 1/2

1 + 2· + 2 (·(· + 1)) = · + (· + 1) =

(· + 1)1/2 ’ · 1/2

226 5. Iterative Methods for Systems of Linear Equations

(1 + 1/·)1/2 + 1

= ,

1/2

’1

(1 + 1/·)

2

which concludes the proof because of 1 + 1/· = κ.

For large κ we have again

κ1/2 ’ 1 2

≈ 1 ’ 1/2 .

κ1/2 + 1 κ

Compared with (5.58), κ has been improved to κ1/2 .

From (5.4) and (5.34) the complexity of the ¬ve-point stencil discretiza-

tion of the Poisson equation on the square results in

1

O κ1/2 O(m) = O(n) O(m) = O m3/2 .

ln

µ

This is the same behaviour as that of the SOR method with optimal re-

laxation parameter. The advantage of the above method lies in the fact

that the determination of parameters is not necessary for applying the

CG method. For quasi-uniform triangulations, Theorem 3.45 implies an

analogous general statement.

A relation to the semi-iterative methods follows from (5.71): The estimate

(5.66) can also be expressed as

¤ p(I ’ A)e(0)

e(k) (5.74)

A A

for any p ∈ Pk with p(1) = 1.

This is the same estimate as (5.42) for the Richardson iteration (5.28) as

basis method, with the Euclidean norm |·|2 replaced by the energy norm ·

A . While the semi-iterative methods are de¬ned by minimization of upper

bounds in (5.42), the CG method is optimal in the sense of (5.74), without

knowledge of the spectrum σ(I ’ A). In this manner the CG method can

be seen as an (optimal) acceleration method for the Richardson iteration.

Exercises

5.4 Let A ∈ Rm,m be a symmetric positive de¬nite matrix.

(a) Show that for x, y with xT y = 0 we have

κ’1

x, y

¤

A

,

x y κ+1

A A

where κ denotes the spectral condition number of A.

Hint: Represent x, y in terms of an orthonormal basis consisting of

eigenvectors of A.

5.3. Preconditioned Conjugate Gradient Method 227

(b) Show using the example m = 2 that this estimate is sharp. To this

end, look for a positive de¬nite symmetric matrix A ∈ R2,2 as well

as vectors x, y ∈ R2 with xT y = 0 and

κ’1

x, y A

= .

x y κ+1

A A

5.5 Prove that the computation of the conjugate direction in the CG

method in the general step k ≥ 2 is equivalent to the three-term recursion

formula

d(k+1) = [±k A + (βk + 1)I] d(k) ’ βk’1 d(k’1) .

5.6 Let A ∈ Rm,m be a symmetric positive de¬nite matrix with spectral

condition number κ. Suppose that the spectrum σ(A) of the matrix A

satis¬es a0 ∈ σ(A) as well as σ(A) \ {a0 } ‚ [a, b] with 0 < a0 < a ¤ b.

Show that this yields the following convergence estimate for the CG

method:

√ k’1

b ’ a0 κ’1

ˆ

√

x(k) ’ x A ¤ 2 x(0) ’ x A ,

a0 κ+1

ˆ

where κ := b/a ( < κ ).

ˆ

5.3 Preconditioned Conjugate Gradient Method

Due to Theorem 5.16, κ(A) should be small or only weakly growing in m,

which is not true for a ¬nite element sti¬ness matrix.

The technique of preconditioning is used ” as already discussed in Sec-

tion 5.1 ” to transform the system of equations in such a way that the

condition number of the system matrix is reduced without increasing the

e¬ort in the evaluation of the matrix vector product too much.

In a preconditioning from the left the system of equations is transformed

to

C ’1 Ax = C ’1 b

with a preconditioner C; in a preconditioning from the right it is transformed

to

AC ’1 y = b ,

such that x = C ’1 y is the solution of (5.1). Since the matrices are generally

sparse, this always has to be interpreted as a solution of the system of

equations Cx = y.

If A is symmetric and positive de¬nite, then this property is generally

violated by the transformed matrix for both variants, even for a symmetric

228 5. Iterative Methods for Systems of Linear Equations

positive de¬nite C. We assume for a moment to have a decomposition of

C with a nonsingular matrix W as

C = WWT .

Then Ax = b can be transformed to W ’1 AW ’T W T x = W ’1 b, i.e., to

B = W ’1 AW ’T , c = W ’1 b .

By = c with (5.75)

The matrix B is symmetric and positive de¬nite. The solution x is then

given by x = W ’T y. This procedure is called split preconditioning.

Due to W ’T BW T = C ’1 A and W BW ’1 = AC ’1 , B, C ’1 A and AC ’1

have the same eigenvalues, and therefore also the same spectral condition

number κ. Therefore, C should be “close” to A in order to reduce the

condition number. The CG method, applied to (5.75) and then back trans-

formed, leads to the preconditioned conjugate gradient method (PCG):

The terms of the CG method applied to (5.75) will all be marked by ˜,

with the exception of ±k and βk .

Due to the back transformation

x = W ’T x

˜

the algorithm has the search direction

d(k) := W ’T d(k)

˜

for the transformed iterate

x(k) := W ’T x(k) .

˜ (5.76)

The gradient g (k) of (5.44) in x(k) is given by

g (k) := Ax(k) ’ b = W B x(k) ’ c = W g (k) ,

˜ ˜

and hence

˜

g (k+1) = g (k) + ±k W B d(k) = g (k) + ±k Ad(k) ,

so that this formula remains unchanged compared with the CG method

with a new interpretation of the search direction. The search directions are

updated by

d(k+1) = ’W ’T W ’1 g (k+1) + βk d(k) = ’C ’1 g (k+1) + βk d(k) ,

so that in each iteration step additionally the system of equations

Ch(k+1) = g (k+1) has to be solved.

Finally, we have

T T

g (k) g (k) = g (k) C ’1 g (k) = g (k) h(k)

T

˜ ˜

and

T

˜T ˜

d(k) B d(k) = d(k) Ad(k) ,

so that the algorithm takes the form of Table 5.3.

5.3. Preconditioned Conjugate Gradient Method 229

Choose any x(0) ∈ Rm and calculate

d(0) := ’h(0) := ’C ’1 g (0) .

g (0) = Ax(0) ’ b ,

For k = 0, 1, . . . put

T

g (k) h(k)

±k = ,

T

d(k) Ad(k)

x(k+1) x(k) + ±k d(k) ,

=

g (k+1) g (k) + ±k Ad(k) ,

=

C ’1 g (k+1) ,

h(k+1) =

T

g (k+1) h(k+1)

βk = ,

T

g (k) h(k)

’h(k+1) + βk d(k) ,

d(k+1) =

up to the termination criterion (“|g (k+1) |2 = 0”) .

Table 5.3. PCG method.

The solution of the additional systems of equations for sparse matrices

should have the complexity O(m), in order not to worsen the complexity

for an iteration. It is not necessary to know a decomposition C = W W T .

Alternatively, the PCG method can be established by noting that C ’1 A

is self-adjoint and positive de¬nite with respect to the energy scalar product

·, · C de¬ned by C:

T

C ’1 Ax, y = C ’1 Ax Cy = xT Ay = xT C(C ’1 Ay) = x, C ’1 Ay

C C

and hence also C ’1 Ax, x C > 0 for x = 0.

Choosing the CG method for (5.75) with respect to ·, · C , we obtain

precisely the above method.

In case the termination criterion “ g (k+1) 2 = 0” is used for the iteration,

the scalar product must be additionally calculated. Alternatively, we may

T

use “ g (k+1) h(k+1) = 0”. Then the residual is measured in the norm

· C ’1 .

Following the reasoning at the end of Section 5.2, the PCG method can be

interpreted as an acceleration of a linear stationary method with iteration

matrix

M = I ’ C ’1 A .

For a consistent method, we have N = C ’1 or, in the formulation (5.10),

W = C. This observation can be extended in such a way that the CG

method can be used for the acceleration of iteration methods, for example

also for the multigrid method, which will be introduced in Section 5.5. Due

230 5. Iterative Methods for Systems of Linear Equations

to the deduction of the preconditioned CG method and the identity

x(k) ’ x = x(k) ’ x

˜ ˜ ,

A B

which results from the transformation (5.76), the approximation properties

for the CG method also hold for the PCG method if the spectral condition

number κ(A) is replaced by κ(B) = κ(C ’1 A). Therefore,

k

κ1/2 ’ 1

’x ¤2 x(0) ’ x

(k)

x A A

κ1/2 + 1

with κ = κ(C ’1 A).

There is a close relation between those preconditioning matrices C, which

keep κ(C ’1 A) small, and well-convergent linear stationary iteration meth-

ods with N = C ’1 (and M = I ’ C ’1 A) if N is symmetric and positive

de¬nite. Indeed,

κ(C ’1 A) ¤ (1 + (M ))/(1 ’ (M ))

if the method de¬ned by M and N is convergent and N is symmetric for

symmetric A (see Exercise 5.7).

From the considered linear stationary methods because of the required

symmetry we may take

• Jacobi™s method:

This corresponds exactly to the diagonal scaling, which means the division

of each equation by its diagonal element. Indeed, from the decomposition

(5.18) we have C = N ’1 = D, and the PCG method is equivalent to the

preconditioning from the left by the matrix C ’1 in combination with the

usage of the energy scalar product ·, · C .

• The SSOR method:

According to (5.38) we have

C = ω ’1 (2 ’ ω)’1 (D + ωL)D’1 (D + ωLT ) .

Hence C is symmetric and positive de¬nite. The solution of the auxiliary

systems of equations needs only forward and backward substitutions with

the same structure of the matrix as for the system matrix, so that the

requirement of lower complexity is also ful¬lled. An exact estimation of

κ(C ’1 A) shows (see [3, pp. 328 ¬.]) that under certain requirements for A,

which re¬‚ect properties of the boundary value problem and the discretiza-

tion, we ¬nd a considerable improvement of the conditioning by using an

estimate of the type

κ(C ’1 A) ¤ const(κ(A)1/2 + 1) .

The choice of the relaxation parameter ω is not critical. Instead of try-

ing to choose an optimal one for the contraction number of the SSOR

5.3. Preconditioned Conjugate Gradient Method 231

method, we can minimize an estimation for κ(C ’1 A) (see [3, p. 337]),

which recommends a choice of ω in [1.2, 1.6].

For the ¬ve-point stencil discretization of the Poisson equation on the

square we have, according to (5.34), κ(A) = O(n2 ), and the above con-

ditions are ful¬lled (see [3, pp. 330 f.]). By SSOR preconditioning this is

improved to κ(C ’1 A) = O(n), and therefore the complexity of the method

is

1 1

O κ1/2 O(m) = ln O n1/2 O(m) = O m5/4 .

ln (5.77)

µ µ

As discussed in Section 2.5, direct elimination methods are not suitable in

conjunction with the discretization of boundary value problems with large

node numbers, because in general ¬ll-in occurs. As discussed in Section 2.5,

L = (lij ) describes a lower triangular matrix with lii = 1 for all i = 1, . . . , m

(the dimension is described there with the number of degrees of freedom

M ) and U = (uij ) an upper triangular matrix. The idea of the incomplete

LU factorization, or ILU factorization, is to allow only certain patterns

E ∈ {1, . . . , m}2 for the entries of L and U , and instead of A = LU , in

general we can require only

A = LU ’ R.

Here the remainder R = (rij ) ∈ Rm,m has to satisfy

for (i, j) ∈ E .

rij = 0 (5.78)

The requirements

m

for (i, j) ∈ E

aij = lik ukj (5.79)

k=1

mean |E| equations for the |E| entries of the matrices L and U . (Notice that

lii = 1 for all i.) The existence of such factorizations will be discussed later.

Analogously to the close connection between the LU factorization and

an LDLT or LLT factorization for symmetric or symmetric positive def-

inite matrices, as de¬ned in Section 2.5, we can use the IC factorization

(incomplete Cholesky factorization) for such matrices. The IC factorization

needs a representation in the following form:

A = LLT ’ R .

Based on an ILU factorization a linear stationary method is de¬ned by

N = (LU )’1 (and M = I ’ N A), the ILU iteration. We thus have an

expansion of the old method of iterative re¬nement.

Using C = N ’1 = LU for the preconditioning, the complexity of the

auxiliary systems depends on the choice of the matrix pattern E. In general,

the following is required:

E := (i, j) aij = 0 , i, j = 1, . . . , m ‚ E , (i, i) i = 1, . . . , m ‚ E .

(5.80)

232 5. Iterative Methods for Systems of Linear Equations

The requirement of equality E = E is most often used. Then, and also in the

case of ¬xed expansions of E , it is ensured that for a sequence of systems

of equations with a matrix A that is sparse in the strict sense, this will also

hold for L and U . All in all, only O(m) operations are necessary, including

the calculation of L and U , as in the case of the SSOR preconditioning

for the auxiliary system of equations. On the other hand, the remainder R

should be rather small in order to ensure a good convergence of the ILU

iteration and also to ensure a small spectral condition number κ(C ’1 A).

Possible matrix patterns E are shown, for example, in [28, pp. 275 ¬.], where

a more speci¬c structure of L and U is discussed if the matrix A is created

by a discretization on a structured grid, for example by a ¬nite di¬erence

method.

The question of the existence (and stability) of an ILU factorization