¬ ·

¬ 0 ’1 0 ’1 4 ’1 0 ’1 0 ·

¬ ·

¬ 0 0 ’1 0 ’1 4 0 0 ’1 ·

¬ ·

¬ 0 0 0 ’1 0 0 4 ’1 0 ·

¬ ·

0 0 0 0 ’1 0 ’1 4 ’1

0 0 0 0 0 ’1 0 ’1 4

m = 3 — 3 : rowwise ordering.

«

’1 ’1 0 0

4 0 0 0 0

¬ ·

’1 0 ’1 0

0 4 0 0 0

¬ ·

¬ ·

’1 ’1 ’1 ’1

0 0 4 0 0

¬ ·

¬ ·

0 ’1 0 ’1

0 0 0 4 0

¬ ·

¬ ·

0 0 ’1 ’1

0 0 0 0 4

¬ ·

¬ ·

¬’1 ’1 ’1 0 0 0·

¬ ·

4 0 0

¬’1 0 ’1 ’1 0 0·

¬ ·

0 4 0

0 ’1 ’1 0 ’1 0

0 0 4

0 0 ’1 ’1 ’1 0 0 0 4

red-black ordering:

red: node 1, 3, 5, 7, 9 from rowwise ordering

black: node 2, 4, 6, 8 from rowwise ordering

Figure 5.2. Comparison of orderings.

that contribute to the corresponding row of the discretization matrix. In

the example of the ¬ve-point stencil, starting with rowwise numbering, one

can combine all odd indices to a block S1 (the “red nodes”) and all even

indices to a block S2 (the “black” nodes). Here we have p = 2. We call this

a red-black ordering (see Figure 5.2). If two “colours” are not su¬cient, one

can choose p > 2.

We return to the SOR method and its convergence: In the following the

iteration matrix will be denoted by MSOR(ω) with the relaxation parameter

ω. Likewise, MJ and MGS are the iteration matrices of Jacobi™s and the

Gauss“Seidel method, respectively. General propositions are summarized

in the following theorem:

5.1. Linear Stationary Iterative Methods 213

Theorem 5.6 (of Kahan; Ostrowski and Reich)

MSOR(ω) ≥ |1 ’ ω| for ω = 0.

(1)

(2) If A is symmetric and positive de¬nite, then

for ω ∈ (0, 2) .

MSOR(ω) < 1

2

Proof: See [16, pp. 91 f.].

Therefore, we use only ω ∈ (0, 2). For a useful procedure we need more

information about the optimal relaxation parameter ωopt , given by

MSOR(ωopt ) = min MSOR(ω) ,

0<ω<2

and about the size of the contraction number. This is possible only if the

ordering of equations and unknowns has certain properties:

De¬nition 5.7 A matrix A ∈ Rm,m is consistently ordered if for the

partition (5.18), D is nonsingular and

C(±) := ±’1 D’1 L + ±D’1 R

has eigenvalues independent of ± for ± ∈ C\{0}.

There is a connection to the possibility of a multi-colour ordering, because

a matrix in the block form (5.39) is consistently ordered if it is block-

tridiagonal (i.e., Aij = 0 for |i ’ j| > 1) and the diagonal blocks Aii are

nonsingular diagonal matrices (see [28, pp. 114 f.]).

In the case of a consistently ordered matrix one can prove a relation

between the eigenvalues of MJ , MGS , and MSOR(ω) . From this we can see

how much faster the Gauss“Seidel method converges than Jacobi™s method:

Theorem 5.8 If A is consistently ordered, then

(MJ )2 = (MGS ) .

2

Proof: For a special case see Remark 5.5.2 in [16].

Due to (5.4) we can expect a halving of the number of iteration steps,

but this does not change the asymptotic statement (5.27).

Finally, in the case that Jacobi™s method converges the following theorem

holds:

Theorem 5.9 Let A be consistently ordered with nonsingular diagonal ma-

trix D, the eigenvalues of MJ being real and β := (MJ ) < 1. Then we have

for the SOR method:

2

(1) ωopt = ,

1 + (1 ’ β 2 )1/2

214 5. Iterative Methods for Systems of Linear Equations

±

2 2 1/2

1 ’ ω + 1 ω 2 β 2 + ωβ 1 ’ ω + ω β

2 4

(2) (MSOR(ω) ) = for 0 < ω < ωopt

ω’1 for ωopt ¤ ω < 2 ,

β2

(3) MSOR(ωopt ) = .

(1 + (1 ’ β 2 )1/2 )2

Proof: See [18, p. 216].

2

ρ( M SOR(ω) )

1

0 ω opt 2ω

1

Figure 5.3. Dependence of MSOR(ω) on ω.

If (MJ ) is known for Jacobi™s method, then ωopt can be calculated. This

is the case in the example of the ¬ve-point stencil discretization on a square:

From (5.26) and Theorem 5.9 it follows that

π2

2

π

+ O(n’4 ) ;

=1’

(MGS ) = cos 2

n n

hence

π

ωopt = 2/ 1 + sin n ,

ωopt ’ 1 = 1 ’ 2 π + O(n’2 ) .

MSOR(ωopt ) = n

Therefore, the optimal SOR method has a lower complexity than all

methods described up to now.

Correspondingly, the number of operations to reach the relative er-

ror level µ > 0 is reduced to ln 1 O(m3/2 ) operations in comparison to

µ

ln 1 O(m2 ) operations for the previous procedures.

µ

Table 5.1 gives an impression of the convergence for the model problem.

It displays the theoretically to be expected values for the numbers of iter-

ations of the Gauss“Seidel method (mGS ), as well as for the SOR method

5.1. Linear Stationary Iterative Methods 215

n mGS mSOR

8 43 8

16 178 17

32 715 35

64 2865 70

128 11466 140

256 45867 281

Table 5.1. Gauss“Seidel and optimal SOR method for the model problem.

with optimal relaxation parameter (mSOR ). Here we use the very moderate

termination criterion µ = 10’3 measured in the Euclidean norm.

The optimal SOR method is superior, even if we take into account the

almost doubled e¬ort per iteration step. But generally, ωopt is not known

explicitly. Figure 5.3 shows that it is probably better to overestimate ωopt

instead of underestimating. More generally, one can try to improve the

relaxation parameter during the iteration:

If (MJ ) is a simple eigenvalue, then this also holds true for the spectral

radius (MSOR(ω) ). The spectral radius can thus be approximated by the

power method on the basis of the iterates. By Theorem 5.9 (3) one can

approximate (MJ ), and by Theorem 5.9 (1) then also ωopt .

This basic principle can be extended to an algorithm (see, for example,

[18, Section 9.5]), but the upcoming overall procedure is no longer a linear

stationary method.

5.1.5 Extrapolation Methods

Another possibility for an extension of the linear stationary methods, re-

lated to the adaption of the relaxation parameter, is the following: Starting

with a linear stationary basic iteration xk+1 := ¦ xk we de¬ne a new

˜ ˜

iteration by

x(k+1) := ωk ¦ x(k) + (1 ’ ωk )x(k) , (5.41)

with extrapolation factors ωk to be chosen. A generalization of this de¬-

nition is to start with the iterates of the basic iteration x(0) , x(1) , . . .. The

˜ ˜

iterates of the new method are to be determined by

k

(k)

±kj x(j) ,

x := ˜

j=0

with ±kj de¬ned by a polynomial pk ∈ Pk , with the property pk (t) =

k j

j=0 ±kj t and pk (1) = 1. For an appropriate de¬nition of such extrapola-

tion or semi-iterative methods we need to know the spectrum of the basic

iteration matrix M , since the error e(k) = x(k) ’ x satis¬es

e(k) = pk (M )e(0) ,

216 5. Iterative Methods for Systems of Linear Equations

where M is the iteration matrix of the basic iteration. This matrix should

be normal, for example, such that

pk (M ) = (pk (M ))

2

holds. Then we have the obvious estimation

¤ pk (M )e(0) ¤ pk (M ) ¤ (pk (M )) e(0) 2 .

e(k) e(0) (5.42)

2 2 2 2

If the method is to be de¬ned in such a way that

(pk (M )) = max |pk (»)| » ∈ σ(M )

is minimized by choosing pk , then the knowledge of the spectrum σ(M ) is

necessary. Generally, instead of this, we assume that suitable supersets are

known: If σ(M ) is real and

a¤»¤b for all » ∈ σ(M ) ,

then, due to

¤ max pk (») e(0) 2 ,

e(k) 2 »∈[a,b]

it makes sense to determine the polynomials pk as a solution of the

minimization problem on [a, b],

max |pk (»)| ’ min p ∈ Pk

for all with p(1) = 1 . (5.43)

»∈[a,b]

In the following sections we will introduce methods with an analogous

convergence behaviour, without control parameters necessary for their

de¬nition.

For further information on semi-iterative methods see, for example, [16,

Chapter 7].

Exercises

5.1 Investigate Jacobi™s method and the Gauss“Seidel method for solving

the linear system of equations Ax = b with respect to their convergence if

we have the following system matrices:

« «

1 2 ’2 2 ’1 1

1

(a) A = 1 1 1 , (b) A = 2 2 2 .

2

’1 ’1 2

221

5.2 Prove the consistency of the SOR method.

5.3 Prove Theorem 5.6, (1).

5.2. Gradient and Conjugate Gradient Methods 217

5.2 Gradient and Conjugate Gradient Methods

In this section let A ∈ Rm,m be symmetric and positive de¬nite. Then the

system of equations Ax = b is equivalent to the problem

1

Minimize f (x) := xT Ax ’ bT x for x ∈ Rm , (5.44)

2

since for such a functional the minima and stationary points coincide, where

a stationary point is an x satisfying

0 = ∇f (x) = Ax ’ b . (5.45)

In contrast to the notation x · y for the “short” space vectors x, y ∈ Rd we

write here the Euclidean scalar product as matrix product xT y.

For the ¬nite element discretization this corresponds to the equivalence

of the Galerkin method (2.23) with the Ritz method (2.24) if A is the

sti¬ness matrix and b the load vector (see (2.34) and (2.35)). More generally,

Lemma 2.3 implies the equivalence of (5.44) and (5.45), if as bilinear form

the so-called energy scalar product

:= xT Ay

x, y (5.46)

A

is chosen.

A general iterative method to solve (5.44) has the following structure:

De¬ne a search direction d(k) .

˜

± ’ f (±) := f x(k) + ±d(k)

Minimize (5.47)

exactly or approximately, with the solution ±k .

x(k+1) := x(k) + ±k d(k) .

De¬ne (5.48)

If f is de¬ned as in (5.44), the exact ±k can be computed from the condition

˜

f (±) = 0 and

T (k)

˜

f (±) = ∇f x(k) + ±d(k) d

as

T

g (k) d(k)

±k = ’ , (5.49)

T

d(k) Ad(k)

where

g (k) := Ax(k) ’ b = ∇f x(k) . (5.50)

The error of the kth iterate is denoted by e(k) :

e(k) := x(k) ’ x .

Some relations that are valid in this general fromework are the following:

Due to the one-dimensional minimization of f , we have

T

g (k+1) d(k) = 0 , (5.51)

218 5. Iterative Methods for Systems of Linear Equations

and from (5.50) we can conclude immediately that

Ae(k) = g (k) , e(k+1) = e(k) + ±k d(k) , (5.52)

(k+1) (k) (k)

g =g + ±k Ad . (5.53)

We consider the energy norm

1/2

:= xT Ax

x (5.54)

A

induced by the energy scalar product. For a ¬nite element sti¬ness matrix

A with a bilinear form a we have the correspondence

= a(u, u)1/2 = u

x A a

m

for u = i=1 xi •i if the •i are the underlying basis functions. Comparing

the solution x = A’1 b with an arbitrary y ∈ Rm leads to

1

f (y) = f (x) + y ’ x 2 , (5.55)

A

2

so that condition (5.44) also minimizes the distance to x in · A . The

energy norm will therefore have a special importance. Measured in the

energy norm we have, due to (5.52),

T T

2

= e(k) g (k) = g (k) A’1 g (k) ,

e(k) A

and therefore due to (5.52) and (5.51),

T

2

e(k+1) = g (k+1) e(k) .

A

The vector ’∇f x(k) in x(k) points in the direction of the locally steepest

descent, which motivates the gradient method, i.e.,

d(k) := ’g (k) , (5.56)

and thus

T

d(k) d(k)

±k = . (5.57)

T

d(k) Ad(k)

The above identities imply for the gradient method

T

d(k) d(k)

(k+1) 2 (k) T (k)

1 ’ ±k

(k)

e(k) 2

e =g + ±k Ad e = A T

d(k) A’1 d(k)

and thus by means of the de¬nition of ±k from (5.57)

±

T (k) 2

d(k) d

2 2

’x A = x ’x A 1’

(k+1) (k)

x .

(k) T Ad(k) d(k) T A’1 d(k)

d

With the inequality of Kantorovich (see, for example, [28, p. 132]),

xT Ax xT A’1 x

2

1 1/2 1 ’1/2

¤ κ +κ ,

2 2 2

(xT x)

5.2. Gradient and Conjugate Gradient Methods 219

where κ := κ(A) is the spectral condition number, and the relation

(a ’ 1)2

4

1’ = for a > 0 ,

2 2

a1/2 + a’1/2 (a + 1)

we obtain the following theorem:

Theorem 5.10 For the gradient method we have

k

κ’1

’x ¤ x(0) ’ x

(k)

x . (5.58)

A A

κ+1

This is the same estimate as for the optimally relaxed Richardson method

(with the sharper estimate M A ¤ κ’1 instead of (M ) ¤ κ’1 ). The

κ+1 κ+1

essential di¬erence lies in the fact that this is possible without knowledge

of the spectrum of A.

Nevertheless, for ¬nite element discretizations we have the same poor

convergence rate as for Jacobi™s or similar methods. The reason for this

T

de¬ciency lies in the fact that due to (5.51), we have g (k+1) g (k) = 0, but

T

in general not g (k+2) g (k) = 0. On the contrary, these search directions are

very often almost parallel, as can be seen from Figure 5.4.

.

m = 2:

x (0)

f = constant

(contour lines)

Figure 5.4. Zigzag behaviour of the gradient method.

The reason for this problem is the fact that for large κ the search di-

rections g (k) and g (k+1) can be almost parallel with respect to the scalar

products ·, · A (see Exercise 5.4), but with respect to · A the distance

to the solution will be minimized (see (5.55)).

The search directions d(k) should be orthogonal with respect to ·, · A ,

which we call conjugate.

De¬nition 5.11 Vectors d(0) , . . . , d(l) ∈ Rm are conjugate if they satisfy

d(i) , d(j) =0 for i, j = 0, . . . , l , i = j .

A

If the search directions of a method de¬ned according to (5.48), (5.49) are

chosen as conjugate, it is called a method of conjugate directions.

Let d(0) , . . . , d(m’1) be conjugate directions. Then they are also linearly

independent and thus form a basis in which the solution x of (5.1) can be

220 5. Iterative Methods for Systems of Linear Equations

represented, say by the coe¬cients γk :

m’1

γk d(k) .

x=

k=0

Since the d(k) are conjugate and Ax = b holds, we have

T

d(k) b

γk = , (5.59)

T

d(k) Ad(k)

and the γk can be calculated without knowledge of x. If the d(k) would by

given a priori, for example by orthogonalization of a basis with respect to

·, · A , then x would be determined by (5.59).

If we apply (5.59) to determine the coe¬cients for x ’ x(0) in the form

m’1

x’x (0)

γk d(k) ,

=

k=0

which means replacing b with b ’ Ax(0) in (5.59), then we get

T

g (0) d(k)

γk = ’ .

T

d(k) Ad(k)

For the kth iterate we have, according to (5.48);

k’1

(k) (0)

±i d(i)

x =x +

i=0

and therefore (see (5.50))

k’1

(k) (0)

±i Ad(i) .

g =g +

i=0

For a method of conjugate directions this implies

T T

g (k) d(k) = g (0) d(k)

and therefore

T

g (k) d(k)

γk = ’ = ±k ,

T

d(k) Ad(k)

which means that x = x(m) . A method of conjugate directions therefore

is exact after at most m steps. Under certain conditions such a method

may terminate before reaching this step number with g (k) = 0 and the

¬nal iterate x(k) = x. If m is very large, this exactness of a method of

conjugate directions is less important than the fact that the iterates can

be interpreted as the solution of a minimization problem approximating

(5.44):

5.2. Gradient and Conjugate Gradient Methods 221

Theorem 5.12 The iterates x(k) that are determined by a method of con-

jugate directions minimize the functional f from (5.44) as well as the error

x(k) ’ x A on x(0) + Kk (A; g (0) ), where

Kk (A; g (0) ) := span d(0) , . . . , d(k’1) .

This is due to

T

for i = 0, . . . , k ’ 1 .

g (k) d(i) = 0 (5.60)

Proof: It is su¬cent to prove (5.60). Due to the one-dimensional mini-

mization this holds for k = 1 and for i = k ’ 1 (see (5.51) applied to k ’ 1).

To conclude the assertion for k from its knowledge for k ’ 1, we note that

(5.53) implies, for 0 ¤ i < k ’ 1,

T T

g (k) ’ g (k’1) = ±k’1 d(i) Ad(k’1) = 0 .

d(i)

2

In the method of conjugate gradients, or CG method, the d(k) are

determined during the iteration by the ansatz

d(k+1) := ’g (k+1) + βk d(k) . (5.61)

Then we have to clarify whether

d(k) , d(i) = 0 for k > i

A

can be obtained. The necessary requirement d(k+1) , d(k) = 0 leads to

A

’ g (k+1) , d(k) ⇐’

+ βk d(k) , d(k) =0

A A

T

g (k+1) Ad(k)

βk = . (5.62)

T

d(k) Ad(k)

In applying the method it is recommended not to calculate g (k+1) directly

but to use (5.53) instead, because Ad(k) is already necessary to determine

±k and βk .

The following equivalences hold:

Theorem 5.13 In case the CG method does not terminate prematurely

with x(k’1) being the solution of (5.1), then we have for 1 ¤ k ¤ m

Kk (A; g (0) ) = span g (0) , Ag (0) , . . . , Ak’1 g (0)

(5.63)

= span g (0) , . . . , g (k’1) .

Furthermore,

T

0 for i = 0, . . . , k ’ 1, and

g (k) g (i) = (5.64)

dim Kk (A; g (0) ) = k.

222 5. Iterative Methods for Systems of Linear Equations

The space Kk (A; g (0) ) = span g (0) , Ag (0) , . . . , Ak’1 g (0) is called the

Krylov (sub)space of dimension k of A with respect to g (0) .

Proof: The identities (5.64) are immediate consequences of (5.63) and

Theorem 5.12. The proof of (5.63) is given by induction:

For k = 1 the assertion is trivial. Let us assume that for k ≥ 1 the identity

(5.63) holds and therefore also (5.64) does. Due to (5.53) (applied to k ’ 1)

it follows that

g (k) ∈ A Kk A; g (0) ‚ span g (0) , . . . , Ak g (0)

and thus

span g (0) , . . . , g (k) = span g (0) , . . . , Ak g (0) ,

because the left space is contained in the right one and the dimension of

the left subspace is maximal (= k + 1) due to (5.64) and g (i) = 0 for all

i = 0, . . . , k. The identity

span d(0) , . . . , d(k) = span g (0) , . . . , g (k)

2

follows from the induction hypothesis and (5.61).

The number of operations per iteration can be reduced to one matrix

vector, two scalar products, and three SAXPY operations, if the following

equivalent terms are used:

T T

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

±k = , βk = . (5.65)

T T

d(k) Ad(k) g (k) g (k)

Here a SAXPY operation is of the form

z := x + ±y

for vectors x, y, z and a scalar ±.

The identities (5.65) can be seen as follows: Concerning ±k we note that

because of (5.51) and (5.61),

T T T

’g (k) d(k) = ’g (k) ’ g (k) + βk’1 d(k’1) = g (k) g (k) ,

and concerning βk , because of (5.53), (5.64), (5.62), and the identity (5.49)

for ±k , we have

T T T T

g (k+1) g (k+1) = g (k+1) g (k) + ±k Ad(k) = ±k g (k+1) Ad(k) = βk g (k) g (k)

and hence the assumption. The algorithm is summarized in Table 5.2.

Indeed, the algorithm de¬nes conjugate directions:

Theorem 5.14 If g (k’1) = 0, then d(k’1) = 0 and the d(0) , . . . , d(k’1) are

conjugate.

5.2. Gradient and Conjugate Gradient Methods 223

Choose any x(0) ∈ Rm and calculate

d(0) := ’g (0) = b ’ Ax(0) .

For k = 0, 1, . . . put

T

g (k) g (k)

±k = ,

T

d(k) Ad(k)

x(k+1) x(k) + ±k d(k) ,

=

g (k+1) g (k) + ±k Ad(k) ,

=

T

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

βk = ,

T

g (k) g (k)

d(k+1) = ’g (k+1) + βk d(k) ,

until the termination criterion (“|g (k+1) |2 = 0”) is ful¬lled.

Table 5.2. CG method.

Proof: The proof is done by induction:

The case k = 1 is clear. Assume that d(0) , . . . , d(k’1) are all nonzero and

conjugate. Thus according to Theorem 5.12 and Theorem 5.13 the identities

(5.60)“(5.64) hold up to index k. Let us ¬rst prove that d(k) = 0:

Due to g (k) + d(k) = βk’1 d(k’1) ∈ Kk (A; g (0) ) the assertion d(k) = 0

would imply directly g (k) ∈ Kk (A; g (0) ). But relations (5.63) and (5.64)

imply for the index k,

T

g (k) x = 0 for all x ∈ Kk (A; g (0) ) ,

which contradicts g (k) = 0.

T

In order to prove d(k) Ad(i) = 0 for i = 0, . . . , k ’ 1, according to (5.62)

we have to prove only the case i ¤ k ’ 2. We have

T T T

d(i) Ad(k) = ’d(i) Ag (k) + βk’1 d(i) Ad(k’1) .

The ¬rst term disappears due to Ad(i) ∈ A Kk’1 A; g (0) ‚ Kk A; g (0) ,

which means that Ad(i) ∈ span d(0) , . . . , d(k’1) , and (5.60). The second

2

term disappears because of the induction hypothesis.

Methods that aim at minimizing the error or residual on Kk A; g (0)

with respect to a norm · are called Krylov subspace methods. Here the

error will be minimized in the energy norm · = · A according to (5.55)

and Theorem 5.12.

Due to the representation of the Krylov space in Theorem 5.13 the

elements y ∈ x(0) + Kk A; g (0) are exactly the vectors of the form

y = x(0) + q(A)g (0) , for any q ∈ Pk’1 (for the notation q(A) see Appendix

224 5. Iterative Methods for Systems of Linear Equations

A.3). Hence it follows that

y ’ x = x(0) ’ x + q(A)A x(0) ’ x = p(A) x(0) ’ x ,

with p(z) = 1 + q(z)z, i.e., p ∈ Pk and p(0) = 1. On the other hand,

any such polynomial can be represented in the given form (de¬ne q by

q(z) = (p(z) ’ 1) /z). Thus Theorem 5.12 implies

x(k) ’ x ¤ y’x = p(A) x(0) ’ x (5.66)

A

A A

for any p ∈ Pk with p(0) = 1.

Let z1 , . . . , zm be an orthonormal basis of eigenvectors, that is,

T

Azj = »j zj and zi zj = δij for i, j = 1, . . . , m . (5.67)

m

Then we have x(0) ’ x = for certain cj ∈ R, and hence

j=1 cj zj

m

’x =

(0)

p(A) x p (»j ) cj zj

j=1

and therefore

m m

2 T 2

’x ’x ’x = »j |cj |

(0) (0) (0) T

x =x Ax ci cj zi Azj =

A

i,j=1 j=1

and analogously

m 2

2 2

2

’x »j |cj p(»j )| ¤ max |p(»i )| x(0) ’ x

(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

remains to be discussed. It is known from (2.56) that also for the existence

of an LU factorization certain conditions are necessary, as for example the

M-matrix property. This is even su¬cient for an ILU factorization.

Theorem 5.17 Let A ∈ Rm,m be an M-matrix. Then for a given pat-

tern E that satis¬es (5.80), an ILU factorization exists. The hereby de¬ned

decomposition of A as A = LU ’ R is regular in the following sense:

(LU )’1 ≥ 0, (R)ij ≥ 0 for all i, j = 1, . . . , m .

ij

2

Proof: See [16, p. 235].

An ILU (or IC) factorization can be de¬ned by solving the equations

(5.78) for lij and uij in an appropriate order. Alternatively, the elimination

or Cholesky method can be used in its original form on the pattern E.

An improvement of the eigenvalue distribution of C ’1 A is sometimes

possible by using an MIC factorization (modi¬ed incomplete Cholesky fac-

torization) instead of an IC factorization. In contrast to (5.79) the updates

in the elimination method for positions outside the pattern are not ignored

here but have to be performed for the corresponding diagonal element.

Concerning the reduction of the condition number by the ILU or IC

preconditioning for the model problem, we have the same situation as for

the SSOR preconditioning. In particular (5.77) holds, too.

The auxiliary system of equations with C = N ’1 , which means that

h(k+1) = N g (k+1) ,

can also be interpreted as an iteration step of the iteration method de¬ned

by N with initial value z (0) = 0 and right-hand side g (k+1) . An expansion

of the discussed possibilities for preconditioning is therefore obtained by

using a ¬xed number of iteration steps instead of only one.

5.4. Krylov Subspace Methods for Nonsymmetric Systems of Equations 233

Exercises

5.7 Let A1 , A2 , . . . , Ak , C1 , C2 , . . . , Ck ∈ Rm,m be symmetric positive

semide¬nite matrices with the property

axT Ci x ¤ xT Ai x ¤ bxT Ci x for x ∈ Rm , i = 1, . . . , k and 0 < a ¤ b .

Prove: If A := k Ai and C := k Ci are positive de¬nite, then the

i=1 i=1

spectral condition number κ of C ’1 A satis¬es

b

κ(C ’1 A) ¤ .

a

5.8 Show that the matrix

«

2 1 1

1 1

2

A :=

1 1 2

is positive de¬nite and its spectral condition number is 4.

Hint: Consider the associated quadratic form.

5.9 Investigate the convergence of the (P)CG method on the basis of

Theorem 3.45 and distinguish between d = 2 and d = 3.

5.4 Krylov Subspace Methods

for Nonsymmetric Systems of Equations

With the di¬erent variants of the PCG method we have methods that

are quite appropriate ” regarding their complexity ” for those systems

of equations that arise from the discretization of boundary value prob-

lems. However, this holds only under the assumption that the system

matrix is symmetric and positive de¬nite, reducing the possibilities of ap-

plication, for example to ¬nite element discretizations of purely di¬usive

processes without convective transport mechanism (see (3.23)). Exceptions

for time-dependent problems are only the (semi-)explicit time discretization

(compare (7.72)) and the Lagrange“Galerkin method (see Section 9.4). For

all other cases the systems of equations that arise are always nonsymmetric

and positive real, which means that the system matrix A satis¬es

A + AT is positive de¬nite.

It is desirable to generalize the (P)CG methods for such matrices. The CG

method is characterized by two properties:

• The iterate x(k) minimizes f (·) = · ’x on x(0) + Kk A; g (0) ,

A

where x = A’1 b.

234 5. Iterative Methods for Systems of Linear Equations

• The basis vectors d(i) , i = 0, . . . , k ’ 1, of Kk A; g (0) do not have to

be calculated in advance (and stored in the computer), but will be

calculated by a three-term recursion (5.61) during the iteration. An

analogous relation holds by de¬nition for x(k) (see (5.48)).

The ¬rst property can be preserved in the following, whereby the norm

of the error or residual minimization varies in each method. The second

property is partially lost, because generally all basis vectors d(0) , . . . , d(k’1)

are necessary for the calculation of x(k) . This will result in memory space

problems for large k. As for the CG methods, preconditioning will be nec-

essary for an acceptable convergence of the methods. The conditions for

the preconditioning matrices are the same as for the CG method with the

exception of symmetry and positive de¬niteness. All three methods of pre-

conditioning are in principle possible. Therefore, preconditioning will not

be discussed in the following; we refer to Section 5.3.

The simplest approach is the application of the CG method to a system

of equations with symmetric positive de¬nite matrix equivalent to (5.1).

This is the case for the normal equations

AT Ax = AT b . (5.81)

The approach is called CGNR (Conjugate Gradient Normal Residual),

because here the iterate x(k) minimizes the Euclidean norm of the residual

on x(0) + Kk AT A; g (0) with g (0) = AT Ax(0) ’ b . This follows from the

equation

y’x = (Ay ’ b)T (Ay ’ b) = |Ay ’ b|2

2

(5.82)

AT A 2

for any y ∈ Rm and the solution x = A’1 b.

All advantages of the CG method are preserved, although in (5.53) and

(5.65) Ad(k) is to be replaced by AT Ad(k) . Additionally to the doubling of

the number of operations this may be a disadvantage if κ2 (A) is large, since

κ2 (AT A) = κ2 (A)2 can lead to problems of stability and convergence. Due

to (5.34) this is to be expected for a large number of degrees of freedom.

Furthermore, in the case of list-based storage one of the operations Ay

and AT y is always very expensive due to searching. It is even possible

that we do not explicitly know the matrix A but that only the mapping

y ’ Ay can be evaluated, which then disquali¬es this method completely

(see Exercise 8.6).

The same drawback occurs if

AAT x = b

˜ (5.83)

with the solution x = A’T x taken instead of (5.81). If x(k) is the kth iterate

˜ ˜

of the CG method applied to (5.83), then the x := AT x(k) minimizes the

(k)

˜

T T (0)

residual in the Euclidean norm on x0 + A Kk AA ; g : Note that

T 2

2

y’x = AT y ’ x AT y ’ x = AT y ’ x

˜˜ ˜ ˜ ˜

AAT 2

5.4. Krylov Subspace Methods for Nonsymmetric Equations 235

Let g (0) ∈ Rm , g (0) = 0 be given, Set

v1 := g (0) / |g (0) |2 .

For j = 1, . . . , k calculate

T

hij := vi Avj for i = 1, . . . , j ,

j

Avj ’

wj := hij vi ,

i=1

|wj |2 .

hj+1,j :=

If hj+1,j = 0, termination; otherwise, set

vj+1 := wj /hj+1,j .

Table 5.4. Arnoldi algorithm.

holds for any y ∈ Rm and g (0) = Ax(0) ’ b. This explains the terminology

˜

CGNE (with E for Error).

Whether a method minimizes the error of the residual obviously depends

on the norm used. For a symmetric positive de¬nite B ∈ Rm,m , any y ∈ Rm ,

and x = A’1 b, we have

Ay ’ b = y’x .

AT BA

B

For B = A’T and a symmetric positive de¬nite A we get the situation of

the CG method:

Ay ’ b = y’x .

A’T A

For B = I we get again (5.82):

|Ay ’ b|2 = y ’ x AT A .

The minimization of this functional on x(0) + Kk A; g (0) (not

Kk AT A; g (0) ) leads to the GMRES method (Generalized Minimum

RESidual).

This (and other) methods are founded algorithmically on the recur-

sive construction of orthonormal bases of Kk A; g (0) by Arnoldi™s method.

This method combines the generation of a basis according to (5.61) and

Schmidt™s orthonormalization (see Table 5.4).

If Arnoldi™s method can be performed up to the index k, then

hij := 0 for j = 1, . . . , k, i = j + 2, . . . , k + 1 ,

(hij )ij ∈ Rk,k ,

Hk :=

(hij )ij ∈ Rk+1,k ,

¯

Hk :=

(v1 , . . . , vk+1 ) ∈ Rm,k+1 .

Vk+1 :=

The matrix Hk is an upper Hessenberg matrix (see Appendix A.3). The

basis for the GMRES method is the following theorem:

236 5. Iterative Methods for Systems of Linear Equations

Theorem 5.18 If Arnoldi™s method can be performed up to the index k,

then

(1) v1 , . . . , vk+1 form an orthonormal basis of Kk+1 (A; g (0) ).

(2)

¯

AVk = Vk Hk + wk eT = Vk+1 Hk , (5.84)

k

with ek = (0, . . . , 0, 1)T ∈ Rk ,

VkT AVk = Hk . (5.85)

(3) The problem

|Ay ’ b|2 y ∈ x(0) + Kk (A; g (0) )

Minimize for

with minimum x(k) is equivalent to

Hk ξ ’ βe1 ξ ∈ Rk

¯

Minimize for (5.86)

2

with β := ’ g (0) and minimum ξ (k) , and we have

2

x(k) = x(0) + Vk ξ (k) .

If Arnoldi™s method terminates at the index k, then

x(k) = x = A’1 b .

Proof: (1): The vectors v1 , . . . , vk+1 are orthonormal by construction;

hence we have only to prove vi ∈ Kk+1 A; g (0) for i = 1, . . . , k + 1. This

follows from the representation

with polynomials qi’1 ∈ Pi’1 .

vi = qi’1 (A)v1

In this form we can prove the statement by induction with respect to k.

For k = 0 the assertion is trivial. Let the statement hold for k ’ 1. The

validity for k then follows from

k k

hk+1,k vk+1 = Avk ’ Aqk’1 (A) ’

hik vi = hik qi’1 (A) v1 .

i=1 i=1

(2): Relation (5.85) follows from (5.84) by multiplication by VkT , since

VkT Vk = I and VkT wk = hk+1,k VkT vk+1 = 0 due to the orthonormality

of the vi .

The relation in (5.84) is the matrix representation of

j j+1

Avj = hij vi + wj = hij vi for j = 1, . . . , k .

i=1 i=1

(3): Due to (1), the space x(0) + Kk A; g (0) has the parametrisation

ξ ∈ Rk .

y = x(0) + Vk ξ with (5.87)

5.4. Krylov Subspace Methods for Nonsymmetric Equations 237

The assertion is a consequence of the identity

Ay ’ b = A x(0) + Vk ξ ’ b = AVk ξ + g (0)

Vk+1 Hk ξ ’ βv1 = Vk+1 Hk ξ ’ βe1 ,

¯ ¯

=

which follows from (2), since it implies

|Ay ’ b|2 = Vk+1 (Hk ξ ’ βe1 ) = Hk ξ ’ βe1

¯ ¯

2 2

due to the orthogonality of Vk+1 . The last assertion ¬nally can be seen in

this way: If Arnoldi™s method breaks down at the index k, then relation (2)

becomes

AVk = Vk Hk ,

and

¯

AVk = Vk+1 Hk

will further hold with vk+1 chosen arbitrarily (due to hk+1,k = 0). Since A

is nonsingular, this also holds for Hk . Hence the choice

’1

ξ := Hk (βe1 ) ,

which satis¬es

Hk ξ ’ βe1 = |Hk ξ ’ βe1 |2 = 0 ,

¯

2

is possible. Hence the corresponding y ∈ Rm de¬ned by (5.87) ful¬lls y =

2

x(k) = x.

One problem of Arnoldi™s method is that the orthogonality of the vi is

easily lost due to rounding errors. If one substitutes the assignment

j

wj := Avj ’ hij vi

i=1

in Table 5.4 by the operations

wj := Avj ,

for i = 1, . . . , j calculate

T

hij := wj vi ,

wj := wj ’ hij vi ,

which de¬ne the same vector, one obtains the modi¬ed Arnoldi™s method.

From this relation and from (5.86) the GMRES method is constructed in

its basic form. Alternatively, Schmidt™s orthonormalization can be replaced

by the Householder method (see [28, pp. 159 ¬.]). With exact arithmetic

the GMRES algorithm terminates only after reaching the exact solution

(with hk+1,k = 0). This is not always the case for alternative methods

of the same class. For an increasing iteration index k and large problem

dimensions m there may be lack of enough memory for the storage of

238 5. Iterative Methods for Systems of Linear Equations

the basis vectors v1 , . . . , vk . A remedy is o¬ered by working with a ¬xed

number n of iterations and then to restart the algorithm with x(0) := x(n)

and g (0) := Ax(0) ’ b, until ¬nally the convergence criterion is ful¬lled

(GMRES method with restart). There is also a truncated version of the

GMRES method, in which only the last n basis vectors are used. The

minimization of the error in the energy norm (on the vector space K)

as with the CG method makes sense only for symmetric positive de¬nite

matrices A. But the variational equation

(Ay ’ b)T z = 0 for all z ∈ K

that characterizes this minimum in general can be taken as de¬ning con-

dition for y. Further variants of Krylov subspace methods rely on this.

Another large class of such methods is founded on the Lanczos biorthogo-

nalization, in which apart from a basis v1 , . . . , vk of Kk (A; v1 ) another basis

w1 , . . . , wk of Kk (AT ; w1 ) is constructed, such that

T

vj wi = δij for i, j = 1, . . . , k .

The best-known representative of this method is the BICGSTAB method.

For further discussion of this topic see, for example, [28].

Exercises

5.10 Consider the linear system Ax = b, where A = ±Q for some ± ∈

R \ {0} and some orthogonal matrix Q. Show that, for an arbitrary initial

iterate x(0) , the CGNE method terminates after one step with the exact

solution.

5.11 Provided that Arnoldi™s method can be performed up to the index

k, show that it is possible to incorporate a convergence test of the GMRES

method without computing the approximate solution explicitely, i.e., prove

the following formulas:

g (k) := Ax(k) ’ b = hk+1,k eT ξ (k) vk+1 ,

k

|g (k) |2 = hk+1,k |eT ξ (k) | .

k

5.5 The Multigrid Method

5.5.1 The Idea of the Multigrid Method

We discuss again the model problem of the ¬ve-point stencil discretization

for the Poisson equation on the square and use the relaxed Jacobi™s method.

Then due to (5.31) the iteration matrix is

ω

M = ωMJ + (1 ’ ω)I = I ’ A ,

4

5.5. Multigrid Method 239

with A being the sti¬ness matrix according to (1.14). For ω = ω/4 this

˜

coincides with the relaxed Richardson method, which according to (5.35)

has the poor convergence behaviour of Jacobi™s method, even for optimal

choice of the parameter. Nevertheless, for a suitable ω the method has

positive properties. Due to (5.25) the eigenvalues of M are

ω kπ lπ

»k,l = 1 ’ ω + 1 ¤ k, l ¤ n ’ 1 .

cos + cos ,

2 n n

This shows that there is a relation between the size of the eigenvalues and

the position of the frequency of the assigned eigenfunction depending on

the choice of ω: For ω = 1, which is Jacobi™s method, (M ) = »1,1 =

’»n’1,n’1 . Thus the eigenvalues are large if k and l are close to 1 or n.

Hence there are large eigenvalues for eigenfunctions with low frequency as

well as for eigenfunctions with high frequency. For ω = 1 , however, we have

2

(M ) = »1,1 , and the eigenvalues are large only in the case that k and l

are near to 1, which means that the eigenfunctions have low frequency.

In general, if the error of the iterate e(k) had a representation in terms of

orthonormal eigenvectors zν with small eigenvalues, as for example |»ν | ¤

1

2,

e(k) = cν z ν ,

ν:|»ν |¤ 1

2

then according to (5.11) it would follow for the error measured in the

Euclidean vector norm | · |2 that

« 1/2

»2 c2