The Arithmetic Mean method for solving systems

of nonlinear equations in finite differences ∗

Emanuele Galligani

Dipartimento di Matematica Pura e Applicata “G.Vitali”

Universita degli Studi di Modena e Reggio Emilia

Via Campi 213/b, 41100, Modena, Italy

Technical Report n. 68, September 2005

Dipartimento di Matematica, Universita di Modena e Reggio Emilia

Abstract. In this paper we consider the application of additive operatorsplitting methods for solving a finite difference nonlinear system of the formF (u) = (I − τA(u))u − w = 0 generated by the discretization of two dimen-sional diffusion–convection problems with Neumann boundary conditions.Existence and uniqueness of a solution of this system has been proved understandard assumptions on the matrix A(u) and the source term w.Using the fact that the matrix A(u) can be decomposed into different split-tings, we develop a nonlinear Arithmetic Mean method and a two–stage iter-ative method (a fixed–point–Arithmetic Mean method) for solving the systemabove. The convergence of these methods has been analyzed.Numerical experiments show that the fixed–point–Arithmetic Mean method israpidly convergent when the diffusion coefficient is weakly nonlinear.

Key Words: fixed–point iteration, splitting methods, nonlinear diffusion–convection problem, finite differences.

AMS Classification: 65H10, 65M06.

C.R. Categories: G.1.8, G.1.5.

∗This research was supported by the Italian Ministry of Education and Research(MIUR) projects: FIRB–2001 n. RBAU01877P, COFIN/PRIN–2003 n. 2003095748 andCOFIN/PRIN–2004 n. 2004012559.

1

Quaderni del Dipartimento di Matematica, Università di Modena e Reggio Emilia, n. 68, September 2005.

1 Statement of the problem

We are concerned with the numerical solution of nonlinear diffusion equationsby implicit finite difference methods. By an implicit method we mean the ap-proximation of the time derivative by a suitable backward difference quotient.Thus, the parabolic equation is replaced by a sequence of elliptic partial dif-ferential equations. The discretization of the spatial derivatives leads to thesolution of a sequence of systems of nonlinear algebraic equations.For example, the widely used θ–method (see e.g. [23]) is described by thefollowing system (l = 0, 1, ...):

(

I − θ∆tA(u(l+1)))

u(l+1) =(

I + (1− θ)∆tA(u(l)))

u(l)+

+∆t(

θs(l+1) + (1− θ)s(l))

u(0) = y0

(1)

where θ is a real parameter such that 0 ≤ θ ≤ 1; for any θ 6= 0, the method (1)is implicit.Here, u(l) ∈ R

n is an approximation to the restriction on the spatial grid of thesolution of the parabolic equation at the discrete value of time t = l∆t, where∆t is the time step and l is the time level, l = 0, 1, ....The matrix A(u(l)) arises from the discretization of the elliptic partial differen-tial operator at time level l.As a model problem we can use the two dimensional diffusion problem on therectangular domain R

∂ϕ

∂t=

∂

∂x

(

σ(ϕ)∂ϕ

∂x

)

+∂

∂y

(

σ(ϕ)∂ϕ

∂y

)

+ ψ(x, y, t) (2)

with the Neumann condition

∂ϕ

∂ν= 0 on ∂R ν is the normal to ∂R (3)

on the boundary ∂R of R and the initial condition

ϕ(x, y, 0) = ϕ0(x, y) (4)

in the closure R of R.The diffusion coefficient σ(ϕ) and the source term ψ(x, y, t) are real–valuedsufficiently smooth functions; besides σ(ϕ) > 0 in R.On the rectangular domain R we superimpose a uniform rectangular grid ofpoints Rh and we discretize on Rh the spatial derivatives using centered–inspace–difference equations.If R is the unit square, the spatial grid Rh has mesh width h = 1/N in bothdirections, with N the number of grid–points per direction.

2

The standard five–points discretization formula gives rise to a symmetric neg-ative semidefinite matrix A(u(l)), where the vector u(l) is the restriction ofϕ(x, y, t) on Rh at the time t = l∆t.For column ordering (by vertical lines) of the grid points, A(u(l)) is the blocktridiagonal matrix of order n = N ×N where each square block of the diagonalof A(u(l)) is a diagonal dominant tridiagonal matrix of order N and each blockon the subdiagonal and on the superdiagonal of A(u(l)) is a diagonal matrix oforder N .By omitting the index l, we can express the matrix A(u) as

A(u) =

D1(u) C1(u)B2(u) D2(u) C2(u)

. . .. . .

. . .

BN−1(u) DN−1(u) CN−1(u)BN (u) DN(u)

(5)

Using only one index for numbering the grid points Pi, i = 1, ..., n, the elementsaij(u) of the matrix A(u) are given by

ai,j(u) =

σi+σj

2h2 j ∈ N (i)

−∑

k∈N (i)σi+σk

2h2 j = i

0 otherwise

(6)

where σi denotes an approximation to σ(ϕ) with ϕ evaluated at the grid pointsPi and time t = l∆t, and N (i) denotes the four adjacent points of Pi corre-sponding to the five–points difference stencil.Boundary points have only inner points as neighbours, as a result of the bound-ary condition (3).Our central topic is the solvability of the nonlinear algebraic equations (1) whenthe problem (2)–(4) is approximated in a finite dimensional space, by using finitedifferences.Precisely, at each time level l we have to solve a system of nonlinear equationsof the form

(I − τA(u))u = w (7)

where w ∈ Rn is a specified source term and τ = θ∆t.

Here, we have dropped in (1) the subscript l and l+ 1 on u and s for a simplerpresentation.We assume that the n × n matrix A(u), with elements aij(u), i, j = 1, ..., n,satisfies the following properties

C1) irreducibility for all u ∈ Rn (see e.g. [26, p. 18]);

C2) vanishing column sum:

n∑

i=1

ai,j(u) = 0 for j = 1, ..., n and all u ∈ Rn;

3

C3) negative diagonal entries and nonnegative off–diagonal entries for all u ∈R

n:aii(u) < 0 and aij(u) ≥ 0 for i 6= j and all u ∈ R

n;

C4) Lipschitz–continuity of A(u) for every bounded subset of Rn:

‖A(u)−A(v)‖ ≤ Λ‖u− v‖

for all u, v of subset.

A solution u∗ of the system (7) is a root of the nonlinear equation

F (u) = (I − τA(u))u −w = 0 τ > 0 (8)

We will study these equations (8) by taking into account the fact that the matrixA(u) can be decomposed into different splittings.We can express the matrix A(u) as the matrix sum

A(u) = A1(u) +A2(u) (9)

or as the two splittings

A(u) = H1(u) +K1(u) = H2(u) +K2(u) (10)

where A1(u) and PA2(u)PT are two tridiagonal matrices and P is the permu-tation matrix which reorders the vector u as Pu, that is:

u = (u11, ..., u1N , ..., uN1, ..., uNN)T ;Pu = (u11, ..., uN1, ..., u1N , ..., uNN )T

and, if N is even

H1(u) =

0

B

B

B

B

B

B

B

B

B

@

D1(u) C1(u)B2(u) C2(u)

D3(u) C3(u)B4(u) D4(u)

. . .

DN−1(u) CN−1(u)BN (u) DN (u)

1

C

C

C

C

C

C

C

C

C

A

(11)

and, consequentlyK1(u) = A(u)−H1(u) (12)

H2(u) =

0

B

B

B

B

B

B

B

B

B

@

D1(u)D2(u) C2(u)B3(u) D3(u)

. . .

DN−2(u) CN−2(u)BN−1(u) DN−1(u)

DN (u)

1

C

C

C

C

C

C

C

C

C

A

(13)

4

and,K2(u) = A(u)−H2(u) (14)

If N is odd, we can proceed in a similar way.The matrix A(u) satisfies the properties C1)–C4) and, in addition, the property

C5) symmetry

aij(u) = aji(u) for i, j = 1, ..., n and all u ∈ Rn;

Also the matrices A1(u) and A2(u) satisfy the properties C1)–C5).If equation (2) contains the effect of convection with the term v · gradϕ, formesh spacings h sufficiently small the matrices A(u), A1(u) and A2(u) satisfythe properties C1)–C4).The matrices H1(u) and H2(u) are diagonally dominant and have negativediagonal entries and nonnegative off–diagonal entries, K1(u) and K2(u) aretwo nonnegative matrices.The monotonicity property of the discrete operator in (8) and the operatorsplitting strategy play an important role in the study of the nonlinear diffusionproblem (2)–(4) (see e.g. [6], [18]).The organization of this article is as follows. In section 2, using the Brouwer’sfixed–point theorem ([5, p. 232], [16], [21, p. 161]), under the assumption thatthe components of the vectors u and w are nonnegative (u ∈ R

n+, w ∈ R

n+),

we prove a basic existence result for the system of nonlinear equations (7) forarbitrarily large time steps, while uniqueness is proved for limited time steps.Since the Arithmetic Mean method has been applied for the solution of weaklynonlinear systems (see [7], [8], [9], [10]), in sections 3 and 4, we analyze thepossibility of applying the Arithmetic Mean method for solving the nonlinearsystem (7).In particular, in Section 3, using the fact that the matrix A(u) can be expressedas the matrix sum (9), we introduce a nonlinear Arithmetic Mean method forsolving the nonlinear system (7) and a convergence theorem is proved.Moreover, in Section 4, a procedure is presented for solving the nonlinear system(7) without the use of jacobian matrices or their difference approximations,but by the repeated solution of linear systems whose coefficient matrix is amonotone matrix (M–matrix) ([26, p. 93], [20, p. 108]). Such a solution may bedetermined efficiently by the parallel splitting methods described in [3], [4], [19],[24], [25], [14] or by the conjugate gradient method with the parallel additivepreconditioners described in [1], [11] [12], [13] when the matrix A(u) satisfies theproperties C1)–C5). We analyze the possibility of solving these linear systemsonly approximately. A fixed–point–Arithmetic Mean method is introduced.Some numerical experiments, reported in Section 5, show the effectiveness ofthis fixed–point–Arithmetic Mean method.

5

2 An existence and uniqueness theorem

We establish an existence and uniqueness theorem about the solution u∗ of (8)under the assumption that the components of the vector u and w are nonneg-ative: u ∈ R

n+, w ∈ R

n+.

Theorem 1. Let F : Rn+ → R

n+ be a mapping of the form (8) on which the

conditions C1)–C4) hold for the matrix A(u) for all u ∈ Rn+ and the source

term w is nonnegative: w ≥ 0.Let Ω ⊂ R

n+ be the subset

Ω = u ∈ Rn+ : ‖u‖1 = ‖w‖1 (15)

Then, the system (8) has a solution u∗ ∈ Ω.Furthermore, if τΛ‖w‖1 < 1, this solution is unique.Proof. Define the matrix T (u) = I − τA(u) for all u ∈ R

n+. By the properties

C1), C2) and C3) on A(u), T (u) is an irreducibly diagonally dominant matrix([26, p. 23]) with positive diagonal and nonpositive off–diagonal entries for allu ∈ R

n+. Thus, T (u) is an irreducible M–matrix and T (u)−1 ≥ 0 for all u ∈ R

n+

([20, p. 110]).By the property C2), eTA(u) = 0T where e ∈ R

n is the vector whose compo-nents are all 1.Thus, eTT (u) = eT − τeTA(u) = eT and

eT = eTT (u)−1

Define the mapping

u = G(u) with G(u) = (I − τA(u))−1w (16)

for all u ∈ Rn+.

By hypothesis, w ≥ 0. Thus, G(u) = T (u)−1w ≥ 0.We have

eT G(u) = eTT (u)−1w = eT w = ‖w‖1

Therefore,‖G(u)‖1 = ‖w‖1

Consider the set Ω defined by (15). It is a bounded, closed and convex subsetof R

n+.

Thus, the mapping G(u) can be viewed as a continuous mapping from Ω to Ω.By the Brouwer fixed–point theorem, it follows that the mapping G : Ω → Ωhas at least one fixed point u∗ in the set Ω, that is a solution of the system (8).For all u,v ∈ Ω we have

G(u)−G(v) = (I − τA(u))−1w − (I − τA(v))−1w

= (I − τA(u))−1 ((I − τA(v)) − (I − τA(u))) ·

·(I − τA(v))−1w

= τ(I − τA(u))−1(A(u)−A(v)) ·

·(I − τA(v))−1w

6

By definition, the norm ‖ · ‖1 of an n× n matrix H = (hij) is

‖H‖1 = max1≤j≤n

n∑

i=1

|hij |

Since (I − τA(u))−1 ≥ 0 and eT (I − τA(u))−1 = eT , it follows that

‖(I − τA(u))−1‖1 = 1

Therefore,

‖G(u)−G(v)‖1 ≤ τ‖(I − τA(u))−1‖1‖A(u)− A(v)‖1 ·

·‖(I − τA(v))−1‖1‖w‖1

= τ‖A(u)−A(v)‖1‖w‖1

By the property C4) on A(u), we obtain

‖G(u)−G(v)‖1 ≤ (τΛ‖w‖1)‖u− v‖1

If (τΛ‖w‖1) < 1, the mapping G(u) is contractive on the set Ω. Then, G hasa unique fixed point in Ω ([21, p. 120]).Hence, it follows that the solution u∗ of (8) is unique ♯.

3 A nonlinear Arithmetic Mean method

We assume that the matrix A(u) in (7) is extremely large and sparse and thatit can be decomposed in the form (9)

A(u) = A1(u) +A2(u)

where the matrices A1(u) = (a(1)ij (u)) and A2(u) = (a

(2)ij (u)) satisfy the prop-

erties C1)–C4).In this case we analyze the possibility of using the Arithmetic Mean method([24]) for solving the nonlinear system (8).Using the splitting (9), we can write the system (8) in the form (ρ > 0, τ > 0):

F (u) = (I − τA1(u)− τA2(u))u −w = 0

that is

F (u) = ((1 + ρ)I − τA1(u))u − (w + (ρI + τA2(u))u)

= ((1 + ρ)I − τA2(u))u − (w + (ρI + τA1(u))u) = 0

7

We can define the following iterative method

set u(0) = w; ρ > 0

for k = 0, 1, ... until convergence do

z1 = ((1 + ρ)I − τA1(u(k)))−1(w + (ρI + τA2(u

(k)))u(k))

z2 = ((1 + ρ)I − τA2(u(k)))−1(w + (ρI + τA1(u

(k)))u(k))

u(k+1) = 12 (z1 + z2)

(17)

Formula (17) can be written in the form

z1 = ((1 + ρ)I − τA1(u(k)))−1

“

((1 + ρ)I − τA1(u(k)))u(k) − F (u(k))

”

= u(k) − ((1 + ρ)I − τA1(u

(k)))−1F (u(k))

z2 = u(k) − ((1 + ρ)I − τA2(u

(k)))−1F (u(k))

u(k+1) = u

(k) −M(u(k))−1F (u(k))

where

M(u(k))−1 =1

2

“

((1 + ρ)I − τA1(u(k)))−1 + ((1 + ρ)I − τA2(u

(k)))−1”

We state a convergent result for this method.By the definition of ‖ · ‖1 and by the properties C2) and C3) we have

‖A1(u)‖1 = 2 max1≤i≤n

|a(1)ii (u)|

‖A2(u)‖1 = 2 max1≤i≤n

|a(2)ii (u)|

By the property C4) on A1(u) and A2(u), the functions ‖A1(u)‖1 and ‖A2(u)‖1are continuous on the bounded and closed set Ω defined by (15).Thus, we can define

α1 = supu∈Ω

‖A1(u)‖1; α2 = supu∈Ω

‖A2(u)‖1; α ≥1

2maxα1, α2 (18)

By the property C4) on A1(u) and A2(u), we have for all u,v ∈ Ω:

‖A1(u)−A1(v)‖1 ≤ Λ1‖u− v‖1

‖A2(u)−A2(v)‖1 ≤ Λ2‖u− v‖1

(19)

and‖A1(u)u−A1(v)v‖1 ≤ L1‖u− v‖1

‖A2(u)u−A2(v)v‖1 ≤ L2‖u− v‖1

(20)

8

whereL1 = Λ1‖w‖1 + α1

L2 = Λ2‖w‖1 + α2

(21)

Indeed, using (19), we can write for all u,v ∈ Ω:

‖A1(u)u − A1(v)v‖1 = ‖A1(u)u − A1(v)u + A1(v)u −A1(v)v‖1≤ ‖A1(u)− A1(v)‖1‖u‖1 + ‖A1(v)‖1‖u − v‖1≤ (Λ1‖u − v‖1) ‖u‖1 + ‖A1(v)‖1‖u − v‖1≤ (Λ1‖w‖1 + α1) ‖u − v‖1

since ‖u‖1 = ‖w‖1.Similarly, we have for all u,v ∈ Ω:

‖A2(u)u−A2(v)v‖1 ≤ (Λ2‖w‖1 + α2) ‖u− v‖1

Let τ(1)∗ and τ

(2)∗ be the positive roots of the equations

a1τ2 + b1τ − 1 = 0

a2τ2 + b2τ − 1 = 0

respectively, where

a1 = α2Λ1‖w‖1 + α (Λ1‖w‖1 + L2)b1 = (Λ1‖w‖1 + L2)− αa2 = α1Λ2‖w‖1 + α (Λ2‖w‖1 + L1)b2 = (Λ2‖w‖1 + L1)− α

A criterion on the convergence of the method (17) is provided by the followingtheorem.Theorem 2. Let F : R

n+ → R

n+ be a mapping of the form (8)–(9) on which the

conditions C1)–C4) hold for the matrices A1(u) and A2(u) for all u ∈ Rn+ and

the source term w is nonnegative, w ≥ 0.Let Ω ⊂ R

n+ be the subset defined by (15).

If the two positive parameters τ and ρ satisfy the condition

τ ≤ minτ(1)∗ , τ

(2)∗ ; ρ = ατ (22)

then the iterative method (17) is convergent to a solution u∗ of the system (8).Proof. Define the matrices T1(u) = (1 + ρ)I − τA1(u) and T2(u) = (1 + ρ)I −τA2(u) for all u ∈ R

n+.

By the properties C1), C2) and C3) on A1(u) and A2(u), T1(u) and T2(u) aretwo irreducibly diagonally dominant matrices with positive diagonal entries andnonpositive off–diagonal entries for all u ∈ R

n+. Thus, T1(u) and T2(u) are two

irreducible M–matrices and T1(u)−1 ≥ 0 and T2(u)−1 ≥ 0 for all u ∈ Rn+ [20,

p. 110].

9

By the property C2) we have eTA1(u) = 0T and eTA2(u) = 0T . Thus,eTT1(u) = (1 + ρ)eT − τeTA1(u) = (1 + ρ)eT and eT = (1 + ρ)eTT1(u)−1.Similarly, eT = (1 + ρ)eTT2(u)−1.Consider the set Ω defined by (15). Define the mapping

u = G(u) with G(u) =1

2(G1(u) + G2(u)) (23)

andG1(u) = ((1 + ρ)I − τA1(u))−1 (w + (ρI + τA2(u))u)

G2(u) = ((1 + ρ)I − τA2(u))−1 (w + (ρI + τA1(u))u)(24)

for all u ∈ Rn+.

By hypothesis w ≥ 0. Besides, for ρ = ατ , the matrices (ρI+τA2(u)) and (ρI+τA1(u)) are nonnegative for all u ∈ Ω. Therefore, the vectors (ρI + τA2(u))uand (ρI + τA1(u))u are nonnegative for all u ∈ Ω. Thus, for all u ∈ Ω:

G1(u) = T1(u)−1(w + (ρI + τA2(u))u) ≥ 0

G2(u) = T2(u)−1(w + (ρI + τA1(u))u) ≥ 0

We have

eT G1(u) = eTT1(u)−1 (w + ρu + τA2(u)u)

=1

1 + ρ

(

eT w + ρeT u + τeTA2(u)u)

=1

1 + ρ

(

eT w + ρeT u + τ0T u)

Then, since u ∈ Ω,

‖G1(u)‖1 =1

1 + ρ(‖w‖1 + ρ‖u‖1)

=1

1 + ρ(‖w‖1 + ρ‖w‖1) = ‖w‖1

Similarly‖G2(u)‖1 = ‖w‖1

and‖G(u)‖1 = ‖w‖1

Thus, the mapping (23)–(24) can be viewed as a continuous mapping from Ω toΩ.The set Ω is a bounded, closed and convex subset of R

n+. By the Brouwer fixed–

point theorem, it follows that the mapping G : Ω → Ω has at least one fixedpoint u∗ in the set Ω.

10

This point u∗ is a solution of the system (8). Indeed, if u∗ is a solution of thesystem (8), u∗ is a solution of the systems

F 1(u) = ((1 + ρ)I − τA1(u))u− (w + (ρI + τA2(u))u) = 0

F 2(u) = ((1 + ρ)I − τA2(u))u− (w + (ρI + τA1(u))u) = 0(25)

that is, a fixed point of the mapping (23)–(24).Conversely, if u∗ is a fixed point of the mapping (23)–(24), it is a solution ofthe system

(u −G1(u)) + (u −G2(u)) = 0

orT1(u)−1F 1(u) + T2(u)−1F 2(u) = 0

Since T1(u∗)−1 ≥ 0, T2(u

∗)−1 ≥ 0, F 1(u∗) = F 2(u

∗) = F (u∗), we obtainF (u∗) = 0.For all u,v ∈ Ω we have

G1(u)−G1(v) = T1(u)−1 (w + (ρI + τA2(u))u)− T1(v)−1 (w+

(ρI + τA2(v))v)

= T1(u)−1 (w + (ρI + τA2(u))u)− T1(v)−1 (w+

+(ρI + τA2(u))u) + T1(v)−1 (w + (ρI + τA2(u))u)−−T1(v)−1 (w + (ρI + τA2(v))v)

= T1(u)−1 (T (v)− T (u)) T1(v)−1 (w + (ρI + τA2(u))u) +

+T1(v)−1 ((ρI + τA2(u))u − (ρI + τA2(v))v)

= τT1(u)−1 (A1(u)− A1(v)) T1(v)−1 (w + (ρI + τA2(u))u) +

+T1(v)−1 (ρ(u − v) + τ (A2(u)u − A2(v)v))

Similarly, for all u,v ∈ Ω

G2(u)−G2(v) = τT2(u)−1 (A2(u)− A2(v)) T2(v)−1 (w + (ρI + τA1(u))u) +

+T2(v)−1 (ρ(u − v) + τ (A1(u)u − A1(v)v))

Since T1(u)−1 ≥ 0 and (1 + ρ)eTT1(u)−1 = eT , it follows that

‖T1(u)−1‖1 =1

1 + ρ

Similarly

‖T2(u)−1‖1 =1

1 + ρ

Using (19) and (20), for all u,v ∈ Ω we obtain

‖G1(u)−G1(v)‖1 ≤ τ‖T1(u)−1‖1‖A1(u)−A1(v)‖1‖T1(v)−1‖1 ·

· (‖w‖1 + ρ‖u‖1 + τ‖A2(u)‖1‖u‖1) + ρ‖T1(v)−1‖1 ·

·‖u− v‖1 + τ‖T1(v)−1‖1‖A2(u)u−A2(v)v‖1

11

=τ

(1 + ρ)2‖A1(u)−A1(v)‖1 ((1 + ρ)‖w‖1 + τ‖A2(u)‖1‖w‖1) +

+ρ

1 + ρ‖u− v‖1 +

τ

1 + ρ‖A2(u)u−A2(v)v‖1

≤

(

τΛ1‖w‖11 + ρ

+τ2Λ1α2‖w‖1

(1 + ρ)2+

ρ

1 + ρ+

τL2

1 + ρ

)

‖u− v‖1

and

‖G2(u)−G2(v)‖1 ≤ τ‖T2(u)−1‖1‖A2(u)−A2(v)‖1‖T2(v)−1‖1 ·

· (‖w‖1 + ρ‖u‖1 + τ‖A1(u)‖1‖u‖1) + ρ‖T2(v)−1‖1 ·

·‖u− v‖1 + τ‖T2(v)−1‖1‖A1(u)u−A1(v)v‖1

≤

(

τΛ2‖w‖11 + ρ

+τ2Λ2α1‖w‖1

(1 + ρ)2+

ρ

1 + ρ+

τL1

1 + ρ

)

‖u− v‖1

If(

τΛ1‖w‖11 + ρ

+τ2Λ1α2‖w‖1

(1 + ρ)2+

ρ

1 + ρ+

τL2

1 + ρ

)

< 1

and(

τΛ2‖w‖11 + ρ

+τ2Λ2α1‖w‖1

(1 + ρ)2+

ρ

1 + ρ+

τL1

1 + ρ

)

< 1

the mapping G(u) is contractive on the set Ω.When τ and ρ satisfy the condition (22), it is easy to verify that these inequalitieshold.Therefore, if the positive parameters τ and ρ are chosen so that (22) holds, theiterative method (17) is convergent to the solution u∗ ∈ Ω of the system (8)[15, p. 67]. ♯

4 A fixed–point–Arithmetic Mean method

Theorem 1 suggests the use of the mapping (16) as an iteration mapping toapproximate the solution u∗ of the nonlinear system (7) ([15, p. 66], [20, p.153]).The procedure looks like

set u(0) = w;

for k = 0, 1, ... until convergence do

u(k+1) = (I − τA(u(k)))−1w

(26)

By the assumption that the continuous mapping G : Ω→ Ω is a contraction onΩ, that is

‖G(u)−G(v)‖1 ≤ λ‖u− v‖1

12

for all u ∈ Ω and λ = τΛ‖w‖1 < 1, the sequence of iterates

u(k+1) = G(u(k)) = (I − τA(u(k)))−1w k = 0, 1, ... (27)

satisfies

‖u(k+1) − u(k)‖1 = ‖G(u(k))−G(u(k−1))‖1 ≤ λ‖u(k) − u(k−1)‖1

and‖u(k+1) − u∗‖1 = ‖G(u(k))−G(u∗)‖1 ≤ λ‖u(k) − u∗‖1

where u∗ is the solution of (7) in Ω.Hence, if the system of linear equations in (26)

(I − τA(u(k)))u = w (28)

are solved exactly, the sequence u(k) will converge to the fixed point u∗ ofG(u) in Ω with linear convergence factor λ.For k ≥ 1, λ can be estimated by

λ ≃ λk = ‖∆u(k)‖1/‖∆u(k−1)‖1

where ∆u(k) = u(k+1) − u(k).The following error estimate may be obtained

‖u(k+1) − u∗‖1 = ‖u(k+1) − u

(k+2) + u(k+2) − u

(k+3) + u(k+3) − ...− u

∗‖1(29)

≤`

λ + λ2 + λ

3 + ...´

‖u(k) − u(k+1)‖1 =

λ

1− λ‖∆u

(k)‖1

This estimate is often used in practice for defining the convergence test of theiterative process (26) [20, p. 153].At each step of the iterative method (26), the linear system (28) must be solved.Computation of the exact solution u = u(k+1) can be too expensive if n islarge and, for any n, may not be justified when u(k) is relatively far from u∗.Therefore, one might prefer to compute some approximate solution of (28).There exists a considerable body of literature on the solvability of symmetric andnonsymmetric linear systems of the form (28), in which the coefficient matrix isan M–matrix.For this class of linear systems of monotone kind, splitting methods have beenstudied extensively, as linear solvers and as iterative preconditioners. Also theadaptability of these methods to vector–parallel computers has been taken intoaccount. For example, an approximate solution of (28) for many problems ofpractical interest may be efficiently determined on parallel computers by theoperator splitting methods described in [3], [4], [19], [24], [25], [14], or by theconjugate gradient method with the parallel additive polynomial preconditionerdescribed in [1] [12], [13].We assume that the matrix A(u) in (7) can be decomposed into two splittings(10). Then, we can determine an approximate solution of (28) with the Arith-metic Mean solver.

13

This leads to the following two–stage iterative method, that is the fixed–point–Arithmetic Mean method:

set u(0) = w; ρ > 0

for k = 0, 1, ... until convergence do

z(0) = 0

for j = 1, ..., jk do

(

(I − τH1(u(k))) + ρI

)

z1 = (ρI + τK1(u(k)))z(j−1) + w

(

(I − τH2(u(k))) + ρI

)

z2 = (ρI + τK2(u(k)))z(j−1) + w

z(j) = 12 (z1 + z2)

u(k+1) = z(jk)

(30)

The iterative solver defined by the loop over j is convergent (see [25, Theor. 1]).If we set

M(u)−1 =1

2

(

((I − τH1(u)) + ρI)−1 + ((I − τH2(u)) + ρI)−1)

H(u) = I −M(u)−1 (I − τA(u))

B(u) =

jk−1∑

j=0

H(u)jM(u)−1

we haveu(k+1) = B(u(k))w

The matrix B(u) is an approximant to (I − τA(u))−1.When we consider a two–stage iterative method the main interest is focusedon the analysis of the effect of the inner iteration on convergence of the outeriteration and on the development of strategies for a proper coordination of theinner and outer iterations in order to achieve a significant reduction of the totalcomputational effort.When the system (28) is solved only approximately, we have an inexact sequenceof iterates

u(k+1) = G(u(k)) = G(u(k)) + ηk+1 (31)

with u(0) = u(0).We assume that all u(k) belong to Ω.

14

Thus, the first question is concerned with the possibility to approximate, un-der proper conditions, a fixed point of G(u) in Ω to an accuracy determinedessentially by the accuracy of G(u) to G(u), u ∈ Ω.Since

ηk+1 = u(k+1) −G(u(k)) = u(k+1) − (I − τA(u(k)))−1w

= (I − τA(u(k)))−1(

(I − τA(u(k)))u(k+1) −w)

the accuracy of G(u(k)) to G(u(k)) can be expressed as

ηk+1 = (I − τA(u(k)))−1r(k) (32)

wherer(k) = (I − τA(u(k)))u(k+1) −w

is the residual error at the k–th outer iteration.The aim is to determine an error estimate for ‖u(k) − u∗‖1.We have

‖u(k) − u∗‖1 = ‖(u(k) − u(k)) + (u(k) − u∗)‖1

≤ ‖u(k) − u(k)‖1 + ‖u(k) − u∗‖1

Using (29) and the inequality

‖u(k+1) − u(k)‖1 ≤ λ‖u(k) − u(k−1)‖1

we can write

‖u(k) − u∗‖1 ≤λ

1− λ‖∆u(k−1)‖1 ≤

λ

1− λλ‖∆u(k−2)‖1 ≤ ...

≤λ

1− λλk−1‖∆u(0)‖1

Thus,

‖u(k) − u∗‖1 ≤ ‖u(k) − u(k)‖1 +

λk

1− λ‖∆u(0)‖1 (33)

Now, we determine an estimate for the deviation ‖u(k) − u(k)‖1.

We assume that ηk+1 = G(u(k))−G(u(k)) satisfies the condition (k = 1, 2, ...)

‖ηk+1‖1 ≤ γk+1λk‖∆u(0)‖1; ‖η1‖1 ≤ γ1‖∆u(0)‖1 (34)

where γk is a positive number less than one (such as γk = γξk with 0 < ξ < 1

and γ > 0), and ∆u(0) = u(1) − u(0).

Thus, starting with ‖u(0) − u(0)‖1 = 0, we find by mathematical induction fork ≥ 1

‖u(k) − u(k)‖1 ≤ βkλk−1‖∆u(0)‖1 (35)

15

where

βk =

k∑

j=1

γj

Indeed, for k = 1 we have by (31) and (27)

‖u(1) − u(1)‖1 = ‖η1 + G(u(0))−G(u(0))‖1 = ‖η1‖1

≤ γ1‖∆u(0)‖1 = β1λ0‖∆u(0)‖1

Assume that (35) holds up to index k − 1.For k we have by (31) and (27)

‖u(k) − u(k)‖1 = ‖ηk + G(u(k−1))−G(u(k−1))‖1

≤ ‖ηk‖1 + ‖G(u(k−1))−G(u(k−1))‖1

≤ ‖ηk‖1 + λ‖u(k−1) − u(k−1)‖1

Using (34) and (35) we can write

‖u(k) − u(k)‖1 ≤ γkλk−1‖∆u(0)‖1 + λ(βk−1λ

k−2‖∆u(0)‖1)

= λk−1(γk +

k−1∑

j=1

γj)‖∆u(0)‖1

= βkλk−1‖∆u(0)‖1

Thus, from (33) and (35) we obtain

‖u(k) − u∗‖1 ≤ βkλk−1‖∆u(0)‖1 +

λk

1− λ‖∆u(0)‖1 (36)

From (31) we have

u(1) = G(u(0)) + η1 = G(u(0)) + η1 = u(1) + η1

that is∆u(0) = u(1) − u(0) = u(1) + η1 − u(0) = ∆u(0) + η1

Hence, from (34)

‖∆u(0)‖1 ≤ ‖∆u(0)‖1 + ‖η1‖1 ≤ (1 + γ1)‖∆u(0)‖1

To conclude, we have obtained by (36) the following error estimate under theassumption (34):

‖u(k) − u∗‖1 ≤ λk−1

(

βk +λ(1 + γ1)

1− λ

)

‖∆u(0)‖1 (37)

where βk =∑k

j=1 γj is a convergent series when k diverges.

16

When λ is very close to 1, this error estimate is very large. This may cause poorperformance of the methods (26) and (30).Assumption (34) gives a condition on the precision to which the linear system(28) should be solved by the Arithmetic Mean method (defined by the loop overj in (30)) at the outer iteration k in order to have an error estimate (37).Actually, if we execute the loop over j in (30) up to obtain a residual error

r(k) = (I − τA(u(k)))z(jk) −w (38)

such that‖r(k)‖1 ≤ γk+1λ

k‖∆u(0)‖1 (39)

we have from (32) the satisfaction of condition (34)

‖ηk+1‖1 ≤ ‖(I − τA(u(k)))−1‖1‖r(k)‖1 ≤ γk+1λk‖∆u

(0)‖1

5 A Numerical Example

By the analysis above on the convergence of the two–stage iterative method (30),it is possible to develop a strategy for a proper coordination of the inner andouter iterations. However, this strategy has a high computational complexity.In many problems of practical interest, numerical experiments show that theadoption of the following strategy:

• to use the following termination criterion for the outer iteration k

‖u(k+1) − u(k)‖1 ≤ ϑk‖u(k) − u(k−1)‖1 ϑk ≤ ϑ < 1

‖F (u(k+1))‖1 ≤ k‖F (u(k))‖1 k ≤ ¯< 1

• to perform jk inner iterations with the Arithmetic Mean method until theratio between the inner and the outer residuals is less than a term lessthan one (forcing term);

makes the two–stage iterative method (30) robust and rapidly convergent forsolving diffusion problems of the type (2)–(4) with a diffusion coefficient σ(ϕ)weakly nonlinear.By this, we report the following theorem for the convergence of a sequencev(k) ([17, §6.3]).Theorem 3. Let K ⊂ R

n be a compact set1 and F : K → Rn be a continuous

mapping.Consider a sequence v(k) in K satisfying

‖v(k+1) − v(k)‖ ≤ ϑk‖v(k) − v(k−1)‖ ϑk ≤ ϑ < 1 (40)

‖F (v(k+1))‖ ≤ k‖F (v(k))‖ k ≤ ¯< 1 (41)

1For the computation it is no restriction to assure the entire iterative process remains in acompact, convex set K ⊂ R

n ([22, p. 71]).

17

Then, the sequence v(k) converges in K to a zero v∗ of F 2.Proof. By the triangle inequality and from (40), the sequence v(k) is aCauchy sequence:

‖v(k+p) − v(k)‖ = ‖

p−1X

i=0

(v(k+p−i) − v(k+p−i−1))‖ ≤

p−1X

i=0

‖v(k+p−i) − v(k+p−i−1))‖

≤p−1X

i=0

ϑp−i−1‖v(k+1) − v

(k))‖ ≤ 1− ϑp

1− ϑ‖v(k+1) − v

(k))‖

≤ ϑk

1− ϑ‖v(1) − v

(0))‖

Since the Cauchy sequence stays in the set K which is compact, hence complete([2, p. 74]), it must converge to a certain v∗ ∈ K.From

‖F (v(k))‖ ≤ k‖F (v(0))‖

with < 1, we get limk→∞ F (v(k)) = 0.By continuity of F we have

0 = limk→∞

F (v(k)) = F ( limk→∞

v(k)) = F (v∗) ♯

Note that in this theorem we have made no assumption on the differentiabilityof F and on how the sequence v(k) is generated.The first condition (40) ensures that the iterates v(k) converge.The second condition (41) ensures that these iterates converge to a zero of F (v).

Consider the system (7) which arises in solving the initial–boundary value prob-lem (2)–(3)–(4) by the θ–method, with θ = 1.The domainR is the square (0, 1)×(0, 1); the grid–points (xi, yj), (i, j = 1, ..., N)of Rh are xi = xi−1 + h and yi = yi−1 + h for i = 2, ..., N , with x1 = y1 = h/2and xN+1 = xN + h/2, yN+1 = yN + h/2.By using box–integration method ([26, §6.3, p. 204]) for the spatial term, thematrix A(u) has the form (5); its elements are given as in (6).In the numerical experiment, we assume that the diffusion coefficient σ(ϕ) hasthe expression σ(ϕ) = βϕp with p = 1

3 ,12 , 1 for different values of β.

The vector u∗ is the restriction of the solution

ϕ(x, y, t) = γ cos2(π

2x) cos2(

π

2y)

on Rh at fixed time t = l∆t.The parameter γ is chosen to satisfy the constraint ‖u∗‖1 = 1.In (7) the source term w is set

w = (I − τA(u∗))u∗

2Here ‖ · ‖ denotes any vector norm.

18

and the time–step τ is chosen in order to have w ≥ 0 and ‖w‖1 = 1.We consider the splittings (10) for the matrix A(u), where H1(u), K1(u) andH2(u), K2(u) are given by (11)–(12) and (13)–(14) respectively.

In the iterative method (30), the iterate u(k+1) is obtained by solving approxi-mately the linear system (28) with the Arithmetic Mean method.By using the 2N × 2N permutation matrix P , such that

Pv = (v1, vN+2, v3, ..., vN−1, v2N , vN+1, v2, vN+3, ..., v2N−1, vN )T

to the systems in (30), then the matrices

P

(

Di(u(k)) Ci(u

(k))

Bi+1(u(k)) Di+1(u

(k))

)

PT i = 1, ..., N − 1

have the diagonal blocks which are diagonal and tridiagonal off–diagonal blocks.These systems are factorized by 2 × 2 block gaussian elimination and, at eachiteration j, the solutions of the tridiagonal and pentadiagonal (Schur comple-ment) systems are computed by gaussian elimination.We will find the approximate solution z(jk) when the 1–norm of the residualr(k) in (38) is less than ς‖F (u(k))‖1. The values of ς used in the experiment

are 0.5 and 0.05. We define u(k+1) = z(jk) with z(0) = 0.The initial guess u(0) is set equal to the source w or to a vector w whosecomponents are 1/n.The used outer stopping rule is (see [15, §5.2])

‖F (u(k))‖1 ≤ τa + τr‖F (u(0))‖1

where τa and τr are the absolute and the relative tolerance respectively equalto 10−5 and F (u) is defined in (8).In all the experiments we have N = 30. We have different tables for differentvalues of the diffusion coefficient σ. All the experiments have been obtained bya double precision Fortran code carried out on Workstation Alpha 21264 EV6with 667 MHz.In the tables, we report for each outer iteration k, the number of inner iterationj needed to satisfy the inner stopping rule and the values F = ‖F (u(k))‖1,

d = ‖u(k)−u(k−1)‖1, u = ‖u(k)‖1 and the upper bound of formula (39), denotedas r3, the value of τ and the initial vector used; we observe in Table 4 thatr = ‖r(k)‖1 satisfies formula (39). Here, the notation 2.2(−4) means 2.2 · 10−4.In the tables, at the last outer iterations the values of the function F , of theresidual and of the error are the same. In all the cases, the computer time isO(10−2) seconds.From the numerical experience we can draw the following conclusions:

• when τ is small, r has nearly the same order of the inner residual or ofς‖F (u(k))‖1;

3We set γk+1 = 0.999 and λk = λkk, for k ≥ 1 and λ0 = 0.9.

19

• for the case σ(ϕ) = βϕ1/3 and ς = 0.05, the inner stopping rule guaranteesthe satisfaction of formula (39), that is ‖r(k)‖1 < r;

Therefore, the used adaptive inner stopping rule permits to obtain the conver-gence in all the cases examined. When we consider an a priori fixed number ofinner iterations as in the weakly nonlinear case (see [7]), we could not expectthe convergence. In the following, for the case σ = βϕ, we report the values ofF , d and u for jk = k and jk = 10; even in these cases the final values of thefunction, of the residual and of the error are the same. For the case jk = 5 theconvergence to a zero of F (u) is not obtained and we report the values of F , dand u for the first seven iterates; the code stops because the maximum number100 of outer iterations is reached.

References

[1] Adams L.M., Ong E.G.: Additive polynomial preconditioners for parallelcomputers, Parallel Computing 9 (1988/89), 333–345.

[2] Apostol T.M.: Mathematical Analysis, Second Edition, Addison–WesleyPubl., Reading MA, 1974.

[3] Evans D.J.: Group explicit iterative methods for solving large linear sys-tems, Intern. J. Computer Math. 17 (1985), 81–108.

[4] Evans D.J., Yousif W.S.: The solution of two–point boundary value prob-lems by the alternating group explicit (AGE) method, SIAM J. Sci. Stat.Comput. 9 (1988), 474–484.

[5] Franklin J.: Methods of Mathematical Economics: Linear and NonlinearProgramming, Fixed–Point Theorems, Springer–Verlag, New York, 1980.

[6] Fuhrmann J.: Existence and uniqueness of solutions of certain systems ofalgebraic equations with off–diagonal nonlinearity, Appl. Numer. Math. 37(2001), 359–370.

[7] Galligani E.: A two–stage iterative method for solving a weakly nonlinearparametrized system, Intern. J. Computer Math., 79 (2002), 1211–1224.

[8] Galligani E.: A two–stage iterative method for solving weakly nonlinearsystems, Atti Sem. Mat. Fis. Univ. Modena, L (2002), 195–215.

[9] Galligani E.: The Newton–Arithmetic Mean method for the solution of sys-tems of nonlinear equations, Appl. Math. Comput., 134 (2003), 9–34.

[10] Galligani E.: On solving a special class of weakly nonlinear finite differencesystems, Intern. J. Computer Math., to appear 2005.

20

[11] Galligani E., Ruggiero V.: The Arithmetic Mean preconditioner for multi-vector computers, Intern. J. Computer Math., 44 (1992), 207–222.

(Also in: Preconditioned Iterative Methods, Topics in Computer Mathe-matics 4 (D.J. Evans ed.), Gordon & Breach Science Publ., Lausanne, 1994,273–288)

[12] Galligani E., Ruggiero V.: A polynomial preconditioner for block tridiagonalmatrices, Parallel Algorithms Applic., 3 (1994), 227–237.

[13] Galligani E., Ruggiero V.: A parallel preconditioner for block tridiagonalmatrices, Parallel Computing: Trends and Applications (G.R. Joubert, D.Trystram, F.J. Peters, D.J. Evans eds.), Elsevier Science B.V., Amsterdam,1994, 113–120.

[14] Galligani E., Ruggiero V.: The two–stage arithmetic mean method, Appl.Math. Comput., 85 (1997), 245–264.

[15] Kelley C.T.: Iterative Methods for Linear and Nonlinear Equations, SIAM,Philadelphia, 1995.

[16] Kellogg R.B., Li T.Y., Yorke J.: A constructive proof of the Brouwer fixed–point theorem and computational results, SIAM J. Numer. Anal., 13 (1976),473–483.

[17] Lakshmikantham V., Trigiante D.: Theory of Difference Equations: Nu-merical Methods and Applications, Second Edition, Marcel Dekker Inc.,New York, 2002.

[18] Marchuk G.I.: Splitting and Alternating Direction Methods, Handbook ofNumerical Analysis (P.G. Ciarlet, J.L. Lions eds.), vol. I, North–Holland,Amsterdam, 1990, 197–462.

[19] Neumann M., Plemmons R.J.: Convergence of parallel multisplitting itera-tive methods for M–matrices, Linear Algebra Appl. 88/89 (1987), 559–573.

[20] Ortega J.M.: Numerical Analysis, A Second Course, Academic Press, NewYork, 1972.

[21] Ortega J.M., Rheinboldt W.C.: Iterative Solution of Nonlinear Equationsin Several Variables, Academic Press, New York, 1970.

[22] Rheinboldt W.C.: Methods for Solving Systems of Nonlinear Equations,Second Edition, SIAM, Philadelphia, 1998.

[23] Rosen J.B.: Stability and bounds for nonlinear systems of difference anddifferential equations, J. Math. Anal. Applic., 2 (1961), 370–393.

[24] Ruggiero V., Galligani E.: An iterative method for large sparse linear sys-tems on a vector computer, Computers Math. Applic., 20 (1990), 25–28.

21

[25] Ruggiero V., Galligani E.: A parallel algorithm for solving block tridiagonallinear systems, Computers Math. Applic., 24 (1992), 15–21.

[26] Varga R.: Matrix Iterative Analysis, Second Edition, Springer, Berlin,2000.

22

σ = βϕ; β = 1; ς = 0.5

k j F d u r τ u(0)

0 2.2(-4) 7.6(-6) 0.01 w

1 3 1.5(-5) 2.3(-4) 0.99998 4.8(-7)2 4 4.5(-7) 1.5(-5) 0.99999

0 2.2(-3) 0.1 w

1 5 3.9(-4) 2.5(-3) 0.999612 6 9.1(-5) 3.0(-4) 0.999903 7 2.2(-5) 6.9(-5) 0.999974 8 5.3(-6) 1.7(-5) 0.99999

0 1.1(-2) 0.5 w

1 9 5.6(-3) 1.6(-2) 0.994432 11 1.8(-3) 1.8(-3) 0.998183 13 6.7(-4) 1.1(-3) 0.999324 15 2.5(-4) 4.3(-4) 0.999755 17 9.1(-5) 1.6(-4) 0.999906 19 3.3(-5) 5.8(-5) 0.999967 21 1.2(-5) 2.1(-5) 0.999988 23 4.6(-6) 7.8(-6) 0.99999

0 0.92797 3.1(-2) 0.01 w

1 1 9.6(-3) 0.92160 0.99042 3.0(-4)2 2 5.3(-4) 9.1(-3) 0.99947 9.7(-5)3 3 1.5(-5) 5.1(-4) 0.99998

0 0.92903 2.5(-2) 0.1 w

1 1 8.8(-2) 0.87220 0.912102 2 3.0(-2) 5.8(-2) 0.970443 3 6.6(-3) 2.3(-2) 0.993364 4 1.6(-3) 5.0(-3) 0.998395 5 3.8(-4) 1.2(-3) 0.999616 6 9.1(-5) 2.9(-4) 0.999907 7 2.2(-5) 6.9(-5) 0.999978 8 5.3(-6) 1.7(-5) 0.99999

0 0.93387 0.5 w

1 1 0.32332 0.75068 0.676842 3 8.0(-2) 0.24369 0.920363 5 3.5(-2) 4.5(-2) 0.965374 7 1.3(-2) 2.2(-2) 0.986985 9 4.9(-3) 8.1(-3) 0.995076 11 1.8(-3) 3.1(-3) 0.998167 13 6.7(-4) 1.2(-3) 0.999328 15 2.5(-4) 4.3(-4) 0.999759 17 9.1(-5) 1.6(-4) 0.9999010 19 3.3(-5) 5.8(-5) 0.9999611 21 1.2(-5) 2.1(-5) 0.99998

Table 1: results of fixed–point AM method

23

σ = β√

ϕ; β = 0.1; ς = 0.5

k j F d u r τ u(0)

0 4.5(-4) 1.6(-5) 0.01 w

1 3 7.7(-5) 5.1(-4) 0.99992 2.3(-6)2 4 3.6(-6) 7.4(-5) 0.99999

0 4.6(-3) 0.1 w

1 6 9.0(-4) 5.3(-3) 0.999092 7 2.9(-4) 6.1(-4) 0.999703 8 9.8(-5) 2.0(-4) 0.999904 9 3.2(-5) 6.5(-5) 0.999965 10 1.1(-5) 2.2(-5) 0.999986 11 3.6(-6) 7.2(-6) 0.99999

0 0.92820 3.0(-2) 0.01 w

1 1 2.8(-2) 0.90957 0.97184 8.5(-4)2 2 1.7(-3) 2.7(-2) 0.99833 1.0(-4)3 3 7.7(-5) 1.6(-3) 0.99992 2.8(-6)4 4 3.6(-6) 7.3(-5) 0.99999

0 0.93142 0.1 w

1 1 0.22352 0.79593 0.776592 2 7.2(-2) 0.15116 0.927643 3 2.5(-2) 4.8(-2) 0.975444 4 8.2(-3) 1.6(-2) 0.991815 5 2.7(-3) 5.5(-3) 0.997296 6 8.9(-4) 1.8(-3) 0.999107 7 2.9(-4) 6.0(-4) 0.999708 8 9.8(-5) 2.0(-4) 0.999909 9 3.2(-5) 6.5(-5) 0.9999610 10 1.1(-5) 2.2(-5) 0.99998

Table 2: results of fixed–point AM method

24

σ = β 3√

ϕ; β = 0.01; ς = 0.5

k j F d u r τ u(0)

0 1.3(-3) 3.2(-5) 0.1 w

1 4 1.3(-4) 1.4(-3) 0.99987 2.6(-6)2 5 1.4(-5) 1.1(-4) 0.99998 3.9(-7)3 6 1.6(-6) 1.3(-5) 0.99999

0 6.4(-3) 0.5 w

1 6 2.6(-3) 8.4(-3) 0.997412 7 9.8(-4) 1.6(-3) 0.999013 8 3.8(-4) 6.0(-4) 0.999624 9 1.5(-4) 2.3(-4) 0.999855 10 5.7(-5) 8.9(-5) 0.999946 11 2.2(-5) 3.5(-5) 0.999977 12 8.5(-6) 1.3(-5) 0.99999

0 0.92896 2.3(-2) 0.1 w

1 1 8.2(-2) 0.87539 0.91755 1.7(-3)2 2 1.0(-2) 7.2(-2) 0.98987 3.2(-4)3 3 1.1(-3) 9.0(-3) 0.99886 2.9(-5)4 4 1.3(-4) 1.0(-3) 0.99987 3.1(-6)5 5 1.4(-5) 1.1(-4) 0.99998

0 0.93341 0.5 w

1 1 0.30828 0.75553 0.691832 2 0.11044 0.19780 0.889553 3 4.4(-2) 6.6(-2) 0.955764 4 1.7(-2) 2.7(-2) 0.982805 5 6.6(-3) 1.1(-2) 0.993366 6 2.6(-3) 4.1(-3) 0.997447 7 9.8(-4) 1.6(-3) 0.999018 8 3.8(-4) 6.0(-4) 0.999629 9 1.5(-4) 2.3(-4) 0.9998510 10 5.7(-5) 8.9(-5) 0.9999411 11 2.2(-5) 3.5(-5) 0.9999712 12 8.5(-6) 1.3(-5) 0.99999

Table 3: results of fixed–point AM method

σ = β 3√

ϕ; β = 0.01; ς = 0.05

k j F d u r r τ u(0)

0 1.3(-3) 1.4(-5) 3.0(-5) 0.1 w

1 5 1.4(-5) 1.3(-3) 0.99998 1.8(-7) 3.3(-7)2 7 1.8(-7) 1.4(-5) 0.99999

0 0.92896 6.9(-3) 2.7(-2) 0.1 w

1 2 7.1(-3) 0.92340 0.99309 1.3(-4) 1.6(-4)2 4 1.3(-4) 6.9(-3) 0.99987 1.6(-6) 6.9(-6)3 6 1.6(-6) 1.2(-4) 0.99999

Table 4: results of fixed–point AM method

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σ = βϕ; β = 1; ς = 0.5; τ = 0.1; u(0) = w

k j F d u

0 2.2(-3)1 1 0.16991 0.16991 0.830082 2 2.3(-2) 0.14671 0.976803 3 6.8(-3) 1.6(-2) 0.993184 4 1.6(-3) 5.2(-3) 0.998395 5 3.8(-4) 1.2(-3) 0.999616 6 9.1(-5) 2.9(-4) 0.999907 7 2.2(-5) 6.9(-5) 0.999978 8 5.3(-6) 1.7(-5) 0.99999

0 2.2(-3)1 10 7.7(-6) 2.2(-3) 0.99999

Table 5: results of fixed–point AM method with a priori stopping rule

σ = βϕ; β = 1; ς = 0.5; τ = 0.1; u(0) = w

k j F d u

0 2.2(-3)1 5 3.89588521(-4) 2.5(-3) 0.99961042 5 3.84883589(-4) 1.2(-5) 0.99961513 5 3.84905551(-4) 6.6(-8) 0.99961504 5 3.84905461(-4) 4.3(-10) 0.99961505 5 3.84905462(-4) 3.1(-12) 0.99961506 5 3.84905462(-4) 2.6(-14) 0.99961507 5 3.84905462(-4) 2.7(-16) 0.9996150

Table 6: results of fixed–point AM method with a priori stopping rule

26