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Some simple characterizations of contractivity regions
Journal of Computational North-Holland
and Applied
Mathematics
25 (1989) 27-32
Some simple characterizations contractivity regions * Paula
DE OLIVEIRA
Departamento
27
of
*
de Matemritica,
Universidade de Coimbra, Coimbra, Portugal
Received 5 November 1987 Revised 30 May 1988 Abstract: We present necessary and sufficient sector, and in disk of the complex plane.
conditions
Keywords:
contractivity.
Multiderivative
multistep
method,
for a multiderivative
multistep
method
to be contractive
in a
1. Introduction We investigate some contractivity properties of linear multistep multiderivative methods. For linear multistep methods contractivity has been introduced by Dahlquist [2] and studied for a particular norm, depending on a certain positive definite matrix. Nevanlinna and Liniger [9], [lo] studied contractivity for method independent norms. Many other authors have studied contractivity questions. We can mention without being exhaustive [l], [3], [4], [5], [6], 7[], [II], [12] and [13]. From an analytical viewpoint contractivity is a very interesting property because we can obtain very simple characterizations of certain methods, as we will see in later sections. In Section 2 some basic definitions are introduced. In Section 3 we give purely algebraic necessary and sufficient conditions, by means of the coefficients of the method, for a multiderivative multistep method to be contractive in a sector of the complex plane. We also present necessary and sufficient conditions for a multiderivative method to be contractive in a disk of the complex plane. Finally in Section 4 we identify the parameters that control the size of some subregions of the contractivity region.
2. Preliminaries For solving initial value problems u’(t)
=f(t,
i Y? fE
c’*,
r(t)>,
~(0) given,
* This work was supported by the Instituto Investiga@o Cientifica e Technolbgica. 0377-0427/89/$3.50
0 1989, Elsevier
(2.1)
fJ’ 0, National
Science Publishers
de InvestigacZo
B.V. (North-Holland)
Cientifica
and
the Junta
National
de
P. de Oliveira / Characterizations
28
we consider
multistep
multiderivative
k
II
I=1
j=O
where (YJ’ p!” J
j
’
I
methods
of contractivity regions
of form
k
(2.2)
j=o
O(l)k and I= l(1) p, are real coefficients
=
with (Ye# 0 and
J=o
i
jaj=
j=O
2 fi/“)=
(2.3) 1.
J=o
In (2.2) h represents a positive We consider the test equation
steplenght
and y,‘yj an approximation
of (dL/dx’)(x,+,).
y’ = xy.
(2.4)
and from (2.2) and (2.4) we have
j=O
(2.5)
J=o
‘=I
where z = hX. Let x = (y,,
)T.
. . . , .%+k-1
We recall the following
Definition 2.1. Let 1). (1 be a norm in c”. Then solutions of the difference equation (2.5) satisfy V’y,EEk, II Xl+, II G IIr, II) The set S (1. I( will be called contractivity
definition.
S,, ,, consists
s, = {z EC;
A(z)
z E E for which
the
V’n>O. region.
In what follows we will use the max-norm and we will represent We can easily prove the following theorem. Theorem 2.1. The contractivity
of those
S,, ,, by S,.
region of (2.2) with respect to the max-norm
is given by
GO}
where
(2.6) From (2.6) we can deduce Theorem 2.2. (i) Method
cYk> 0 and (ii) Method
another
theorem.
(2.2) is contractive at z = 0 a,
j=O(l)k-1.
(2.2) is contractive at z = 00 k-l
iff
iff
P. de Oliveira / Characterizations
We observe that the second derivative contractive at z = 0 and z = 00.
of contractiuity regions
multistep
methods
presented
29
by Emight
in [8] are
3. Contractivity in a sector of the complex plane We study in this section the necessary and sufficient conditions the complex plane. We remark that for first derivative methods sector of the complex plane has been made in [9]. Let us introduce the following notations S(B,
-a)={z~E;
J,=
{j~lV;c~,#f}.
IIT--argzl
<0,Rez>
of contractivity in a sector of a study of contractivity in a
-a},
(3-I) (34
We can then state the following
theorem.
Theorem 3.1. There exist constants a > 0, 0 < 8 < 71such that S(8,
-a)
C S,
(3.3)
iff the method is contractive at z = 0 and c /!yj=J, Proof. Analysing (y,
-
c py /eJo separately
>o.
(34
the terms in (2.6) for j E Jo and j ~5J, we have respectively
5 z’/y’ I-- ]a;][l-RezFj
(3.5)
+O(]z(2),
I=1
ff,-,p$q= Iz(i~B~‘)~+wzI)).
(34
From (3.5) and (3.6) we deduce A(z)=Rez Let in (3.7) z = pe” A(P ei”) =
c
/3:‘)+
JE-J,
(z]
c
i%
I~,o)~+O(]z]‘).
(3.7)
with 8 fixed 0 < 8 < IT and p + 0. We have
~[COSe C p,(l)+ C Ip:“i] + JEJ,, Jo-6
and the result easily follows.
0(p2).
(34
•I
Remark 3.1. If Ci e J,, ] ,$I’ ) = 0 condition have
(3.4) is always satisfied.
In fact as X:=ip,“’
= 1 (2.3) we
30
P. de Oliveira / Characterizations
of contractivity
regions
Remark 3.2. When p = 1, i.e. in the case of first derivatives multistep characterization has been obtained in [9].
methods,
the same
If methods of form (2.2) are used to integrate differential equations whose eigenvalues are “almost” imaginary it would be interesting to characterize methods such that 3r>o
p(-Y,
Y)CSoo.
(3.9)
where B(-r,
r)=
{zEQ=;
]z+y]
(3.10)
Theorem 3.2. There exist r > 0 such that q-r,
r)cs,
(3.11)
iff the method is contractiue c
at z = 0 and
(p,c’)I = 0.
(3.12)
iEJu
Proof. Suppose that (3.11) holds. Then from Theorem 3.1 we have c
p,“’ > 0.
(3.13)
i~Jo
Let us define for 0 < l3 < 71. fP>
=cos
~;~JOp:l)+
(3.14)
,E,iP:“i
and let 0, be a root of f(0) = 0. As f(0) 0 G 0 < 0, and p small enough
IS . a decreasing function in [0, ~1 we deduce that for
A( pe”) > 0.
(3.15)
Therefore from (2.6), (3.11) and (3.15) we must have 19,< &r. Considering that 0, is such that = 0, we conclude that (3.12) holds. For the sufficient part we note that the parametric equations of (3.10) are
f (0,)
z=2r(cos8]eie,
(3.16)
0<8<2a.
With z given by (3.16) we have for j E J, P
aj
-
c. zfy)
= ( (Y, ( + 2r cos%p:‘)~
I=1
J
(3.17)
+ o(9),
and finally, A(2r
lcos 0 I eie) =
C
(a,) +2r
i% j#k
+ 0(
r’) -
1ffk
I (3.18)
1 - 2r c~s~Bp~‘)~ iakt.
P. de Oliveira / Characterizations
From (2.3) we can given (3.18) the following A(2r (cos8 1eie) = -2r
of contractivity
regions
31
form
cos2f3 + 0(r2).
(3.19)
Observing that the explicit form of 0( r*) is C lcos 8 ) r2, where C depends the method, the results easily follows. 0 Remark 3.3. We note that the second conditions of Theorems 3.1 and 3.2.
derivative
methods
presented
on the coefficients
by Enright
of
in [8] satisfy the
4. Some qualitative results Let us introduce
the polynomial k-l
fTl4)=
IzIP I&?) i
-
\
c Ip, J I) (P)
;=o
k
(4.1)
J=o
If we suppose
k
that k-l
L=(ppI
c
-
I$o)I > 0,
(4-2)
J=o
we can easily prove {tEQ=,
pr(l
zI>~O}
cs,.
“l/L,
i = l(l)p
(4.3)
Set a;=
c
(py
-
- 1
j=O
(4.4
and k
a,= The polynomial
c bjl/L* ;=o
(4.5)
I’“( ) z I) takes then the form
P”(Izj)=L
Iz(B-
&z,,-i r=l
(
Using Young’s inequality
(4.6) we obtain
the following
estimate
for the largest root x,, of P”( I z I): (4.7)
32
P. de Oliveira / Characterizations
Fixing
cllj and fij(‘) , I= l(l)p
of contractivity regions
- 1, and taking limits in (4.7) when
L + co, we obtain
lim x,=0.
(4.8)
L-tCC
We can then conclude CB(0,
that for all Y> 0 it is possible
Y) C s,.
If p = 2, the largest positive x2=
to choose b/cP) such that
-
(4.9) root of P*( 1z I) is
5 [Bj”I - [ [ 5 IPjr’~/*;=o j=O
L,cO
(4.10)
Iajl]1’2/L
and so lim,,, x2 = 0, i.e., it is always possible to choose /3J’2’such that (4.9) holds. From the above considerations we conclude that with parameters b,“‘, j = O(l)k, I = l(l)p - 1 we can control the contractivity region “till” a neighborhood of the origin. This last conclusion together with Theorems 2.2, 3.1 and 3.2 furnishes a “relief” of contractivity regions of multistep multiderivative methods in the sense that they describe how the parameters “control” the contractivity region.
Acknowledgments We would like to thank an anonymous the first version of the paper.
referee
for some important
comments,
which improve
References VI K. Burrage and J. Butcher, Stability criteria for implicit Runge-Kutta
methods, X4&f J. Numer. Anal. 16 (1979) 46-57. PI G. Dahlquist, Error analysis for a class of methods for stiff non-linear initial value problems, Numerical Analysis Dundee, 1975, Lecture Notes Math. 506 (Springer, Berlin, 1976) 60-74. G. Dahlquist, On the relation of G-stability to other stability concepts for linear multistep methods, in: J.J.H. 131 Miller, Ed., Topics in Numerical Analysis ZIZ (Academic Press, London, 1977) 349-362. [41 G. Dahlquist, G-stability is equivalent to A-stability, BIT 18 (1978) 384-401. 151 G. Dahlquist, Some properties of linear multistep and one-leg methods for ordinary differential equations, Report TRITA-NA-7904. Royal Inst. Techn., Stockholm, 1979. 161 G. Dahlquist, Some contractivity questions for one-leg and linear multistep methods, Report TRITA-NA-7905, Royal Inst. Techn., Stockholm, 1979. G. Dahlquist and R. Jeltsch, Generalized disks of contractivity for explicit and implicit Runge-Kutta methods, [71 Report TRITA-NA-7906, Royal Inst. Techn., Stockholm, 1979. 181 W.H. Enright, Second derivative multistep methods for stiff ordinary equations, SZAMJ. Numer. Anal. 11 (1974). 191 0. Nevanlinna and W. Liniger, Contractive methods for stiff differential equations. PART I, BIT 18 (1978) 457-474. [IO1 0. Nevanlinna and W. Liniger, Contractive methods for stiff differential equations, PART II, BIT 19 (1979) 53-12. Royal Inst. [III J. Sand, On one-leg and linear multistep formulas with variable step-sizes, Report TRITA-NA-8112, Techn., Stockholm, 1981. [I21 G. SBderlind, Multiple-step G-contractivity with applications to slowly varying linear systems, Report TRITANA-8107, Royal Inst. Techn., Stockholm, 1982. [I31 M.N. Spijker, Contractivity in the numerical solution of initial value problems, Numer. Math. 42 (1983) 271-290.
and Applied
Mathematics
25 (1989) 27-32
Some simple characterizations contractivity regions * Paula
DE OLIVEIRA
Departamento
27
of
*
de Matemritica,
Universidade de Coimbra, Coimbra, Portugal
Received 5 November 1987 Revised 30 May 1988 Abstract: We present necessary and sufficient sector, and in disk of the complex plane.
conditions
Keywords:
contractivity.
Multiderivative
multistep
method,
for a multiderivative
multistep
method
to be contractive
in a
1. Introduction We investigate some contractivity properties of linear multistep multiderivative methods. For linear multistep methods contractivity has been introduced by Dahlquist [2] and studied for a particular norm, depending on a certain positive definite matrix. Nevanlinna and Liniger [9], [lo] studied contractivity for method independent norms. Many other authors have studied contractivity questions. We can mention without being exhaustive [l], [3], [4], [5], [6], 7[], [II], [12] and [13]. From an analytical viewpoint contractivity is a very interesting property because we can obtain very simple characterizations of certain methods, as we will see in later sections. In Section 2 some basic definitions are introduced. In Section 3 we give purely algebraic necessary and sufficient conditions, by means of the coefficients of the method, for a multiderivative multistep method to be contractive in a sector of the complex plane. We also present necessary and sufficient conditions for a multiderivative method to be contractive in a disk of the complex plane. Finally in Section 4 we identify the parameters that control the size of some subregions of the contractivity region.
2. Preliminaries For solving initial value problems u’(t)
=f(t,
i Y? fE
c’*,
r(t)>,
~(0) given,
* This work was supported by the Instituto Investiga@o Cientifica e Technolbgica. 0377-0427/89/$3.50
0 1989, Elsevier
(2.1)
fJ’ 0, National
Science Publishers
de InvestigacZo
B.V. (North-Holland)
Cientifica
and
the Junta
National
de
P. de Oliveira / Characterizations
28
we consider
multistep
multiderivative
k
II
I=1
j=O
where (YJ’ p!” J
j
’
I
methods
of contractivity regions
of form
k
(2.2)
j=o
O(l)k and I= l(1) p, are real coefficients
=
with (Ye# 0 and
J=o
i
jaj=
j=O
2 fi/“)=
(2.3) 1.
J=o
In (2.2) h represents a positive We consider the test equation
steplenght
and y,‘yj an approximation
of (dL/dx’)(x,+,).
y’ = xy.
(2.4)
and from (2.2) and (2.4) we have
j=O
(2.5)
J=o
‘=I
where z = hX. Let x = (y,,
)T.
. . . , .%+k-1
We recall the following
Definition 2.1. Let 1). (1 be a norm in c”. Then solutions of the difference equation (2.5) satisfy V’y,EEk, II Xl+, II G IIr, II) The set S (1. I( will be called contractivity
definition.
S,, ,, consists
s, = {z EC;
A(z)
z E E for which
the
V’n>O. region.
In what follows we will use the max-norm and we will represent We can easily prove the following theorem. Theorem 2.1. The contractivity
of those
S,, ,, by S,.
region of (2.2) with respect to the max-norm
is given by
GO}
where
(2.6) From (2.6) we can deduce Theorem 2.2. (i) Method
cYk> 0 and (ii) Method
another
theorem.
(2.2) is contractive at z = 0 a,
j=O(l)k-1.
(2.2) is contractive at z = 00 k-l
iff
iff
P. de Oliveira / Characterizations
We observe that the second derivative contractive at z = 0 and z = 00.
of contractiuity regions
multistep
methods
presented
29
by Emight
in [8] are
3. Contractivity in a sector of the complex plane We study in this section the necessary and sufficient conditions the complex plane. We remark that for first derivative methods sector of the complex plane has been made in [9]. Let us introduce the following notations S(B,
-a)={z~E;
J,=
{j~lV;c~,#f}.
IIT--argzl
<0,Rez>
of contractivity in a sector of a study of contractivity in a
-a},
(3-I) (34
We can then state the following
theorem.
Theorem 3.1. There exist constants a > 0, 0 < 8 < 71such that S(8,
-a)
C S,
(3.3)
iff the method is contractive at z = 0 and c /!yj=J, Proof. Analysing (y,
-
c py /eJo separately
>o.
(34
the terms in (2.6) for j E Jo and j ~5J, we have respectively
5 z’/y’ I-- ]a;][l-RezFj
(3.5)
+O(]z(2),
I=1
ff,-,p$q= Iz(i~B~‘)~+wzI)).
(34
From (3.5) and (3.6) we deduce A(z)=Rez Let in (3.7) z = pe” A(P ei”) =
c
/3:‘)+
JE-J,
(z]
c
i%
I~,o)~+O(]z]‘).
(3.7)
with 8 fixed 0 < 8 < IT and p + 0. We have
~[COSe C p,(l)+ C Ip:“i] + JEJ,, Jo-6
and the result easily follows.
0(p2).
(34
•I
Remark 3.1. If Ci e J,, ] ,$I’ ) = 0 condition have
(3.4) is always satisfied.
In fact as X:=ip,“’
= 1 (2.3) we
30
P. de Oliveira / Characterizations
of contractivity
regions
Remark 3.2. When p = 1, i.e. in the case of first derivatives multistep characterization has been obtained in [9].
methods,
the same
If methods of form (2.2) are used to integrate differential equations whose eigenvalues are “almost” imaginary it would be interesting to characterize methods such that 3r>o
p(-Y,
Y)CSoo.
(3.9)
where B(-r,
r)=
{zEQ=;
]z+y]
(3.10)
Theorem 3.2. There exist r > 0 such that q-r,
r)cs,
(3.11)
iff the method is contractiue c
at z = 0 and
(p,c’)I = 0.
(3.12)
iEJu
Proof. Suppose that (3.11) holds. Then from Theorem 3.1 we have c
p,“’ > 0.
(3.13)
i~Jo
Let us define for 0 < l3 < 71. fP>
=cos
~;~JOp:l)+
(3.14)
,E,iP:“i
and let 0, be a root of f(0) = 0. As f(0) 0 G 0 < 0, and p small enough
IS . a decreasing function in [0, ~1 we deduce that for
A( pe”) > 0.
(3.15)
Therefore from (2.6), (3.11) and (3.15) we must have 19,< &r. Considering that 0, is such that = 0, we conclude that (3.12) holds. For the sufficient part we note that the parametric equations of (3.10) are
f (0,)
z=2r(cos8]eie,
(3.16)
0<8<2a.
With z given by (3.16) we have for j E J, P
aj
-
c. zfy)
= ( (Y, ( + 2r cos%p:‘)~
I=1
J
(3.17)
+ o(9),
and finally, A(2r
lcos 0 I eie) =
C
(a,) +2r
i% j#k
+ 0(
r’) -
1ffk
I (3.18)
1 - 2r c~s~Bp~‘)~ iakt.
P. de Oliveira / Characterizations
From (2.3) we can given (3.18) the following A(2r (cos8 1eie) = -2r
of contractivity
regions
31
form
cos2f3 + 0(r2).
(3.19)
Observing that the explicit form of 0( r*) is C lcos 8 ) r2, where C depends the method, the results easily follows. 0 Remark 3.3. We note that the second conditions of Theorems 3.1 and 3.2.
derivative
methods
presented
on the coefficients
by Enright
of
in [8] satisfy the
4. Some qualitative results Let us introduce
the polynomial k-l
fTl4)=
IzIP I&?) i
-
\
c Ip, J I) (P)
;=o
k
(4.1)
J=o
If we suppose
k
that k-l
L=(ppI
c
-
I$o)I > 0,
(4-2)
J=o
we can easily prove {tEQ=,
pr(l
zI>~O}
cs,.
“l/L,
i = l(l)p
(4.3)
Set a;=
c
(py
-
- 1
j=O
(4.4
and k
a,= The polynomial
c bjl/L* ;=o
(4.5)
I’“( ) z I) takes then the form
P”(Izj)=L
Iz(B-
&z,,-i r=l
(
Using Young’s inequality
(4.6) we obtain
the following
estimate
for the largest root x,, of P”( I z I): (4.7)
32
P. de Oliveira / Characterizations
Fixing
cllj and fij(‘) , I= l(l)p
of contractivity regions
- 1, and taking limits in (4.7) when
L + co, we obtain
lim x,=0.
(4.8)
L-tCC
We can then conclude CB(0,
that for all Y> 0 it is possible
Y) C s,.
If p = 2, the largest positive x2=
to choose b/cP) such that
-
(4.9) root of P*( 1z I) is
5 [Bj”I - [ [ 5 IPjr’~/*;=o j=O
L,cO
(4.10)
Iajl]1’2/L
and so lim,,, x2 = 0, i.e., it is always possible to choose /3J’2’such that (4.9) holds. From the above considerations we conclude that with parameters b,“‘, j = O(l)k, I = l(l)p - 1 we can control the contractivity region “till” a neighborhood of the origin. This last conclusion together with Theorems 2.2, 3.1 and 3.2 furnishes a “relief” of contractivity regions of multistep multiderivative methods in the sense that they describe how the parameters “control” the contractivity region.
Acknowledgments We would like to thank an anonymous the first version of the paper.
referee
for some important
comments,
which improve
References VI K. Burrage and J. Butcher, Stability criteria for implicit Runge-Kutta
methods, X4&f J. Numer. Anal. 16 (1979) 46-57. PI G. Dahlquist, Error analysis for a class of methods for stiff non-linear initial value problems, Numerical Analysis Dundee, 1975, Lecture Notes Math. 506 (Springer, Berlin, 1976) 60-74. G. Dahlquist, On the relation of G-stability to other stability concepts for linear multistep methods, in: J.J.H. 131 Miller, Ed., Topics in Numerical Analysis ZIZ (Academic Press, London, 1977) 349-362. [41 G. Dahlquist, G-stability is equivalent to A-stability, BIT 18 (1978) 384-401. 151 G. Dahlquist, Some properties of linear multistep and one-leg methods for ordinary differential equations, Report TRITA-NA-7904. Royal Inst. Techn., Stockholm, 1979. 161 G. Dahlquist, Some contractivity questions for one-leg and linear multistep methods, Report TRITA-NA-7905, Royal Inst. Techn., Stockholm, 1979. G. Dahlquist and R. Jeltsch, Generalized disks of contractivity for explicit and implicit Runge-Kutta methods, [71 Report TRITA-NA-7906, Royal Inst. Techn., Stockholm, 1979. 181 W.H. Enright, Second derivative multistep methods for stiff ordinary equations, SZAMJ. Numer. Anal. 11 (1974). 191 0. Nevanlinna and W. Liniger, Contractive methods for stiff differential equations. PART I, BIT 18 (1978) 457-474. [IO1 0. Nevanlinna and W. Liniger, Contractive methods for stiff differential equations, PART II, BIT 19 (1979) 53-12. Royal Inst. [III J. Sand, On one-leg and linear multistep formulas with variable step-sizes, Report TRITA-NA-8112, Techn., Stockholm, 1981. [I21 G. SBderlind, Multiple-step G-contractivity with applications to slowly varying linear systems, Report TRITANA-8107, Royal Inst. Techn., Stockholm, 1982. [I31 M.N. Spijker, Contractivity in the numerical solution of initial value problems, Numer. Math. 42 (1983) 271-290.