- Email: [email protected]
Error bounds for the solution of Volterra and delay equations
Applied Numerical Mathematics 9 (1992) 201-207 North-Holland
Peter Linz Di~v3hxr of Computer
201
Science, University of California, Davis, CA 95616, USA
Richard L.C. Wang Lockheed Missiles and Space Company Inc., Sunnyvale, CA 94086, USA
Abstract
Linz, P. and R.L.C. Wang, Error bounds for the solution of Volterra and delay equations, Applied Numerical Mathematics 9 (1992) 201-207. The construction of efficient and reliable numerical software requires a theory for two distinct issues: a convergence analysis and a way of bounding the error in the computed results. While the first of these questions has received much attention, less is known about the second one. Here we describe a method for bounding the error in the solution of Prcdhoh;i integral equations and show how the approach can be extended to Volterra and delay equations.
.
Introduction
In the error analysis of numerical methods, there are two major, complementary issues: first, the questrtin rsf convergence and stability, and second, the need for computable error bounds. For the sake of discussion, we consider the abstract operator equation LX ‘Y, (1 1I where L : X -+ Y is a linear operator between the normed spaces X and Y. Approximate solutions are obtained by discretization which produces an approximation x,, to x. In convergence analysis we establish results like lim ~~x--xJ~ =0,
fzdW
(2 I1
or the stronger
II x -x, ll = O(?P).
(3)
Convergence analysis is prominent in the literature or‘ numerical analysis; what we can conclude from (2) is that the method works in the sense that we can obtain arbitrarily good answers by making a fine enough discretization. If we also ave resuhs of type (3L then we can Predict the 0168-9274/92/$05.00
0 1992 - Elscvicr Science Publishers B.V. All rights rcscrved
P. Linz, R.L. C. Warlg / Error bounds for Volterra and delay equations
202
expected efficiency of the method. Convergence analysis is therefore indispensable in the selection of effective numerical algorithms. Usually, though, convergence analysis gives little information on the results of a specific computation. For this, we need a theory that produces actual computable error bounds for finite values of n, that is, we want results of the type Iix--&J
G77n,
(4)
where qn is computable for any given n. Recently, there has been considerable interest in this second aspect. Collatz [2], for example, has investigated the use of monotonicity arguments, while others (e.g. Kaucher and Miranker [3]) have used fixed-point arguments. On the whole, however, methods for computing error bounds have not been studied widely. In our work, we have investigated the effective computation of error bounds, using the following elementary idea. Suppose that we have found, by whatever means, an approximate solution x,, of (1). We substitute this into (1) and compute the residual ’
Pn
(5)
=I&,-y.
If (1) is not singular, that is, if the operator L has a bounded inverse, we get immediately that
II x -x, II < II L - l I! II p,,II
(6)
l
For this to be useful it is necessary that both ]IL- ’ 11and IIp,, II can be bounded. Additionally, we wan.t such bounds to be realistic, otherwise the use of (6) leads to a pessimistic estimate of the error. Often there are bounds available, but they are so poor that they are useless in practice. For computational efficiency, we would like a bound that approaches the true value closely as n increases. efinition 1. Let Q be an element of some normal linear space. Then the sequence {B,(Q)) is said to be a precise upper bound of IIQ II if
B,(Q) 2 II Q II for all n,
(7)
lim B,(Q) = II Q Il.
(8)
II -+m
While the approach we investigate here has been known for a long time, it has generally been ignored because it was felt that getting precise bounds on IIL- ’ II was quite difficult. However, our recent work has shown that this belief is not justified for many equations of practical interest. In this paper, we demonstrate how the method can be applied to Volterra integral equations and some delay-differential equations.
2. Frecise error bounds for Fred
integral equations
Consider as specific case of (1) the Fredholm intczral equation of the second kind
x(t)-/‘k(l, s)x(s)
ds=y(t), 0 which we write in operator notation as (I-K)i=y.
O,
(9) (10)
P, Linz, R.L. C. Wang / Error bounds for Vo1:c~r.ra and delay eqtlatiom
203
As the underlying spaces we will take C[O, I] with the naximum norm. In [S] we developed a method for computing realistic bounds on ll(Z -K)- : // when the kernel k(t, s) qf (9) is smooth. The technique uses a degenerate kernel ap roximation based on piecewise linear polynomials on a mesh with knots [t 1, t,, . . . , t,], wher ir = I/(n - I), ti = (i - 1)/r. As a first step, we replace (10) with X-KK,x=y,
PI) where K,, is the integral operator in which k(t, s) is replaced by its bilinear degenerate kernel approximation k,,(t7 s, = i i=l
i
k(tiY
tj)+i(f)@j(s)*
(12)
j=l
Here we use pi to denote the piecewise linear hat function basis for approximation and K and D to denote n x n matrices with respective elements Kij = k(ti, tj~,
Since K, is a bounded perturbation of K, the relati elementary one. It is a little harder to establish the and the easily computed inverse of I - DK. The foll recently established.
(Z-K)-’ and (I--K,)-’ is an between the inverse of (I - K,) ts relating these quantities were
Theorem 1. Let K be some integral operator which can be approximated by operators K, whose kernels have the form (12) and are such that 11K - K,, &I+ 0. Then II(I-K)-*
II < -
lI(I-K)-‘II
3
1 +hIIK(I-DK)-‘(I IIK-K,J(l+hIIK(I-DK)-‘Ilj’
l-
(13)
III+K(I-DK)-‘DII
l+ IIK-K,il
I(I+K(I-DK)-‘DII’
(14)
provided that the denominator in (13) is positive. Furthermore, the upper bound (13) is precise. Proof. The upper bound was first discussed in [5], later results in [4] provided the lower bound.
The fact that (13) is precise was established in [6].
0
The bounds established in this theorem are useful in the design of efficient computational methods for Fredholm equations with smooth kernels. If k(s, t) is twice continuously differentiable, then 11K - K,, II < :h’( k,, + kOZ+ $h”k,,),
(15)
where I’ G -‘ii
-
#+j
at’&
II
4
W) II
l
P. Linz, R.L.C. Wang / Error bounds for Volterra and delay equations
204
With a bound on I]K - K, 11everything in (13) and (14) is completely computable. How this is used in the construction of an actual algorithm for the effective solution of Fredholm integral equations is fully explained in [4]. For some functional equations the smoothness conditions assumed in (35) do not hold, and the theory must be modified to cover this case. We illustrate this with some specific cases.
3. Application to Volterra equations We now consider the Volterra integral equation x(t) -
/‘k(t, s)x(s)
0~ t d T,
ds =y(t),
(17)
0
which can be viewed as a special case of a Fredholm equation involving a kernel with a finite jump discontinuity along t = S, and k(t, s)=O,
fors>t.
Although degenerate kernel approximations are not natural for such kernels, we can still use them. Let D, be the set of grid squares along the diagonal t = s, that is, D, ={(t, s)I It--s1 d), and let D, = [0, l] x [0, l] - D,. In D, standard b ounds for the approximation functions apply. In D, we have max I!<(& s)-&Jt, (t,+=D,
of smooth
s)] < max ]k(t, s)]. O
If we assume sufficient smoothness for the kernel in 0 G s < t, then
II K - K, II \< $h2( k*[j + gz2k,, + I&) + h II k II0 P;Vhilesuch a first-order approximation is not useful for the computation of the solution of (17), it still works for bounding the operator inverse. xample 1. Consider equation (17) with k( t, s) = - t
IIK- K, II < $P(l
*s*and
T = 1. Then
+ ih2) + h.
The computed bounds for ]I(I - K)- ’ II are listed in Table 1. Note that in the case of Volterra equations there is an analytical upper bound for the inverse ]](I-K)-‘I]
but this often gives quite pessimistic results. Table 1
Table 2
n
Lower bound
Upper bound
t1
Lower bound
Upper bound
10 20 30
1.147 1.279 1.315
1.858 1.539 1.476
11 21 31
2.09 2.34 2.42
3.88 3.05 2.86
P. Linz, R.L.C. Wang / Error bounds for Volterra and delay equations
ounds for some delay di
205
rential equations
Consider as example the delay equation
I
Lx(t) =x’(t)
-c(t)x(t
-a)
=y(t),
a!
(18)
with initial condition x(t)=O,
Ogt
This equation is a very simple example, representative of the class of equations with fixed delay. We can integrate (18) to obtain the integral equation x(t)
-
jTk(t,
s)x(s)
ds
=z(t),
0
where
and z(t)
=
t >,a!.
After some elementary computations we then get IIL-‘II G (T-a)ll(l-K)-‘II, where K is the integral operator with kernel k(t, s) =
i
Tc(a + sT)~ o 9
t>a/T,O
We now have exactly the same situation as for Volterra equations, that is a kernel which is sectionally smooth with a finite jump discontinuity along the line s = t - a/T. Therefore, the approach developed for Volterra equations also works. Numerical results for a typical case are given in the next example. Example 2. With c(t) = e-‘, a = 0.4, and T = 2, the bounds listed in Table 2 were obtained. Example 3. As a final example, consider the equation Lx(t) =x’(t)
-x($t)
=y(t),
0
(1%
with initial condition x(0) = 0. This is a simple prototype for a class of equations with variable delay. While the variable delay can create some difficulties in the numerical solution, it does not affect the error analysis. Integrating (19) as before, we find after some elementary calculations that IIPII
G TIl(I-K)-‘II,
P. Linz, R.L.C. Wang / Error bounds for Volterra and delay equations
206
where K is an integral operator with kernel k(t,
s) =
(
2, 0,
0
Another simple argument shows that if K is approximated by a piecewise linear degenerate kernel K,, then I!K-K,
II &h,
amd Theorem I is applicable. An actual numerical computation for T = 1, with y1= 41, yielded 2.08 6 I]L -’ 11< 2.72. The upper bound for IIL- ’ II appears to be within about 20% of the actual value (which is approximately 2.3). This is quite sufficient for practical purposes. To show what role error bounding plays in the complete solution of a problem, we solved the equation .‘(t)=x($)+l,
O
by a rather simple-minded method, approximating x(t) by an nth-degree polynomial and collocating at n + 1 equally spaced points in [0, I]. With y1= 4 we obtained the approximate solution x,(t)
=
0.999996t
+ 0.250016t2 + 0.0208G4t3
+ 0.000667t4
(20)
with residual
II p,, II < 4 x lo-“. Thus (20) represents a quite good solution, with rigorous error bound
II *r-x, ll < 1.2 x lo-? This example illustrates one advantage of our error-bounding approach. Since the error bounding method is applicable regardless of the algorithm by which the approximate solution was obtained, it is possible at times to get good solutions cheaply. While the method used in this example is not suitable as a general algorithm, it is very satisfactory in many instances. Since we can verify the accuracy after the computation, the way in which the approximate solution was obtained can be tailored to the requirements and characteristics of an individual problem.
5. Summary
The method described here allows us to compute realistic error bounds for some Volterra and functional differential equations. The approach involves a conversion to a Fredholm equation, and works wherever such a conversion is possible. While the technique is not particularly useful for the computation of the approximate solution, it does provide us with a means for computing good error bounds with little additional work.
P. Linz, R. L.C. Wang / Error bouds for Volterru and delay equa!iorzs
207
There are two other problems that must be considered when we want to convert these theoretical results into useful numerical software. The first concerns the lack of precision in computer arithmetic. Inequality (13) is strict and its use leads to guaranteed error bounds if its value can be obtained. But when we evaluate it on a computer with floating point arithmetic, round-off error will affect the computation and possibly invalidate the results. We have used interval arithmetic successfully to overcome this difficulty. The second issue comes from the fact that the implementation of any algorithm is subject to programming errors. This is a concern in all software; normally it is addressed by thorough testing. But while extensive testing does uncover most mistakes, it is not completely reliable and some errors can slip through. The reliability of software can often be increased significantly by proving its correctness. This is an issue of substantial research interest in computer science. The results presented here allow us to make precise specifications of the result of a piece of numerical software and thereby open the possibility of using formal correctness proving methods to verify numerical software. That this can be done, and that it can catch some subtle errors, is demonstrated in [l]. We arc now extending this work to more complicated problems. The major stumbling block to the construction of reliable numerical software for larger problems is a lack of theoretical results along the lines of (13). For integral and delay equations, however, the problem is solved, so that in this area it is possible to construct efficient numerical methods that deliver strictly guaranteed results.
References Proceedings SCAN-90 Conference, Aibena, Bulgaria ( 1990). L. Collatz, Flc:lctional Analysis and h’umerical Mathematics (Academic Press, New York, 1966). E.W. Kaucher and W. Miranker, Self-Validating Numerics for Function Space Problems (Academic Press, New York, 1984). C. Kenney, P. Linz, and R.L.C. Wang, Effective error estimates for the numerical solartion of Fredholm integral equations, Computing 42 (1989) 353-362. P. Linz, Precise bounds for inverses of integral equation operators, Internat. J. Comput. Math. 24 (1988) 73-81. P. Linz, Bounds and estimates for condition numbers of integral equations, SLAM J. Numer. Anal. 28 (1991)
M. Archer and P. Linz, On the verification of numerical software, in:
227-235.
Peter Linz Di~v3hxr of Computer
201
Science, University of California, Davis, CA 95616, USA
Richard L.C. Wang Lockheed Missiles and Space Company Inc., Sunnyvale, CA 94086, USA
Abstract
Linz, P. and R.L.C. Wang, Error bounds for the solution of Volterra and delay equations, Applied Numerical Mathematics 9 (1992) 201-207. The construction of efficient and reliable numerical software requires a theory for two distinct issues: a convergence analysis and a way of bounding the error in the computed results. While the first of these questions has received much attention, less is known about the second one. Here we describe a method for bounding the error in the solution of Prcdhoh;i integral equations and show how the approach can be extended to Volterra and delay equations.
.
Introduction
In the error analysis of numerical methods, there are two major, complementary issues: first, the questrtin rsf convergence and stability, and second, the need for computable error bounds. For the sake of discussion, we consider the abstract operator equation LX ‘Y, (1 1I where L : X -+ Y is a linear operator between the normed spaces X and Y. Approximate solutions are obtained by discretization which produces an approximation x,, to x. In convergence analysis we establish results like lim ~~x--xJ~ =0,
fzdW
(2 I1
or the stronger
II x -x, ll = O(?P).
(3)
Convergence analysis is prominent in the literature or‘ numerical analysis; what we can conclude from (2) is that the method works in the sense that we can obtain arbitrarily good answers by making a fine enough discretization. If we also ave resuhs of type (3L then we can Predict the 0168-9274/92/$05.00
0 1992 - Elscvicr Science Publishers B.V. All rights rcscrved
P. Linz, R.L. C. Warlg / Error bounds for Volterra and delay equations
202
expected efficiency of the method. Convergence analysis is therefore indispensable in the selection of effective numerical algorithms. Usually, though, convergence analysis gives little information on the results of a specific computation. For this, we need a theory that produces actual computable error bounds for finite values of n, that is, we want results of the type Iix--&J
G77n,
(4)
where qn is computable for any given n. Recently, there has been considerable interest in this second aspect. Collatz [2], for example, has investigated the use of monotonicity arguments, while others (e.g. Kaucher and Miranker [3]) have used fixed-point arguments. On the whole, however, methods for computing error bounds have not been studied widely. In our work, we have investigated the effective computation of error bounds, using the following elementary idea. Suppose that we have found, by whatever means, an approximate solution x,, of (1). We substitute this into (1) and compute the residual ’
Pn
(5)
=I&,-y.
If (1) is not singular, that is, if the operator L has a bounded inverse, we get immediately that
II x -x, II < II L - l I! II p,,II
(6)
l
For this to be useful it is necessary that both ]IL- ’ 11and IIp,, II can be bounded. Additionally, we wan.t such bounds to be realistic, otherwise the use of (6) leads to a pessimistic estimate of the error. Often there are bounds available, but they are so poor that they are useless in practice. For computational efficiency, we would like a bound that approaches the true value closely as n increases. efinition 1. Let Q be an element of some normal linear space. Then the sequence {B,(Q)) is said to be a precise upper bound of IIQ II if
B,(Q) 2 II Q II for all n,
(7)
lim B,(Q) = II Q Il.
(8)
II -+m
While the approach we investigate here has been known for a long time, it has generally been ignored because it was felt that getting precise bounds on IIL- ’ II was quite difficult. However, our recent work has shown that this belief is not justified for many equations of practical interest. In this paper, we demonstrate how the method can be applied to Volterra integral equations and some delay-differential equations.
2. Frecise error bounds for Fred
integral equations
Consider as specific case of (1) the Fredholm intczral equation of the second kind
x(t)-/‘k(l, s)x(s)
ds=y(t), 0 which we write in operator notation as (I-K)i=y.
O,
(9) (10)
P, Linz, R.L. C. Wang / Error bounds for Vo1:c~r.ra and delay eqtlatiom
203
As the underlying spaces we will take C[O, I] with the naximum norm. In [S] we developed a method for computing realistic bounds on ll(Z -K)- : // when the kernel k(t, s) qf (9) is smooth. The technique uses a degenerate kernel ap roximation based on piecewise linear polynomials on a mesh with knots [t 1, t,, . . . , t,], wher ir = I/(n - I), ti = (i - 1)/r. As a first step, we replace (10) with X-KK,x=y,
PI) where K,, is the integral operator in which k(t, s) is replaced by its bilinear degenerate kernel approximation k,,(t7 s, = i i=l
i
k(tiY
tj)+i(f)@j(s)*
(12)
j=l
Here we use pi to denote the piecewise linear hat function basis for approximation and K and D to denote n x n matrices with respective elements Kij = k(ti, tj~,
Since K, is a bounded perturbation of K, the relati elementary one. It is a little harder to establish the and the easily computed inverse of I - DK. The foll recently established.
(Z-K)-’ and (I--K,)-’ is an between the inverse of (I - K,) ts relating these quantities were
Theorem 1. Let K be some integral operator which can be approximated by operators K, whose kernels have the form (12) and are such that 11K - K,, &I+ 0. Then II(I-K)-*
II < -
lI(I-K)-‘II
3
1 +hIIK(I-DK)-‘(I IIK-K,J(l+hIIK(I-DK)-‘Ilj’
l-
(13)
III+K(I-DK)-‘DII
l+ IIK-K,il
I(I+K(I-DK)-‘DII’
(14)
provided that the denominator in (13) is positive. Furthermore, the upper bound (13) is precise. Proof. The upper bound was first discussed in [5], later results in [4] provided the lower bound.
The fact that (13) is precise was established in [6].
0
The bounds established in this theorem are useful in the design of efficient computational methods for Fredholm equations with smooth kernels. If k(s, t) is twice continuously differentiable, then 11K - K,, II < :h’( k,, + kOZ+ $h”k,,),
(15)
where I’ G -‘ii
-
#+j
at’&
II
4
W) II
l
P. Linz, R.L.C. Wang / Error bounds for Volterra and delay equations
204
With a bound on I]K - K, 11everything in (13) and (14) is completely computable. How this is used in the construction of an actual algorithm for the effective solution of Fredholm integral equations is fully explained in [4]. For some functional equations the smoothness conditions assumed in (35) do not hold, and the theory must be modified to cover this case. We illustrate this with some specific cases.
3. Application to Volterra equations We now consider the Volterra integral equation x(t) -
/‘k(t, s)x(s)
0~ t d T,
ds =y(t),
(17)
0
which can be viewed as a special case of a Fredholm equation involving a kernel with a finite jump discontinuity along t = S, and k(t, s)=O,
fors>t.
Although degenerate kernel approximations are not natural for such kernels, we can still use them. Let D, be the set of grid squares along the diagonal t = s, that is, D, ={(t, s)I It--s1 d), and let D, = [0, l] x [0, l] - D,. In D, standard b ounds for the approximation functions apply. In D, we have max I!<(& s)-&Jt, (t,+=D,
of smooth
s)] < max ]k(t, s)]. O
If we assume sufficient smoothness for the kernel in 0 G s < t, then
II K - K, II \< $h2( k*[j + gz2k,, + I&) + h II k II0 P;Vhilesuch a first-order approximation is not useful for the computation of the solution of (17), it still works for bounding the operator inverse. xample 1. Consider equation (17) with k( t, s) = - t
IIK- K, II < $P(l
*s*and
T = 1. Then
+ ih2) + h.
The computed bounds for ]I(I - K)- ’ II are listed in Table 1. Note that in the case of Volterra equations there is an analytical upper bound for the inverse ]](I-K)-‘I]
but this often gives quite pessimistic results. Table 1
Table 2
n
Lower bound
Upper bound
t1
Lower bound
Upper bound
10 20 30
1.147 1.279 1.315
1.858 1.539 1.476
11 21 31
2.09 2.34 2.42
3.88 3.05 2.86
P. Linz, R.L.C. Wang / Error bounds for Volterra and delay equations
ounds for some delay di
205
rential equations
Consider as example the delay equation
I
Lx(t) =x’(t)
-c(t)x(t
-a)
=y(t),
a!
(18)
with initial condition x(t)=O,
Ogt
This equation is a very simple example, representative of the class of equations with fixed delay. We can integrate (18) to obtain the integral equation x(t)
-
jTk(t,
s)x(s)
ds
=z(t),
0
where
and z(t)
=
t >,a!.
After some elementary computations we then get IIL-‘II G (T-a)ll(l-K)-‘II, where K is the integral operator with kernel k(t, s) =
i
Tc(a + sT)~ o 9
t>a/T,O
We now have exactly the same situation as for Volterra equations, that is a kernel which is sectionally smooth with a finite jump discontinuity along the line s = t - a/T. Therefore, the approach developed for Volterra equations also works. Numerical results for a typical case are given in the next example. Example 2. With c(t) = e-‘, a = 0.4, and T = 2, the bounds listed in Table 2 were obtained. Example 3. As a final example, consider the equation Lx(t) =x’(t)
-x($t)
=y(t),
0
(1%
with initial condition x(0) = 0. This is a simple prototype for a class of equations with variable delay. While the variable delay can create some difficulties in the numerical solution, it does not affect the error analysis. Integrating (19) as before, we find after some elementary calculations that IIPII
G TIl(I-K)-‘II,
P. Linz, R.L.C. Wang / Error bounds for Volterra and delay equations
206
where K is an integral operator with kernel k(t,
s) =
(
2, 0,
0
Another simple argument shows that if K is approximated by a piecewise linear degenerate kernel K,, then I!K-K,
II &h,
amd Theorem I is applicable. An actual numerical computation for T = 1, with y1= 41, yielded 2.08 6 I]L -’ 11< 2.72. The upper bound for IIL- ’ II appears to be within about 20% of the actual value (which is approximately 2.3). This is quite sufficient for practical purposes. To show what role error bounding plays in the complete solution of a problem, we solved the equation .‘(t)=x($)+l,
O
by a rather simple-minded method, approximating x(t) by an nth-degree polynomial and collocating at n + 1 equally spaced points in [0, I]. With y1= 4 we obtained the approximate solution x,(t)
=
0.999996t
+ 0.250016t2 + 0.0208G4t3
+ 0.000667t4
(20)
with residual
II p,, II < 4 x lo-“. Thus (20) represents a quite good solution, with rigorous error bound
II *r-x, ll < 1.2 x lo-? This example illustrates one advantage of our error-bounding approach. Since the error bounding method is applicable regardless of the algorithm by which the approximate solution was obtained, it is possible at times to get good solutions cheaply. While the method used in this example is not suitable as a general algorithm, it is very satisfactory in many instances. Since we can verify the accuracy after the computation, the way in which the approximate solution was obtained can be tailored to the requirements and characteristics of an individual problem.
5. Summary
The method described here allows us to compute realistic error bounds for some Volterra and functional differential equations. The approach involves a conversion to a Fredholm equation, and works wherever such a conversion is possible. While the technique is not particularly useful for the computation of the approximate solution, it does provide us with a means for computing good error bounds with little additional work.
P. Linz, R. L.C. Wang / Error bouds for Volterru and delay equa!iorzs
207
There are two other problems that must be considered when we want to convert these theoretical results into useful numerical software. The first concerns the lack of precision in computer arithmetic. Inequality (13) is strict and its use leads to guaranteed error bounds if its value can be obtained. But when we evaluate it on a computer with floating point arithmetic, round-off error will affect the computation and possibly invalidate the results. We have used interval arithmetic successfully to overcome this difficulty. The second issue comes from the fact that the implementation of any algorithm is subject to programming errors. This is a concern in all software; normally it is addressed by thorough testing. But while extensive testing does uncover most mistakes, it is not completely reliable and some errors can slip through. The reliability of software can often be increased significantly by proving its correctness. This is an issue of substantial research interest in computer science. The results presented here allow us to make precise specifications of the result of a piece of numerical software and thereby open the possibility of using formal correctness proving methods to verify numerical software. That this can be done, and that it can catch some subtle errors, is demonstrated in [l]. We arc now extending this work to more complicated problems. The major stumbling block to the construction of reliable numerical software for larger problems is a lack of theoretical results along the lines of (13). For integral and delay equations, however, the problem is solved, so that in this area it is possible to construct efficient numerical methods that deliver strictly guaranteed results.
References Proceedings SCAN-90 Conference, Aibena, Bulgaria ( 1990). L. Collatz, Flc:lctional Analysis and h’umerical Mathematics (Academic Press, New York, 1966). E.W. Kaucher and W. Miranker, Self-Validating Numerics for Function Space Problems (Academic Press, New York, 1984). C. Kenney, P. Linz, and R.L.C. Wang, Effective error estimates for the numerical solartion of Fredholm integral equations, Computing 42 (1989) 353-362. P. Linz, Precise bounds for inverses of integral equation operators, Internat. J. Comput. Math. 24 (1988) 73-81. P. Linz, Bounds and estimates for condition numbers of integral equations, SLAM J. Numer. Anal. 28 (1991)
M. Archer and P. Linz, On the verification of numerical software, in:
227-235.