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Euler’s method
For the initialvalue problem
y
0
=
f
(
t
,
y
)
,
y
(
t
0
) =
y
0
,
use the formulas
t
n
+
1
=
t
n
+
h
,
y
n
+
1
=
y
n
+
hf
(
t
n
,
y
n
)
,
to compute iteratively the points
(
t
1
,
y
1
)
,...,
(
t
n
,
y
n
)
, using step
size
h
. The piecewiselinear function connecting these points is
the Euler approximation to the solution
y
(
t
)
of the IVP for
t
0
≤
t
≤
t
n
.
M.I. Bueno
Differential Equations and Linear Algebra
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Differential Equations and Linear Algebra
Example
y
0
=
t
+
y
,
y
(
0
) =
1
[
0
,
2
]
,
h
=
0
.
5
.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
2
4
6
8
10
12
M.I. Bueno
Differential Equations and Linear Algebra
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View Full Document Example
y
0
=
t
+
y
,
y
(
0
) =
1
[
0
,
1
]
,
h
=
0
.
1
.
n
t
n
Approx.
y
n
Real
y
(
t
n
)
Error

y
(
t
n
)

y
n

Percent. erro
r
0
0
1
1
0
0
%
1
0.1
1.1
1.1103
0.0103
0
.
93
%
2
0.2
1.22
1.2428
0.0228
1
.
84
%
3
0.3
1.362
1.3997
0.0377
2
.
69
%
4
0.4
1.5282
1.5836
0.0554
3
.
5
%
5
0.5
1.721
1.7974
0.0764
4
.
25
%
6
0.6
1.9431
2.0442
0.1011
4
.
95
%
7
0.7
2.1974
2.3275
0.1301
5
.
59
%
8
0.8
2.4872
2.6511
0.1639
6
.
18
%
9
0.9
2.8159
3.0192
0.2033
6
.
73
%
10
1
3.1875
3.4366
0.2491
7
.
25
%
M.I. Bueno
Differential Equations and Linear Algebra
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
1.5
2
2.5
3
3.5
M.I. Bueno
Differential Equations and Linear Algebra
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View Full Document Error in Euler’s method
How do we estimate the error produced by Euler’s method
when we don’t have an analytic solution, the case of real
interest for using numerical methods?
There are two kinds of error that can arise:
Roundoff error,
Discretization error.
M.I. Bueno
Differential Equations and Linear Algebra
Roundoff error
Produced by rounding or chopping numbers at each stage in
the computation.
In practice, all calculators and computers have limitations in
their computational accuracy.
Roundoff errors can accumulate signiﬁcantly after many steps.
M.I. Bueno
Differential Equations and Linear Algebra
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View Full Document Discretization errors
Results from the process itself.
In Euler’s method, this process is the use of the linear
approximation of the tangent line instead of the true solution
curve in stepping from one value to the next.
Local discretization error
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This note was uploaded on 03/14/2010 for the course MATH 3C taught by Professor Jacobs during the Fall '08 term at UCSB.
 Fall '08
 JACOBS
 Formulas

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