2
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
0.750
0.875
1.000
1.125
exact
computed
1-(2x-x )
1.250
x
Figure 3.1: Evaluation of 1 − (2x − x2) exactly (dashed curve)
and in a floating-point representation with β = 2 and p = 6
(solid line).
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
6
5
4
3
2
x − 6x + 15x − 20x + 15x − 6x + 1 vs x
20
IEEE single precision
value × 10
10 7 (x−1)6
7
10
0
-10
-20
0.99990
0.99995
1.00000
Figure 3.2: Numerical evaluation of a polynomial in the power form
(1 − x)6 (dashed line and right vertical axis) and in the expanded form
x6 − 6x5 + 15x4 − 20x3 + 15x2 − 6x + 1 (solid line and left vertical axis),
using IEEE single-precision floating-point arithmetic in both cases.
The computed values plotted on the vertical axes have been multiplied
by 107. The value of mach is 1.192 × 10−7; the value of one ulp is
5.96 × 10−8 in the range shown.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
Roots of (z−1)20 = 2 −24
0.50
m
[z − 1]
0.25
0.00
-0.25
-0.50
-0.50
-0.25
0.00
0.25
0.50
[zm − 1]
Figure 3.3: The points represent the error in the mth root, zm − 1,
of the polynomial equation (z − 1)20 = 2−24, for m ∈ (0 : 19).
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
y
4.0
3.5
y = 1–x – b
3.0
2.5
2.0
1.5
1.0
0.5
0.0
–0.5
x0
x1
x2
1.0
2.0
3.0
4.0
x
Figure 3.4: Illustration of the first two steps of Newton-Raphson
iteration for the function f(x) = 1/x − b, with b = 0.5.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
f(x4)
f(x3)
f(x2)
f(x1)
x0 x 1 x2 x3 x4
Figure 3.5: Approximate evaluation of the integral
the right-hand rectangle rule.
x4
x0
f(x) dx using
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
Global error in Euler’s method
0.00100
Error
0.00075
0.00050
0.00025
0.00000
0.00000
0.00002
0.00005
0.00007
0.00010
Step size
Figure 3.6: Absolute value of the global error in Euler’s method for L = 1 and a = 1.
The computation was performed using IEEE-754 single-precision arithmetic. The global
truncation error predicted by Eq. (3.111) is shown as a dashed line. For step sizes larger
than approximately h = 0.00001, the error decreases roughly linearly in h, with considerable
scatter due to rounding error. For step sizes much smaller than h = 0.00002, the error
increases rapidly because of an accumulation of rounding errors.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
(ha)
–1
1
(ha)
Figure 3.7: The region of stability of Euler’s method, computed
using the test equation y = ay, is a disk of radius 1 centered at
z = −1, where z = ha.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
cn
100
80
60
40
20
0
-20
-40
-60
-80
-100
50
100
x
Figure 3.8: The solution computed to the equation y = − y by Euler’s method with a step
size h = 2.1, which is outside of the region of stability. The initial condition is y(0) = 1.
The analytical solution is so heavily damped, and is of such a small magnitude, that on the
scale of this graph it nearly coincides with the horizontal axis.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
cn
1.00
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.0
0.00
0.5
1.0
1.5
yn
1.00
2.0
x
Figure 3.9: The computed (solid line) and exact (dashed line)
solutions of the differential equation y = −y, illustrating a
weak instability. The first step of the computed solution was
taken using Euler’s method; the remaining steps were taken
with the midpoint method.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
(ha)
–1
1
(ha)
Figure 3.10: The region of stability of the backward Euler method,
computed using the test equation y = ay, is the set of points outside,
and on the circumference of, a disk of radius 1 centered at z = 1, where
z = ha.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press
(ha)
2
1
(ha)
0
–1
–2
–1.00
–0.75
–0.50
–0.25
0.00
0.25
Figure 3.11: The region of absolute stability of the midpoint-trapezoidal
predictor-corrector method, computed using the test equation y = ay,
in the plane of complex z = ha.
From: Modern Mathematical Methods for Physicists and Engineers, by C. D. Cantrell © 2000 Cambridge University Press