Download Numerical Approximations for Calculus and Differential Equations I - Prof. J. Lega and more Study notes Differential Equations in PDF only on Docsity!
Calculus and Differential Equations I
MATH 250 ANumerical approximations Numerical approximations^ Calculus and Differential Equations I
Numerical approximation of definite integrals^ You should already be familiar with the left and right-handRiemann sums used in the definition of the definite integral:^ I^ ≡
∫^ b^ f^ (x)^ dx^ =a
n−^1 ∑ lim^ f^ (xi n→∞^ i= ) ∆x^ =^ lim n→∞
n∑^ f^ (x) ∆x,i^ i=
where ∆x^ = (
b^ −^ a)/n^ and
x=^ a^ +^ i^ ∆xi^
The left and right rules respectively approximate the integral
I
with LEFT(n) and RIGHT(
n), where n− LEFT(n) = (^1) ∑^ f^ (x) ∆x,^ i^ i=
RIGHT(n) =
n∑^ f^ (x) ∆x,i^ i=
with ∆x^ and^
xdefined as above.i^
Link to Geogebra Numerical approximations
Calculus and Differential Equations I
Numerical approximation of definite integrals (continued)^ The midpoint rule consists in approximating the definiteintegral by evaluating
f^ at the midpoint between
xand^ x:i^ i+ n− MID(n) = (^1 ∑x+^ xi^ i+1 f^2 i= )^ ∆x.
The trapezoid rule approximates the area under the graph of
f
between^ xandi^
xwith the area of the correspondingi+^ trapezoid:TRAP(
(^ n− (^1) ∑f^ (n) = i= x) +^ f^ (x)i^ i+1^2
)^ ∆x.
From the above formula, one can see thatTRAP(
(^1) n) = (LEFT( 2 n) + RIGHT(
n))^. Numerical approximations
Calculus and Differential Equations I
Overestimates and underestimates^1 Assume that
f^ is increasing between
a^ and^ b^ and that we approximate^ I
∫^ b = f^ (x)^ dxa^ with LEFT(n
). Which of the
following statements is correct?^1 LEFT(n) is an underestimate^2 LEFT(n) is an overestimate^3 LEFT(n) is exact 2 If^ f^ is increasing on [
a,^ b], then^ ∫^ b LEFT(n) ≤^ f^ (x)^ dxa
≤^ RIGHT(n)
3 Similarly, if
f^ is decreasing on [
a,^ b], then RIGHT(n)^ ≤
∫^ b^ f^ (x)^ dx^ ≤a
LEFT(n). Numerical approximations
Calculus and Differential Equations I
Overestimates and underestimates (continued)^ In order to ensure that TRAP(
n) is an overestimate, which of the following requirements do we need?^1 f^ is increasing^2 f^ is concave up^3 f^ is concave down^4 f^ is decreasingIf^ f^ is concave up on [
a,^ b], then^ ∫^ b MID(n) ≤^ f^ (x)^ dx^ a
≤^ TRAP(n). Similarly, if^ f^
is concave down on [
a,^ b], then ∫ TRAP(n) ≤ b^ f^ (x)^ dx^ ≤^ a
MID(n). Numerical approximations
Calculus and Differential Equations I
Example of application^ Assume that the function
f^ is positive, decreasing, and concave down on [
a,^ b]. Let^ I^ =
∫^ b^ f^ (x)^ dx. a Assume that the values of LEFT(10), RIGHT(10), TRAP(10),and MID(10) are, in random order, given by
0.^703 ,^0.^724 ,
0.^735 ,^0.^745
Use the above to assign a value to each of LEFT(10),RIGHT(10), TRAP(10), and MID(10).Then, indicate which of the statements below is correct:^1 0.^703 ≤^ I
≤^0.^7242 0. 724 ≤ I ≤^0.^7353 0. 735 ≤ I ≤^0.^745 Numerical approximations
Calculus and Differential Equations I
Approximation errors^ If we use a numerical method, say the left rule, to approximatea definite integral
I^ , we define the absolute error
E(n), asL E(n) =^ I^ −^ LEFT(L
n). One can show that
|E(n)|^ and^ |L
E(n)|^ are^ linear functions ofR^ 1 /n. Similarly,^ |ET
(n)|^ and^ |E M^ (n)|^ decrease quadratically as
n^ is
increased.This can be improved by using Simpson’s rule, given by^ SIMP
(^1) (n) = (2 MID( 3 n) + TRAP(n
))^.
One can show that the error
|E(n)|^ decreases like 1S^
(^4) /n.
Numerical approximations
Calculus and Differential Equations I
Numerical integration of ODEs
dy=^ g^ (x,^ y^ )dx^ The above differential equation may formally be integrated as^ y^ (
x^ +^ h)^ −^ y^ (x) =
∫^ x+h^ g^ (t,^ y^ (t x
))^ dt.
If we know^ y^ (
x), a numerical approximation of
y^ (x^ +^ h) may
thus be obtained by finding an estimate of the integral in theright-hand-side of the above equation.Euler’s method consists in assuming that
g^ (t,^ y^ (t)) is
constant on the interval [
x,^ x^ +^ h], and equal to
g^ (x,^ y^ (x)),
where^ x^ is the left end-point of the interval.We thus have
y^ (x^ +^ h)^ ^ y^
(x) +^ h g^ (x,^ y
(x)). Numerical approximations
Calculus and Differential Equations I
Taylor polynomials (continued)^ The Taylor polynomial of degree
n, centered at
x^ =^ a, of a
function^ f^ is given by P(x) =^ f^ (a) + (x^ −n
(x^ −′ a)f (a) + (^2) a)′′f^ (a) +^ · · · 2!
n(x − a)( + f^ n! n)(a).
Of course, the above assumes that
f^ has is at least
n^ times
differentiable near
a. In what follows, we assume that
f^ is
smooth, for simplicity.One can show that the error made by replacing
f^ by its Taylor
polynomial of degree
n^ is given by f^ (x) =^ P(x) +n
R(x),^ Rn
(x^ −^ a(x) = n n+1)(n+1)f^ (ξ) (n + 1)!^
where^ ξ^ ∈^ (a,
x).
Link to d’Arbeloff Taylor Polynomials software Numerical approximations
Calculus and Differential Equations I
Numerical approximation of definite integrals revisited^ ∫^ b^ f^ (x)^ dxa
∫^ n− 1 x∑i+1 =^ xi i= f^ (x)^ dx,^ xi^
=^ a^ +^ i∆x,^
b^ −^ a∆x = .n^
For the left rule, for each
x^ ∈^ [x,^ x], we havei^ i+ f^ (x) =^ f^ (x) + (i^
′x − x) f (ξ(xi )),^ ξ(x)^ ∈
(x,^ x).i^
From this formula, we see that if
′^ f is positive and bounded by M between
xand^ x, theni^ i+1^0 ≤^ f^ (x)^ −^
f^ (x)^ ≤^ M(x^ i^
−^ x),i^
which gives that LEFT is an underestimate and ∣∫^ xi+1∣∣^ f^ (x)^ dx∣^ xi
∣∣∣ − f (x) ∆xi ∣^ ∫^ xi+1≤^ M^ (x^ xi −x)^ dx^ =^ M^ i^
(^2) (∆x). 2
Numerical approximations
Calculus and Differential Equations I
Approximation errors^ For the midpoint rule, we have, with
x+^ xi^ i+1 m = 2
f^ (x) =^ f^ (m)+(
′m−x) f (m)+ 12 ′′(x^ −m)f^ ( 2
ξ(x)),^ ξ(
x)^ ∈^ (x,^ x).i^
We can check that^ ∫^ xi+1^ xi
(f^ (m) + (m^ −
)^ ′ x) f (m)dx^ =^ f^ (m) ∆x, ′′^ so that if f is positive and bounded by M between
xandi^
x, theni+1^0 ≤^ f^ (x
[) − f^ (m) + (
′m − x) f (m) ]^ M≤^ (x^ −^ m^2
which gives that MID is an underestimate and that the larger′′ |f^ |, the larger the error.^ Numerical approximations
Calculus and Differential Equations I
Approximation errors (continued)^ Finally, for the trapezoid rule, we have (by integration byparts)^ ∫^ xi+1^ xi
f^ (x)^ dx^ = [(x
xi+1 − m)f (x)] xi^ ∫^ xi+1−^ (x^ −^ xi
′m) f (x)^ dx.
We can use a Taylor expansion for
′ f (x) near^ x^ =^ m^ to see
′′^ that if f is positive between
xand^ x, then TRAP gives ani^ i+ overestimate.Moreover, the error over an interval of length ∆
x^ is bounded (^3) by M(∆x)/12, where M^ is the maximum of
′′ |f |^ over that
interval.For all of these methods, if the error on the integral between xand^ xis of order (∆i^ i+^
p+1x), then the error on the integral between^ a^ and
b^ is of order 1
p^ /n, where^ n^ is the number of
sub-intervals.^ Numerical approximations
Calculus and Differential Equations I
Numerical integration of ODEs revisited^ A numerical method is consistent if the local discretizationerror goes to zero as
h^ →^ 0. A numerical method is convergent if the global discretizationerror goes to zero as
h^ →^ 0. Typically, one uses Taylor expansions to decide whether anumerical method is consistent and convergent.A numerical method may also be unstable, in the sense that anumerical solution to
′^ y =^ λy^ with λ <^ 0 can display growth. These are topics typically discussed in an introductory courseon numerical analysis, such as MATH 475.Finally, one should keep in mind that a numerical method is amap of the form
y=^ G^ (yn+1^ n
,^ n), and that if
G^ is nonlinear,
chaos may be observed.
Link to Chaos on the Web Numerical approximations
Calculus and Differential Equations I
