math 151a - spring 2020 - week 10

Math 151A - Spring 2020 - Week 10

Today: Numerical Integration

• Composite Rules

• Degree of Precision

• Gaussian Quadrature


Composite Trapezoidal Rule

Z b

af (x)dx = h

0

@1

2f (a) +

n�1X

j=1

f (xj) +1

2f (b)

1

A� b � a

12h2f 00(⇠)

h =b � an

mesh x0, ..., xn

Tmr


Composite Simpson’s Rule

Z b

a

f (x)dx =h3

0

@f (a) + 2(n/2)�1X

j=1

f (x2j) + 4n/2X

j=1

f (x2j�1) + f (b)

1

A�b � a180

h4f (4)(⇠)

h =b � an

mesh x0, ..., xn n even

Error


Composite Midpoint Rule

Z b

af (x)dx = 2h

n/2X

j=0

f (x2j) +b � a

6h2f 00(⇠)

h =b � an + 2

mesh x�1, x0, ..., xn+1 n even

Gmr

T grimsinink

ormyeloidmesh


Example 1. Determine the value of n required to approximateR 20 e2x sin 3xdx with an accuracy of 1⇥ 10

�4using the Composite

Midpoint Rule.

t.mn b ah2f S c 1 10 4 III h bnz n z

6fix e sm3x

f x 2e sin3 3e cos3X

f X 4e sin3 6e ca3 62 3 9e2 sih3X

Se2sin3x the 2 as3

1BI h f G e 151421 max tf x I6 b XEfo23

26 212ffgox.gg 5e2Xsin3Xtl2e2Xas3xI




Midpoint Rule.

Iknew f 5 I 5 f 212 xmefox.gg l5e2xsin3X i112e2xas3x1

EKITI.ES l 5e2xltll2e2xl

E f 212 5e4 the n 12 n

f 212 Hell

En l7eseenext slide

piety 12

n 4




Midpoint Rule.

f 1 Che c if8117M I6th1212 104

Yate 104 c In1212

n12 4 I 3517.894 7 n1223518

nTr3


Example 2. Write a program to approximate the integral

Z 2

0x2e�x2

dx

using the Composite Trapezoidal rule with n = 8, Composite Simpson’s

rule with n = 8, and the Composite Midpoint rule with n = 6.

See codes


The degree of precision of a quadrature formula is the largest positive

integer n such that formula is exact for xk for each k = 0, 1, ..., n.

Trapezoidal rule has degree of precision 1.

Simpson’s rule has degree of precision 3.


Example 3. A program was written that implements the composite

trapezoidal method to approximate integrals of the formR 10 f (x)dx . This

program was tested on monomials of increasing degrees using double

precision arithmetic and the number of panels N = 1000. The results are

as follows. Is this program working correctly? Explain your answer.

f (x) Computed Approximation

1 1.000500000000

x 0.500500000000

x2 0.333833500000

x3 0.250500250000

x4 0.200500333333

TrapezoidalruleDop I not Iexactfor Axl I not'zand fix _X

ft dx Xl I 0 1 program is not Wahabcorrectly because not

fixDX Ix lot I 0 12 exact fr fix L fix X


Gaussian Quadrature

Z 1

�1f (x)dx ⇡

nX

i=1

ciP(xi )

Degree of precision is 2n � 1.

xi values come from the roots of Legendre polynomials. Some Legendre

polynomials:

P0(x) = 1, P1(x) = x , P2(x) = x2 � 1

3

P3(x) = x3 � 3

5x , P4(x) = x4 � 6

7x2 +

3

35

Sune Anduin

1 If


Example 4.

(a) Derive the 3-point (Legendre) Gaussian Quadrature to approximateR 1�1 f (x)dx .

(b) What is the degree of precision in (a)?

(c) Show that the 3-point Gaussian quadrature can be used for

approximatingR ba f (x)dx by doing a simple change of variables and

applying this to approximate

Z 4

0

cos x

x + 1dx

b 3 point n 32n 1 213 1 50

a fi fix DX C FIX tGfCXz tczfCX3


Example 4. continued


P X3 ZX O fifandx Iff ff t f o tGffffX x23g 0 few 1 fCx7 x fCX7 X2X o tf

Xi Pg 112 0 X3 ff IfidXXl 2

G Itcz g 15CITE 15 2

HX X fixdx zxyj oz ffcr FEG

09ffflG.co fftG 0

4 5




fun _x2 fix'dx XIIIafrtttecoy.ia.fr i2Ia

Icz3atczgIqtCztCz2 cz Kg 2

I 3 city24 11 If flxldX2 fffE 185ft 1 Igf Ff

94 59103 59






Z 4

0

cos x

x + 1dx

fi fix DXa Ffl Ff Igflo t Ff ff

fabflydy L f z DX l l l Ca b

i i iy bzaxtb.at

2

y b a X a

dy KID






Z 4

0

cos x

x + 1dx

figfxdxegfl.FI t8qflol Igf ff

fabfly1dy f f 4 4 112 bazdx

BEf if Hatta DXbzafigfflbFH.tn t f f HI

Eff fEt






Z 4

0

cos x

x + 1dx

bfmdfzafqffcbalt.FI btoy 8qffbtay a o b 4u I fly cost

Igf lba1FsIb x11

fo af dxa2 Eff.tk t8qffz tIgff4ffz4

0.2798850


Example 5. Show that the formula Q(P) =Pn

i=1 ciP(xi ) cannot have

degree of precision greater than 2n � 1, regardless of the choice of

c1, ..., cn and x1, ..., xn. (Hint: construct a polynomial that has double

root at each of the xi values.)

Suppose by antradichui that QLP has degreeofprecision2h

Consider PIX x Xi X Xa X Xn whichhasdegree2h

f if DX a ftCiflXoSince DOP 2h then f pcxtdx f.ciPHIonRHS CiPui

on LHS f pcxjdx f.it xixXiTdx OZO

Antrudillin 070 Therefore DOP is at mist 2n 1

math 151a - spring 2020 - week 10

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