Andrzej Pownuk - Math 4329 (Numerical Analysis)

Andrzej Pownuk - Math 4329 (Numerical Analysis)

Table of Contents

1Numerical integration

1.1Introduction

1.2Rectangle method

1.2.1Introduction

1.2.2Example calculate

1.2.3Calculate the following integral by using the rectangle method (use the left sum).

1.2.4Calculate the following integral by using the rectangle method for (use the right sum).

1.2.5Calculate the following integral by using the rectangle method (use the mid sum).

1.2.6Calculate the following integral by using rectangle method (use right sum).

1.2.7Error estimates

1.1.1(*) Calculate the integral by using the Reimann sums.

1.3Trapezoidal method

1.3.1Introduction

1.3.2Calculate the following integral by using the trapezoidal method .

1.3.3Calculate the following integral by using the trapezoidal method .

1.3.4Calculate the following integral by using trapezoidal method .

1.3.5Calculate the following integral by using trapezoidal method .

1.3.6Calculate the following integral by using trapezoidal method .

1.3.7Error estimation ,

1.3.8Error estimation ,

1.4Simpson method

1.4.1Introduction

1.4.2Calculate the following integral by using the Simpson's method .

1.4.3Calculate by using Simpson method for . (*)

1.4.4Calculate by using Simpson method for .

1.4.5Calculate by using Simpson method for .

1.4.6Calculate the following integral by using the Simpson's method .

1.4.7Calculate the following integral by using the Simpson's method .

1.4.8Calculate the following integral by using the Simpson's method .

1.5(*) Newton's Cotes method for n=3 (Simpson's 3/8 rule)

1.1.2Calculate the following integral

1.6(*) Forth order interpolation

1.1.3Calculate the following integral

1.1.4Calculate the following integral

1.7(*) Newton-Cotes integration

1.8Error in numerical integration

1.8.1(*) Generalized Mean Value Theorem

1.8.2Rectangle method - error for one interval (left or right sums)

1.8.3Rectangle method - error for n-intervals (left or right sums)

1.1.4.1Calculate the following integral by using rectangle method and estimate the error.

1.1.4.2Calculate the following integral by using rectangle method (right sum) and estimate the error.

1.1.4.3Calculate the following integral by using rectangle method (right sum) and estimate the error.

1.1.4.4Estimate the error of numerical integration for , n=4 and the Rectangle Method.

1.1.4.5Estimate the error of numerical integration for , n=4 and the Rectangle Method.

1.8.4Trapezoidal method -error for one interval

1.8.5Trapezoidal method -error for n intervals

1.1.4.6Calculate the following integral by using trapezoid method and estimate the error.

1.1.4.7Calculate the following integral by using trapezoid method and estimate the error.

1.1.4.8Estimate the error of numerical integration for , n=4 and the Trapezoid Method.

1.8.6Simpson's method-error for one interval

1.8.7Simpson's method-error for n intervals

1.1.4.9Estimate the error of numerical integration for , n=4 and the Simpson’s Method.

1.1.4.10Estimate the error of numerical integration for , n=4 and the Simpson’s Method.

1.9Gauss Numerical integration

1.9.1Table of Gauss integration points

1.9.2(*) How to derive Gauss integration formula

1.9.3Calculate by using 1 point Gauss integration

1.9.4Calculate by using 2 points Gauss integration

1.9.5Calculate by using 3 points Gauss integration

1.9.6Calculate by using 1 point Gauss integration

1.9.7Calculate by using 2 points Gauss integration

1.9.8Calculate by using 2 points Gauss integration

1.9.9Calculate by using 2 points Gauss integration

1.9.10Calculate by using 3 points Gauss integration

1.9.11Calculate by using 3 points Gauss integration

1.9.12Calculate by using 3 points Gauss integration

1.9.13Calculate by using 4 points Gauss integration

1.9.14Calculate by using 3 points Gauss integration

1.9.15Calculate by using 2 points Gauss integration

1.9.16Calculate by using two points Gauss integration.

1.9.17Calculate by using two points Gauss integration

1.9.18Calculate by using two points Gauss integration

1.9.19Calculate by using three points Gauss integration.

1.9.20Calculate by using 4 points Gauss integration.

1.9.21Calculate by using 4 points Gauss integration.

1.9.22Calculate by using 5 points Gauss integration.

1.9.23Calculate by using 5 points Gauss integration.

1.10Review

1.10.1Summer 2015

1Numerical integration

1.1Introduction

Method 1

Method 2

Plot[Exp[x^2],{x,0,1},Filling->Axis]

Integrate[Exp[x]*Sin[x]*Log[x]/(1+x^2) ,x]=?

~

Presented methods do not work for more complicated mathematical functions.

Function is not Riemann integrable.

1.2Interpolation methods

If then

Select nodes

Find and interpolation polynomial

1.3Method of undetermined coeficients

For the following formula

Test functions

For we have

For we have

For we have

Finally, it is necessary to solve the system of equations

With the solution

This is the formula for the Simpson’s method.

1.4Rectangle method

1.4.1Introduction

Left sums

Right sums

Mid sums

Rectangle method

Mean value theorem for the integral

Left sums

Right sums

Mid sums

1.4.2Example calculate

% left sum

a=0;

b=1;

n=10;

dx=(b-a)/n;

sum=0;

for i=1:n

x=a+(i-1)*dx;

sum=sum+x*x*dx;

end

disp(sum);

n / /
10 / 0.28500 / 0.048333
100 / 0.32835 / 0.0049833
1000 / 0.33283 / 4.9983e-004
10000 / 0.33328 / 4.9998e-005

/*right sum*/

a=0;

b=1;

n=1000;

dx=(b-a)/n;

sum=0;

for i=1:n

x=a+i*dx;

sum=sum+x*x*dx;

end

disp(sum);

n / /
10 / 0.38500 / 0.051667
100 / 0.33835 / 0.0050167
1000 / 0.33383 / 5.0017e-04
10000 / 0.33338 / 5.0002e-05

/*mid sum*/

a=0;

b=1;

n=1000;

dx=(b-a)/n;

sum=0;

for i=1:n

x=a+(i-0.5)*dx;

sum=sum+x*x*dx;

end

disp(sum);

n / /
10 / 0.33250 / 8.3333e-04
100 / 0.33333 / 8.3333e-06
1000 / 0.33333 / 8.3333e-08
10000

1.4.3Calculate the following integral by using the rectangle method (use the left sum).

1.4.4Calculate the following integral by using the rectangle method for (use theright sum).

1.4.5Calculate the following integral by using the rectangle method (use the mid sum).

1.4.6Calculate the following integral by using rectangle method (use right sum).

1.4.7(*) Error estimates

Plot[{1/x,1/Sqrt[x]},{x,0,1}]

------

a=0;

b=1;

n=1000;

dx=(b-a)/n;

sum=0;

for i=1:n

x=a+(i)*dx;

sum=sum+(1/x)*dx;

end

disp(sum);

------

n /
100 / 5.1874
1000 / 7.4855
10000 / 9.7876
100000 / 12.090

1.1.1(*) Calculate the integral by using the Reimann sums.

Conclusion

1.5Trapezoidal method

1.5.1Introduction

f[x_]=a+b*x;

XX=Solve[{f[x0]f0,f[x0+dx]f1},{a,b}];

g[x_]=Extract[f[x]/.XX,1]

Integrate[g[x],{x,x0,x0+dx}]

-(((f0-f1) x)/dx)-(-dx f0-f0 x0+f1 x0)/dx

(dx f0)/2+(dx f1)/2

f[x_] = x^2;

n = 10;

a = 0;

b = 1;

dx = (b - a)/n;

XX = Table[0 + i*dx, {i, 0, n}]

N[(Sum[f[XX[[i]]], {i, 2, n}] + (f[XX[[1]]] + f[XX[[n + 1]]])/2)*dx]

Result=

1.5.2Calculate the following integral by using the trapezoidal method .

1.5.3Calculate the following integral by using the trapezoidal method .

1.5.4Calculate the following integral by using trapezoidal method .

1.5.5Calculate the following integral by using trapezoidal method .

1.5.6Calculate the following integral by using trapezoidal method .

1.5.7Error estimation ,

1.5.8Error estimation ,

1.6Simpson method

1.6.1Introduction

f[x_] = a + b*x + c*x^2;

x1 = x0 + dx;

x2 = x0 + 2*dx;

Sol = Solve[{f[x0] == f0, f[x1] == f1, f[x2] == f2}, {a, b, c}];

g[x_] = Extract[f[x] /. Sol, 1]

Integrate[g[x], {x, x0, x2}]

Out[39]= -(((-f0 + 2 f1 - f2) x^2)/(2 dx^2)) - (

x (3 dx f0 - 4 dx f1 + dx f2 + 2 f0 x0 - 4 f1 x0 + 2 f2 x0))/(

2 dx^2) - (-2 dx^2 f0 - 3 dx f0 x0 + 4 dx f1 x0 - dx f2 x0 -

f0 x0^2 + 2 f1 x0^2 - f2 x0^2)/(2 dx^2)

Out[40]= (dx f0)/3 + (4 dx f1)/3 + (dx f2)/3------

(dx f0)/3+(4 dx f1)/3+(dx f2)/3

------

------

Different form of THE SAME formula.

for

1.6.2Calculate the following integral by using the Simpson's method .

1.6.3Calculate by using Simpson method for . (*)

1.6.4Calculate by using Simpson method for .

1.6.5Calculate by using Simpson method for .

In[340]:= N[(Sin[1]/1+4 Sin[2]/2+2 Sin[3]/3+4 Sin[4]/4+2 Sin[5]/5+4 Sin[6]/6+Sin[7]/7)*(1/3)]

Out[340]= 0.507117

1.6.6Calculate the following integral by using the Simpson's method .

1.6.7Calculate the following integral by using the Simpson's method .

1.6.8Calculate the following integral by using the Simpson's method .

Plot[Exp[x^2]*Sin[x],{x,0,4}]

1.6.9Calculate the following integral by using the Simpson's method .

1.6.10(*) Simpson method for cubic polynomial

Error of the composite rule

The Simpson method is accurate up to the polynomials of the degree 3.

1.7(*) Newton's Cotes method for n=3 (Simpson's 3/8 rule)

Clear[x0,x1,x2,x3,x,dx];

f[x_]=a+b*x+c*x^2+d*x^3;

x1=x0+dx;

x2=x0+2*dx;

x3=x0+3*dx;

Sol=Solve[{f[x0]f0,f[x1]f1,f[x2]f2,f[x3]f3},{a,b,c,d}];

g[x_]=Extract[f[x]/.Sol,1]

Integrate[g[x],{x,x0,x3}]

"Interpolation polynomial"

-(((f0-3 f1+3 f2-f3) x3)/(6 dx3))-(x2 (-2 dx f0+5 dx f1-4 dx f2+dx f3-f0 x0+3 f1 x0-3 f2 x0+f3 x0))/(2 dx3)-(x (11 dx2 f0-18 dx2 f1+9 dx2 f2-2 dx2 f3+12 dx f0 x0-30 dx f1 x0+24 dx f2 x0-6 dx f3 x0+3 f0 x02-9 f1 x02+9 f2 x02-3 f3 x02))/(6 dx3)-((-4 dx5 x04-6 dx4 x05-2 dx3 x06) ((x03 (dx+x0)2-x02 (dx+x0)3) (-f3 x03+f0 (3 dx+x0)3)-(-f1 x03+f0 (dx+x0)3) (x03 (3 dx+x0)2-x02 (3 dx+x0)3))-((x03 (dx+x0)2-x02 (dx+x0)3) (-f2 x03+f0 (2 dx+x0)3)-(-f1 x03+f0 (dx+x0)3) (x03 (2 dx+x0)2-x02 (2 dx+x0)3)) ((x03 (dx+x0)2-x02 (dx+x0)3) (x03 (3 dx+x0)-x0 (3 dx+x0)3)-(x03 (dx+x0)-x0 (dx+x0)3) (x03 (3 dx+x0)2-x02 (3 dx+x0)3)))/((-4 dx5 x04-6 dx4 x05-2 dx3 x06) ((x03 (dx+x0)2-x02 (dx+x0)3) (x03-(3 dx+x0)3)-(x03-(dx+x0)3) (x03 (3 dx+x0)2-x02 (3 dx+x0)3))-((x03 (dx+x0)2-x02 (dx+x0)3) (x03-(2 dx+x0)3)-(x03-(dx+x0)3) (x03 (2 dx+x0)2-x02 (2 dx+x0)3)) ((x03 (dx+x0)2-x02 (dx+x0)3) (x03 (3 dx+x0)-x0 (3 dx+x0)3)-(x03 (dx+x0)-x0 (dx+x0)3) (x03 (3 dx+x0)2-x02 (3 dx+x0)3)))

"integration scheme"

(3 dx f0)/8+(9 dx f1)/8+(9 dx f2)/8+(3 dx f3)/8

1.1.2Calculate the following integral

1.8(*) Forth order interpolation

Clear[dx,f0];

f[x_]=a+b*x+c*x^2+d*x^3+e*x^4;

x0=0;

x1=x0+dx;

x2=x0+2*dx;

x3=x0+3*dx;

x4=x0+4*dx;

Sol=Solve[{f[x0]f0,f[x1]f1,f[x2]f2,f[x3]f3,f[x4]f4},{a,b,c,d,e}]

g[x_]=Extract[f[x]/.Sol,1]

Integrate[g[x],{x,x0,x4}]

{{af0,b-((25 f0-48 f1+36 f2-16 f3+3 f4)/(12 dx)),c-((-35 f0+104 f1-114 f2+56 f3-11 f4)/(24 dx2)),d-((5 f0-18 f1+24 f2-14 f3+3 f4)/(12 dx3)),e-((-f0+4 f1-6 f2+4 f3-f4)/(24 dx4))}}

f0-((25 f0-48 f1+36 f2-16 f3+3 f4) x)/(12 dx)-((-35 f0+104 f1-114 f2+56 f3-11 f4) x2)/(24 dx2)-((5 f0-18 f1+24 f2-14 f3+3 f4) x3)/(12 dx3)-((-f0+4 f1-6 f2+4 f3-f4) x4)/(24 dx4)

(14 dx f0)/45+(64 dx f1)/45+(8 dx f2)/15+(64 dx f3)/45+(14 dx f4)/45

1.1.3Calculate the following integral

----

NIntegrate[Sin[x]/x,{x,0,4}]

1.7582

1.1.4Calculate the following integral

f[x_]=Sin[x]/x;

x0=0;

x1=1/2;

x2=1;

x3=3/2;

x4=2;

x5=5/2;

x6=3;

x7=7/2;

x8=4;

dx=1/2;

f0=1;

N[(14*f0+64*f[x1]+24*f[x2]+64*f[x3]+14*f[x4])*dx/45

+(14*f[x4]+64*f[x5]+24*f[x6]+64*f[x7]+14*f[x8])*dx/45]

1.7582

1.9Fifth order interpolation

Clear[dx, f0, f1, f2, f3, f4, f5];

f[x_] = a1 + a2*x + a3*x^2 + a4*x^3 + a5*x^4 + a6*x^5;

x0 = 0;

x1 = x0 + dx;

x2 = x0 + 2*dx;

x3 = x0 + 3*dx;

x4 = x0 + 4*dx;

x5 = x0 + 5*dx;

Sol = Solve[{f[x0] == f0, f[x1] == f1, f[x2] == f2, f[x3] == f3,

f[x4] == f4, f[x5] == f5}, {a1, a2, a3, a4, a5, a6}]

g[x_] = Extract[f[x] /. Sol, 1]

Integrate[g[x], {x, x0, x5}]

Out[305]= {{a1 -> f0,

a2 -> -((137 f0 - 300 f1 + 300 f2 - 200 f3 + 75 f4 - 12 f5)/(

60 dx)),

a3 -> -((-45 f0 + 154 f1 - 214 f2 + 156 f3 - 61 f4 + 10 f5)/(

24 dx^2)),

a4 -> -((17 f0 - 71 f1 + 118 f2 - 98 f3 + 41 f4 - 7 f5)/(24 dx^3)),

a5 -> -((-3 f0 + 14 f1 - 26 f2 + 24 f3 - 11 f4 + 2 f5)/(24 dx^4)),

a6 -> -((f0 - 5 f1 + 10 f2 - 10 f3 + 5 f4 - f5)/(120 dx^5))}}

Out[306]= f0 - ((137 f0 - 300 f1 + 300 f2 - 200 f3 + 75 f4 -

12 f5) x)/(

60 dx) - ((-45 f0 + 154 f1 - 214 f2 + 156 f3 - 61 f4 + 10 f5) x^2)/(

24 dx^2) - ((17 f0 - 71 f1 + 118 f2 - 98 f3 + 41 f4 - 7 f5) x^3)/(

24 dx^3) - ((-3 f0 + 14 f1 - 26 f2 + 24 f3 - 11 f4 + 2 f5) x^4)/(

24 dx^4) - ((f0 - 5 f1 + 10 f2 - 10 f3 + 5 f4 - f5) x^5)/(120 dx^5)

Out[307]= (95 dx f0)/288 + (125 dx f1)/96 + (125 dx f2)/144 + (

125 dx f3)/144 + (125 dx f4)/96 + (95 dx f5)/288

1.10(*) Newton-Cotes integration

Closed Newton–Cotes Formulae
Degree / Common name / Formula / Error term
1 / Trapezoid rule / /
2 / Simpson's rule / /
3 / Simpson's 3/8 rule / /
4 / Boole's rule / /

1.11Error in numerical integration

1.11.1(*) Generalized MeanValue Theorem

Let assume that the function are continuous on the closed interval and is nonnegative i.e. on the open interval , then for some point .

1.11.2Rectangle method - error for one interval (left or right sums)

(*) Error formula for the mid sum.

Interpolation error

Formula for one interval

For the interval

Finally, the error for one interval

1.11.3Rectangle method - error for n-intervals (left or right sums)

1.1.4.1Calculate the following integral by using rectangle method and estimate the error.

Error estimate

True error

1.1.4.2Calculate the following integral by using rectangle method (right sum) and estimate the error.

Error estimates

True error

Plot[1/x^2,{x,1,7}]

1.1.4.3Calculate the following integral by using rectangle method (right sum) and estimate the error.

Error estimate.

Plot[1/x^2,{x,0,1}]

1.1.4.4Estimate the error of numerical integration for , n=4 and the Rectangle Method.

Plot[2x+3,{x,-1,1}]

1.1.4.5Estimate the error of numerical integration for , n=4 and the Rectangle Method.

.

1.11.4Trapezoidal method -error for one interval

Mathematica script:

Integrate[((x-a)(x-b))/2,{x,a,b}]

1/12 (a-b)3

1.11.5Trapezoidal method -error for n intervals

Formula for multiple intervals

For continuous from the properties of arithmetic mean we have

for .

Finally

1.1.4.6Calculate the following integral by using trapezoid method and estimate the error.

Exact error

1.1.4.7Calculate the following integral by using trapezoid method and estimate the error.

Error estimates

Plot[2/x^3,{x,1,7}]

True error

1.1.4.8Estimate the error of numerical integration for , n=4 and the Trapezoid Method.

Plot[6x+6,{x,-1,1}]

1.11.6Simpson's method-error for one interval

Integration points

Simplify[Integrate[(x-a)(x-(a+b)/2)(x-b)/6,{x,a,b}]]

0

Let's add a new point . The error of integration for 4 points is the same as in the case of two points:

Simplify[Integrate[(x-a)(x-(a+b)/2)^2(x-b)/4!,{x,a,b}]]

(a-b)5/2880

1.11.7Simpson's method-error for n intervals

We know that

for .

then

Formula for error

1.1.4.9Estimate the error of numerical integration for , n=4 and the Simpson’s Method.

Verification

Clear[f,x];

f[x_]=x^3+3x^2-2;

x0=-1;x1=-1/2;x2=0;x3=1/2;x4=1;

dx=1/2;

(f[x0]+4*f[x1]+2*f[x2]+4*f[x3]+f[x4])*dx/3

Integrate[f[x],{x,-1,1}]

-2

-2

1.1.4.10Estimate the error of numerical integration for , n=4 and the Simpson’s Method.

1.12Gauss Numerical integration

1.12.1Table of Gauss integration points

.

Number of points, n / Points, xi / Weights, wi
1 / 0 / 2
2 / / 1
3 / 0 / 8⁄9
/ 5⁄9
4 / /
5 / 0 / 128⁄225

- Gauss points

15 points (*)

weight - wi / abscissa - xi
1 / 0.2025782419255613 / 0.0000000000000000
2 / 0.1984314853271116 / -0.2011940939974345
3 / 0.1984314853271116 / 0.2011940939974345
4 / 0.1861610000155622 / -0.3941513470775634
5 / 0.1861610000155622 / 0.3941513470775634
6 / 0.1662692058169939 / -0.5709721726085388
7 / 0.1662692058169939 / 0.5709721726085388
8 / 0.1395706779261543 / -0.7244177313601701
9 / 0.1395706779261543 / 0.7244177313601701
10 / 0.1071592204671719 / -0.8482065834104272
11 / 0.1071592204671719 / 0.8482065834104272
12 / 0.0703660474881081 / -0.9372733924007060
13 / 0.0703660474881081 / 0.9372733924007060
14 / 0.0307532419961173 / -0.9879925180204854
15 / 0.0307532419961173 / 0.9879925180204854

Mathematica script for one points Gauss integration

f[x_]=2x+1;

t0=0;

w0=2;

a=1;b=25;

"Exact"

NIntegrate[f[x],{x,a,b}]

"Gauss"

Simplify[N[w0*f[(b-a)*t0/2+(a+b)/2]*(b-a)/2]]

Exact

648.

Gauss

648.

Mathematica script for two points Gauss integration

f[x_]=x^2+1;

t0=-1/Sqrt[3];

t1=1/Sqrt[3];

w0=1;

w1=1;

a=0;b=1;

"Exact"

NIntegrate[f[x],{x,a,b}]

"Gauss"

Simplify[N[w0*f[(b-a)*t0/2+(a+b)/2]*(b-a)/2+w1*f[(b-a)*t1/2+(a+b)/2]*(b-a)/2]]

------

Exact

1.33333

Gauss

1.33333

Mathematica script for three points Gauss integration

f[x_]=x^2+1;

t0=Sqrt[3/5];

t1=0;

t2=-Sqrt[3/5];

w0=5/9;

w1=8/9;

w2=5/9;

a=0;b=1;

"Exact"

NIntegrate[f[x],{x,a,b}]

"Gauss"

Simplify[N[w0*f[(b-a)*t0/2+(a+b)/2]*(b-a)/2+w1*f[(b-a)*t1/2+(a+b)/2]*(b-a)/2]+

w2*f[(b-a)*t2/2+(a+b)/2]*(b-a)/2]

Exact

1.33333

Gauss

1.33333

1.12.2(*) How to derive Gauss integration formula

------

------

f0[x_]=1;

f1[x_]=x;

Solve[{Integrate[f0[x],{x,-1,1}]w0*f0[x0],

Integrate[f1[x],{x,-1,1}]w0*f1[x0]},{w0,x0}]

------

{{w02,x00}}

------

------

f0[x_]=1;

f1[x_]=x;

f2[x_]=x^2;

f3[x_]=x^3;

Solve[{Integrate[f0[x],{x,-1,1}]w0*f0[x0]+w1*f0[x1],

Integrate[f1[x],{x,-1,1}]w0*f1[x0]+w1*f1[x1],

Integrate[f2[x],{x,-1,1}]w0*f2[x0]+w1*f2[x1],

Integrate[f3[x],{x,-1,1}]w0*f3[x0]+w1*f3[x1]

},{w0,x0,w1,x1}]

------

{{w01,x01/,w11,x1-(1/)},{w01,x0-(1/),w11,x11/}}

1.12.3Calculate by using 1 point Gauss integration

Integrate[2x+1,{x,-1,1}]

2

1.12.4Calculate by using 2 points Gauss integration

1.12.5Calculate by using 3 points Gauss integration

1.12.6Calculate by using 1 point Gauss integration

1.12.7Calculate by using 2 points Gauss integration

1.12.8Calculate by using 2 points Gauss integration

1.12.9Calculate by using 2 points Gauss integration

1.12.10Calculate by using 2 points Gauss integration

1.12.11Calculate by using two points Gauss integration.

1.12.12Calculate by using two points Gauss integration

Integrate[Sin[x]/x,x]

------

SinIntegral[x]

1.12.13Calculate by using two points Gauss integration

1.12.14Calculate by using 3 points Gauss integration

1.12.15Calculate by using 3 points Gauss integration

1.12.16Calculate by using 3 points Gauss integration

N[(2*Sin(4))/3. + (5*Sin(4 - 3*Sqrt(0.6)))/(3.*(4 - 3*Sqrt(0.6))) +

(5*Sin(4 + 3*Sqrt(0.6)))/(3.*(4 + 3*Sqrt(0.6)))]

1.12.17Calculate by using 3 points Gauss integration

1.12.18Calculate by using three points Gauss integration.

1.12.19Calculate by using 3 points Gauss integration

Method 2

1.12.20Calculate by using 4 points Gauss integration

1.12.21Calculate by using 4 points Gauss integration.

1.12.22Calculate by using 4 points Gauss integration.

1.12.23Calculate by using 5 points Gauss integration.

1.12.24Calculate by using 5 points Gauss integration.

1.13(*) Monte Carlo method

1.14Review

1.14.1Summer 2015

Section 1.3 (trapezoidal)

Problem 1.3.4

Section 1.4 (Simpson)

Problem 1.4.4

Section 1.8 (Error)

Problem 1.1.4.8

Problem 1.1.4.10

Section 1.7 (Gauss)

Problem 1.9.7

Problem 1.9.10

1