Numerical Methods 262 Cheat Sheet (DRAFT) by Kamva

The formula: ax + by + cz = 1

A= [-1 -1 -3;1 0 0;2 1 2;2 3 6];

b = [1;1;1;1];

x = A\b;

NOTE: Ax = b...backsl­ash(\) to solve for a,b,c

Expone­ntial linearized function
The formula is y = be^mx:

x = [1 2 3 4]; y = [6 3 2 1];   

Given

a = polyfi­t(x­,lo­g(y),1)

>>a = -0.5781 2.3411; 

m = a(1) 

>>m = -0.5781;

b = exp(a(2));

>>b = 10.3923;

The final product: y = 10.39e^-0.5781x

A=[2 -1 0;-1 2 -1;0 -1 2]; a=eig(A);

[V,D] = eig(A)

V - eigenv­ectors

V(:,1)

:get 1st column

v1 = V(:,1);

v1 = v1/v1(1);

: normalise eigenv­ector

a = 0.1; b = 0.002; c = 0.0025; d = 0.2;

f = @(t,x) [-ax(1)+bx(1)*x(2);

dx(2)-cx(1)*x­(2)];

X0 = [20; 100];

Tspan = linspa­ce(­0,1­50,­1000);

[T,X] = ode45(­f,T­spa­n,X0);

plot(T­,X,­'Li­neW­idt­h',2);

a = 0.1; b = 0.002; c = 0.0025; d = 0.2;

f = @(t,x) [-ax(1)+bx(1)*x(2);

dx(2)-cx(1)*x­(2)];

X0 = [20; 100];

Tspan = linspa­ce(­0,1­50,­1000);

[T,X] = ode45(­f,T­spa­n,X0);

plot(T­,X,­'Li­neW­idt­h',2);

f=@(x) 1/2*sq­rt(­16+­(x+­3).^­4)./(­x+3)^2;

G2=f(-­1/s­qrt­(3)­)+f­(1/­sqr­t(3));

>>G2= 1.1289;

and

G3=5/9f(-sqr­t(3­/5)­)+8/9f(0)+5­/9*­f(s­qrt­(3/5));

>>G3= 1.1319;

y0=1;y1=2; % N = 1/h;h = 0.2

i = [1:N-1]';

xi = i*h;    % Interior nodes

A=(h^2­-2)­*ey­e(N-1); % Diagonal part of A 

A=A+di­ag(­1-0.5hxi(2:N­-1)­,-1);% Add sub- and superd­iagonal

A=A+di­ag(­1+0.5hxi(1:N­-2)­,+1);% Add sub- and superd­iagonal

b = 2h^2xi;% RHS refined formula

b(1)=b­(1)­-(1-0.5hxi(1))­*y0;% Add boundary contri­butions

b(N-1)­=b(­N-1­)-(­1+0.5hxi(N-1­))*y1;% Add boundary contri­butions

y = sparse­(A)\b;   % Solve

f = @(x,y) x.^2-y.^2-1;

a = 0; b = 3; g = 0; p = 1;

integr­al2­(f,­a,b­,g,p);

>>ans = 5.0000­000­000­01309;

or

f = @(x,y) y;

a = 0; b = pi/4;

g = @(x) sin(x); p = @(x) cos(x);

m = integr­al2­(f,­a,b­,g,p);

>>m = 0.2500­000­000­00013;

y = a0 + a1x + a2x²:

x = [1; 2; 3; 4]; y = [6; 3; 2; 1];

A = [ones(4,1) x x.ˆ2]; a = A\y

a = 9.5000 -4.1000 0.5000

a2 = polyfi­t(x­,y,2)

xx = linspa­ce(­0,5);

y2 = polyva­l(a­2,xx);

plot(x­,y,­'r.'­,x­x,y­2,'b-')

x = [1; -1; 3; -5];

norm(x­,inf)

||x||i­nf(­sub­script)
||A||2 = (||Ax||2)/(||x||2)
~ cond(A) = (||A||­)(||A^-1||)
~

cond(A)

>> 1:ill-­con­dit­ion­ed&close singular matrix
~

cond(A)

≈ 1:well­-co­ndi­tioned
~ decimals digits that trustw­orthy = 10^-a = 10^c x 10^-d
a = d-c
~r = b - Ax
det (A) = 0; No A^-1; Singular matrix
test singul­arity:

cond,c­ond­est­,rank

f=@(x)­sqr­t(1­+x.^­4)./x.^2;

L=inte­gra­l(f­,1,2);

>>L­=1.1­32­090­393­305918;

or

y=[0 0.3 0.5 1 1.5 2 2.5 3 4 5];

v=[0 0.4 0.5 0.56 0.6 0.63 0.66 0.68 0.71 0.72];

Q=10*t­rap­z(y,v);

>>Q­=30.8000;

f = @(x,y) -x*y;                   %Define rhs

N = 100;                             % Number of steps 

....

for i = 1:N

K1 = f(x(i)­,y(i));

K2 = f(x(i)+0.5h,y(i)+0.5h*K1);

K3 = f(x(i)+0.5h,y(i)+0.5h*K2);

K4 = f(x(i+­1),­y(i­)+h­*K3);

x(i+1) = x(i)+h;

y(i+1) = y(i)+(1/6)h(K1+2K­2+2­K3+K4);

end

f = @(x,y) -x*y;                   %Define rhs

N = 100;                             % Number of steps 

....

for i = 1:N

K1 = f(x(i)­,y(i));

K2 = f(x(i)+0.5h,y(i)+0.5h*K1);

K3 = f(x(i)+0.5h,y(i)+0.5h*K2);

K4 = f(x(i+­1),­y(i­)+h­*K3);

x(i+1) = x(i)+h;

y(i+1) = y(i)+(1/6)h(K1+2K­2+2­K3+K4);

end

f = @(x) sqrt(1­+x.^­4)./x.^2;

a = 1; b = 2;

N = 100; h = (b-a)/N;

x = a+[0:N]*h;

fx = f(x);

L = h/3(fx(1)+4sum(fx­(2:­2:N­))+­2*s­um(­fx(­3:2­:N-­1))­+fx­(N+1))

or

w = ones(s­ize­(x)); w(2:2:N) = 4; w(3:2:N-1) = 2;

L = h/3sum(w.f(x))

>>L = 1.1320­903­946­95845;

f = @(x) sqrt(1­+x.^­4)./x.^2;

a = 1; b = 2;

N = 100; h = (b-a)/N;

x = a+[0:N]*h;

fx = f(x);

L = h3( 0.5fx(1) + sum(fx­(2:N)) + 0.5*fx­(N+1) )

or

w = ones(s­ize­(x)); w(1) = 0.5; w(N+1) = 0.5;

L = hsum(w.f(x));

>>L = 1.1321­016­727­88808;

x = [22 35 48 61 74];

y = [320 490 540 500 480];

yy = spline­(x,­y,42);

not-a-knot

>>yy =  531.1971;

xx = linspa­ce(­20,­80,­1000);

*clamped condit­ions:

[0 y 0] 

ss = spline­(x,­y,xx);

plot(x­,y,­'r.'­,x­x,s­s,'­b-');