Week 4: Numerical Methods and 3D Plotting

Today’s Question

When exact answers are unavailable, how do we compute responsibly?

Week 4 Plan

• Meeting 1: roots, integrals, and checks
• Meeting 2: surfaces, contours, and AI claims

Approximation Is A Claim

Numerical answers depend on:

• method;
• initial guess;
• tolerance;
• step size;
• conditioning.

Meeting 1 Question

What evidence makes a numerical answer trustworthy?

Root-Finding Form

Write the problem as:

f(x) = 0

Then a numerical root is an x where f(x) is close to zero.

Root-Finding Pattern

f = @(x) x.*exp(-x) - 0.2;
x = linspace(0, 6, 300);
plot(x, f(x))
root = fzero(f, [0 1]);
residual = f(root);

Why Plot First?

The plot can reveal:

• no crossing;
• more than one crossing;
• a bad initial guess;
• a numerical answer that needs qualification.

One Root Or All Roots?

left_root = fzero(f, [0 1]);
right_root = fzero(f, [1 5]);

One solver call is not a proof that there is only one root.

Visual Check

Plot before solving. Plot again near the computed answer.

Substitute the answer back into the original function.

Numerical Integration

g = @(x) sin(x)./(1 + x.^2);
area = integral(g, 0, 6);

The answer depends on the method and the function behavior.

Data-Based Integration

x = linspace(0, 6, 13);
y = g(x);
area_data = trapz(x, y);

trapz is useful when the function is known only through sampled data.

Comparison Habit

Ask:

• What input did the method receive?
• What spacing or tolerance was used?
• Does a finer grid change the answer?

Optimization Pattern

h = @(x) (x - 2).^2 + 0.5*sin(5*x);
[xmin, hmin] = fminbnd(h, 0, 4);

fminbnd reports a local minimum on the interval you gave it.

ODE Initial-Value Problem

Write the problem as:

y' = f(t,y),  y(t0) = y0

The solver needs the derivative rule, interval, and initial value.

ODE Solver Pattern

f = @(t,y) -y + sin(t);
tspan = [0 10];
y0 = 1;
[t, y] = ode45(f, tspan, y0);
plot(t, y)

ode45 is the first solver to try for many nonstiff problems.

ODE Check

initial_error = abs(y(1) - y0);

For every ODE solve, check that the reported solution starts where the problem says it should.

Meeting 2 Question

What does a 3D plot reveal, and what can it hide?

3D Line Plot

t = linspace(0, 6*pi, 400);
x = sqrt(t).*sin(2*t);
y = sqrt(t).*cos(2*t);
z = 0.5*t;
plot3(x, y, z)

The three coordinate vectors must have compatible sizes.

Surface Workflow

1. Create a grid in the x,y plane.
2. Evaluate z = f(x,y) on the grid.
3. Plot the surface or contours.

Meshgrid Pattern

x = linspace(-3, 3, 100);
y = linspace(-3, 3, 100);
[X, Y] = meshgrid(x, y);
Z = sin(X).*cos(Y);

Elementwise operations matter.

Surface Plot

surf(X, Y, Z)
shading interp
colorbar
xlabel("x")
ylabel("y")
zlabel("z")

Use the plot to answer a question.

Contour Plot

contourf(X, Y, Z, 20)
colorbar
axis equal tight

Contours are often easier to read than a rotated 3D surface.

AI Check

Substitute any generated numerical answer back into the original problem.

Then ask:

• Did it find all relevant roots?
• Did it state the interval or initial guess?
• Did it check the residual?
• Did it explain the numerical method?

Plausible But Weak AI Code

f = @(x) x.^3 - 6*x.^2 + 11*x - 6;
root = fzero(f, 0);
fprintf("The solution is %.4f\n", root)

What is missing?

Exit Ticket

A numerical answer is trustworthy enough to report when …