Orthogonal Functions Calculator

Calculator Inputs

Function f(x)

Example: 1, x, x^2, sin(x), cos(2*x)

Function g(x)

Example: x, x^3, sin(2*x), cos(x)

Weight w(x)

Use 1 for ordinary inner products.

Lower Limit a

Upper Limit b

Integration Samples

Orthogonality Tolerance

Preset Pair

Plotly Graph

The chart shows f(x), g(x), w(x), and the weighted integrand f(x)g(x)w(x).

Example Data Table

Example	f(x)	g(x)	w(x)	Interval	Expected ⟨f,g⟩w	Orthogonal?
Constant vs Linear	1	x	1	[-1,1]	0	Yes
Sine Modes	sin(x)	sin(2*x)	1	[0,π]	0	Yes
Cosine Modes	cos(x)	cos(2*x)	1	[0,π]	0	Yes
Non-Orthogonal Pair	1+x	1+x^2	1	[-1,1]	Nonzero	No

Formula Used

Weighted Inner Product
⟨f,g⟩_w = ∫_a^b f(x)g(x)w(x) dx

Norm of a Function
||f|| = √⟨f,f⟩_w

Orthogonality Test
f and g are orthogonal when |⟨f,g⟩_w| ≤ tolerance

Projection Coefficient
c = ⟨f,g⟩_w / ⟨g,g⟩_w

Numerical Method
The calculator evaluates definite integrals with Simpson’s rule using an even number of subintervals for stable numerical approximation.

How to Use This Calculator

Enter the first function in the f(x) field.
Enter the second function in the g(x) field.
Provide a weight function. Use 1 when no weighting is needed.
Set the integration interval with lower and upper limits.
Choose enough samples for accurate numerical integration.
Set a tolerance for deciding whether the inner product is effectively zero.
Press Calculate Now to place results above the form.
Review the graph, interpretation, and downloadable summary.

FAQs

1. What does this calculator measure?

It measures the weighted inner product of two functions, their norms, projection coefficient, residual norm, and whether they are numerically orthogonal.

2. What makes two functions orthogonal?

Two functions are orthogonal when their weighted inner product over the chosen interval equals zero, or is very close to zero within the selected tolerance.

3. Why does the weight function matter?

The weight changes the geometry of the function space. A pair may be orthogonal with one weight and not orthogonal with another.

4. Why are more samples sometimes necessary?

Rapid oscillations, steep curves, or difficult weight functions may need more subintervals so Simpson’s rule can better approximate the integral.

5. Can I use trigonometric and exponential functions?

Yes. You can use expressions such as sin(x), cos(2*x), exp(x), sqrt(x+2), log(x+3), and combinations using standard arithmetic operators.

6. What does the projection coefficient mean?

It shows how much of f lies along g. A zero coefficient means f has no component in the direction of g.

7. What is the Gram determinant used for?

It indicates linear independence in the two-function span. A positive value suggests the functions contribute distinct directions in the weighted space.

8. Why might a theoretically orthogonal pair show a tiny nonzero result?

Numerical integration introduces approximation error. Increasing samples or adjusting the tolerance usually reduces the apparent mismatch.