Polynomials

Fit, evaluate, and manipulate polynomials

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Polynomials appear everywhere in scientific computing — from curve fitting and interpolation to control systems and signal processing. Equana provides a complete set of polynomial operations that work on coefficient arrays.

Representing Polynomials

A polynomial is represented as an array of coefficients in low-to-high degree order. The first element is the constant term, the second is the linear coefficient, and so on:

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# p(x) = 1 + 2x + 3x²
p = [1, 2, 3]
println(p)

# q(x) = 5 - x
q = [5, -1]
println(q)

Evaluating Polynomials

Use polyval to evaluate a polynomial at one or more points:

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# p(x) = 1 + 2x + 3x²
p = [1, 2, 3]

# Evaluate at a single point
println(polyval(p, 2.0))   # 1 + 4 + 12 = 17.0

# Evaluate at multiple points
x = [0.0, 1.0, 2.0, 3.0]
println(polyval(p, x))     # [1, 6, 17, 34]

Polynomial Arithmetic

Add, subtract, multiply, and divide polynomials:

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a = [1, 2, 3]   # 1 + 2x + 3x²
b = [1, 1]       # 1 + x

# Addition
println(polyadd(a, b))

# Subtraction
println(polysub(a, b))

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a = [1, 2, 3]
b = [1, 1]

# Multiplication
println(polymul(a, b))   # (1+2x+3x²)(1+x) = 1+3x+5x²+3x³

# Division returns (quotient, remainder)
(q, r) = polydiv(a, b)
println("quotient: ", q)
println("remainder: ", r)

Derivatives and Integration

Compute derivatives with polyder and antiderivatives with polyint:

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p = [1, 2, 3]   # 1 + 2x + 3x²

# First derivative: 2 + 6x
println(polyder(p))

# Second derivative: 6
println(polyder(p, 2))

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p = [1, 2, 3]

# Antiderivative (constant of integration = 0)
println(polyint(p))    # [0, 1, 1, 1] → x + x² + x³

# Double integration
println(polyint(p, 2))

Curve Fitting

Use polyfit to find the best-fit polynomial through data points:

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# Data points
x = [0.0, 1.0, 2.0, 3.0, 4.0]
y = [1.0, 3.0, 9.0, 19.0, 33.0]

# Fit a degree-2 polynomial: p(x) ≈ a₀ + a₁x + a₂x²
coeffs = polyfit(x, y, 2)
println(coeffs)

# Verify the fit
println(polyval(coeffs, x))

Polynomials from Roots

Construct a polynomial whose roots are a given set of values using poly:

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# Polynomial with roots at x = 1, 2, 3
# (x-1)(x-2)(x-3) = -6 + 11x - 6x² + x³
p = poly([1.0, 2.0, 3.0])
println(p)

# Verify: should be ≈ 0 at each root
println(polyval(p, [1.0, 2.0, 3.0]))

Summary

Function	Description	Example
`polyval(p, x)`	Evaluate polynomial	`polyval([1,2,3], 2.0)`
`polyadd(a, b)`	Add polynomials	`polyadd([1,2], [3,4])`
`polysub(a, b)`	Subtract polynomials	`polysub([1,2], [3,4])`
`polymul(a, b)`	Multiply polynomials	`polymul([1,1], [1,1])`
`polydiv(a, b)`	Divide polynomials	`polydiv([1,3,2], [1,1])`
`polyder(p)`	Derivative	`polyder([1,2,3])`
`polyder(p, m)`	m-th derivative	`polyder([1,2,3], 2)`
`polyint(p)`	Antiderivative	`polyint([2,6])`
`polyint(p, m)`	m-th antiderivative	`polyint([6], 2)`
`polyfit(x, y, deg)`	Least-squares fit	`polyfit(x, y, 2)`
`poly(roots)`	From roots	`poly([1.0, 2.0])`

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Next Steps

Working with Matrices

Learn matrix operations and linear algebra

Plotting & Visualization

Visualize polynomial fits and curves