Exercise On Polynomials — SepalSolver

Exercise: Polynomial Operations and Calculus

This set of exercises is designed to solidify your mastery of polynomial manipulation within the SepalSolver ecosystem. You will apply coefficient logic, calculus rules, and numerical fitting to solve engineering-inspired problems.

Instruction: Complete the code blocks by filling in the logic for fitting, arithmetic, and calculus tasks.

Task 1: Polynomial Representation and Evaluation

Example 1C#

// Define the polynomial P(x) = 4x^3 - 2x + 5
// Note: Remember to include the zero coefficient for the x^2 term
double[] P = [ ]; // Fill the array in descending order
// Evaluate P(x) at x = 2.0
double x = 2.0;
double result = ; // call the Evaluate method
Console.WriteLine($":math:`P({x}) = {result}`");

Task 2: Arithmetic Alignment (PolyAdd)

Example 2C#

// P1 = x^2 + 5, P2 = 2x + 1
double[] p1 = [1.0, 0.0, 5.0];
double[] p2 = [2.0, 1.0];
// Add p1 and p2. The library handles the degree mismatch automatically.
double[] sum = ; // use PolyAdd
// Expected result: [1.0, 2.0, 6.0] -> x^2 + 2x + 6
Console.WriteLine($"Sum Coefficients: {string.Join(", ", sum)}");

Task 3: Multiplication via Convolution (Conv)

Example 3C#

// Multiply (x - 1) by (x + 1)
double[] p1 = [1.0, -1.0];
double[] p2 = [1.0, 1.0];
// Use convolution to multiply
double[] product = ; // call Conv
// Expected: [1.0, 0.0, -1.0] -> x^2 - 1
Console.WriteLine($"Product: {string.Join(", ", product)}");

Task 4: Polynomial Long Division (Deconv)

Example 4C#

// Divide (x^2 - 1) by (x - 1)
double[] dividend = [1.0, 0.0, -1.0];
double[] divisor = [1.0, -1.0];
// Deconv returns a tuple (Quotient, Remainder)
var (Q, R) = ; // call Deconv
Console.WriteLine($"Quotient: {string.Join(", ", Q)}"); // Expected: [1.0, 1.0]
Console.WriteLine($"Remainder: {string.Join(", ", R)}"); // Expected: [0.0]

Task 5: Differentiation (Polyder)

Example 5C#

// Define P(x) = 5x^2 + 10x + 2
double[] P = [5.0, 10.0, 2.0];
// Find the derivative dP/dx
double[] dP = ; // call Polyder
// Expected: [10.0, 10.0] -> 10x + 10
Console.WriteLine($"Derivative: {string.Join(", ", dP)}");

Task 6: Integration with Constant (Polyint)

Example 6C#

// Integrate F(x) = 3x^2
double[] F = [3.0, 0.0, 0.0];
// Perform integration with a constant of integration C = 10.0
double[] integral = ; // call Polyint(F, 10.0)
// Expected: [1.0, 0.0, 0.0, 10.0] -> x^3 + 10
Console.WriteLine($"Integral: {string.Join(", ", integral)}");

Task 7: Data Fitting (Polyfit)

Example 7C#

double[] xData = [0, 1, 2, 3];
double[] yData = [1, 3, 9, 19]; // Follows y = 2x^2 + 1 roughly
// Fit a 2nd degree polynomial to the points
int degree = 2;
double[] fit = ; // call Polyfit
Console.WriteLine($"Fitted Coefficients: {string.Join(", ", fit)}");

Task 8: Finding Roots

Example 8C#

// Solve x^2 - 4 = 0
double[] P = [1.0, 0.0, -4.0];
// Find the roots (should return ±2)
var roots = ; // call Roots
foreach(var r in roots) Console.WriteLine($":math:`x = {r.Real}`");

Task 9: Finding Maximums via Calculus

Example 9C#

// P = -x^2 + 6x (A parabola opening downwards)
double[] P = [-1.0, 6.0, 0.0];
// The maximum occurs where the derivative is zero.
double[] dP = Polyder(P);
var criticalPoints = ; // find roots of the derivative
// Expected root: x = 3.0
Console.WriteLine($"Maximum occurs at :math:x = {criticalPoints[0].Real}");

Task 10: Combining Operations (Area Under Curve)

Example 10C#

// Calculate area of y = x under the interval [0, 2]
double[] line = [1.0, 0.0];
// 1. Integrate
double[] areaPoly = Polyint(line);
// 2. Evaluate at bounds: Integral(2) - Integral(0)
double area = ; // Use Evaluate(areaPoly, 2.0) - ...
// Expected: 2.0
Console.WriteLine($"Area: {area}");