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Finds real root in interval [a, b] where f(a) and f(b) have opposite signs by repeated bisection.
Finds real roots of f(x) = 0 using two initial approximations without requiring symbolic derivatives.
Finds real roots of f(x) = 0 using symbolic derivatives and quadratic convergence.
Secant line interpolation method to find roots within bracketed interval [a, b].
Solves x = g(x) iteratively starting from initial guess x0.
Equal interval interpolation formula using forward difference table Δy for target X near table start.
Equal interval backward difference formula using ∇y for target X near table end.
Central difference interpolation formula for values of x lying near the center of the table (0 < u < 1).
Central difference interpolation formula for values of x lying near the center of the table (-1 < u < 0).
Unequal interval polynomial interpolation formula for arbitrary (x_i, y_i) data points.
Unequal interval divided difference interpolation for arbitrary tabulated points.
Computes first and second derivatives f'(x) and f''(x) from forward difference table of tabulated data.
Approximates definite integral ∫[a,b] f(x) dx using N linear trapezoidal subintervals.
Parabolic quadratic interpolation for definite integral over even number of subintervals N.
Cubic interpolation for numerical integration where subintervals N is a multiple of 3.
Higher-order Newton-Cotes formula over subintervals N (multiple of 4) with weights [7, 32, 12, 32, 7].
High-precision 6-interval Newton-Cotes rule using weights [1, 5, 1, 6, 1, 5, 1].
Gauss-Legendre 2-point and 3-point quadrature formula mapping interval [a, b] to [-1, 1].
Iterative technique for solving diagonally dominant linear systems Ax = b using updated values immediately.
Iterative solver for diagonally dominant linear systems Ax = b updating variables simultaneously from previous vector.
Direct solution of 3x3 linear system Ax = b via forward row operations to upper triangular form and back substitution.
Reduces augmented matrix [A|b] directly to reduced row echelon form (identity matrix) to solve linear systems.
Factorizes square matrix A into lower triangular L and upper triangular U to solve Ly = b and Ux = y.
Approximates ODE solution y(x) via Taylor series expansion y(x0+h) = y0 + h y' + (h²/2!) y'' + (h³/3!) y''' + ...
Solves first order initial value problem dy/dx = f(x, y) starting at (x0, y0) with step size h.
Predictor-corrector method: predicts y* = y + h*f(x,y), corrects with average slope at (x_n, y_n) and (x_{n+1}, y*).
High accuracy 4th order numerical solution for initial value differential equation dy/dx = f(x, y).
Finds optimal linear regression line y = a + b*x by minimizing sum of squared residuals.
Calculates linear regression equation y = mx + c with slope m, intercept c, and Pearson correlation coefficient r.
Fits second-degree parabolic curve y = a + b*x + c*x² via normal equations system.
Tests mean difference for large samples (N ≥ 30) with known population standard deviation σ.
Tests mean difference for small samples (N < 30) where population standard deviation is unknown.
Tests goodness of fit between observed frequencies O_i and expected frequencies E_i.
Tests whether two independent normal populations have equal variances: F = S1² / S2² where S1² > S2².