Searchable Numerical Formula Database

Variables: a, b = interval bounds; c = midpoint estimate; f(x) = continuous function

Continuous interval halving formula guaranteeing convergence whenever f(a) and f(b) have opposite signs.

Variables: x_k = current estimate; f(x_k) = function value; f'(x_k) = derivative slope

Rapid quadratic convergence formula using tangent derivative slopes.

Variables: a_{kk} = pivot element; a_{ik} = element to eliminate below pivot

Elementary row operation formula reducing coefficient matrix A to Upper Triangular Form U.

Variables: L = lower triangular matrix; U = upper triangular matrix; b = target vector

Factorization formula allowing fast forward and back substitution for multiple right-hand vectors b.

Variables: (x_i, y_i) = data points; P(x) = interpolating polynomial

Closed-form formula fitting a degree-n polynomial exactly through n+1 distinct data points.

Variables: h = (b - a) / n = step size; n = subintervals count

Summation formula approximating definite integration via trapezoidal areas under f(x).

Variables: h = step size; n = even number of subintervals

Fourth-order accurate integration formula fitting parabolic arcs across consecutive subintervals.

Variables: k_1 = f(x_k, y_k), \; k_2 = f(x_k+\frac{h}{2}, y_k+\frac{h k_1}{2}), \; k_3 = f(x_k+\frac{h}{2}, y_k+\frac{h k_2}{2}), \; k_4 = f(x_k+h, y_k+h k_3)

Gold-standard 4th-order ODE solver formula balancing speed and high precision.