Basics of Numerical Methods for Machine Learning & Engg.

Explore interpolation with equal intervals, distinguishing interpolation from extrapolation by showing how known data in a range estimates values like population in missing years.

Explore various interpolation methods, from graphical interpolation to curve fitting and calculus-based finite differences, including Newton–Gregory techniques for equal, unequal, and central difference methods.

Explore the Newton-Gregory formula, expressed with u, a, h, delta terms, and factorials to compute function values from known data.

Apply Newton-Gregory forward difference interpolation to estimate y at x=3.62 from the data, building the forward difference table to obtain y ≈ 37.338.

Use Newton forward interpolation with the forward-difference table and deltas up to third order to estimate net premium at age 25, yielding f25 ≈ 0.0162543.

Explore the corollary of the Newton–Gregory formula and its forward difference kernel for interpolation at f0 near the initial value, noting forward vs backward differences.

learn how to construct a forward difference table and apply the Newton Gregory formula to derive a cubic polynomial P(x) = x^3 - 2 x^2 + 1 and extrapolate to x = 4.

Understand the Newton–Gregory backward interpolation formula and its comparison with forward interpolation for equally spaced data, using nabla differences and the ending values of x.

Apply the Newton–Gregory backward interpolation method to estimate f(7.5) from a backward difference table and eight data points, yielding approximately 421.875.

Use Newton's backward interpolation to estimate the 1975 population from census data (1941–1981), computing with nabla differences and u = -0.6, yielding about 98.3104 lacs.

Learn interpolation for unequally spaced data, contrasting equidistant and unequal intervals, and apply Lagrange's interpolation formula and divided difference to solve problems.

Explore Lagrange's interpolation formula for constructing a polynomial from given pairs (xi, f(xi)), using the product pattern in the numerator and denominator, including the equidistant x-case.

Apply Lagrange's interpolation formula to compute f(5) from x-values 1, 2, 3, 4, 7 and their f-values, yielding about 32.933 (494/15).

Apply the Lagrange interpolation formula to estimate f(1.2) from x0=1.0, x1=1.1, x2=1.3 with f values 0, 0.09531, and 0.26236.

Apply Lagrange's formula to interpolate y at x = 10 from uneven data points with f-values 12, 13, 14, 16. Obtain y ≈ 14.66.

The lecture demonstrates building a four-point Lagrange interpolation polynomial for (1, -3), (3, 9), (4, 30), (6, 132) by forming the Lagrange basis and summing terms to obtain the interpolant.

Learn the divided difference formula by computing the first divided difference as (f(x0)-f(x1))/(x0-x1) and (f(x1)-f(x0))/(x1-x0). Express it symbolically as delta between x0 and x1.

Apply Newton's divided difference method to build a divided difference table and compute f(2), f(8), and f(15) from the given x and f(x) values.

Learn numerical differentiation from discrete data by applying Newton–Gregory forward interpolation to estimate first and higher order derivatives, selecting Stirling, Lagrange, backward interpolation, or Bessel formulas as needed.

Apply the Newton Gregory forward interpolation formula to build a difference table and compute dy/dx and d2y/dx2 at x = 1.1 using deltas up to delta3.

Apply the Newton Gregory backward interpolation formula to compute dy/dx and d2y/dx2 from equidistant table values, using backward differences and differentiation.

Explore numerical integration using the general quadrature formula, deriving from Newton forward interpolation and applying trapezoidal and Simpson rules to compute difficult definite integrals.

learn the trapezoidal rule for numerical integration, a quadrature formula using the mean of the first and last values and assuming a linear form y equals a plus b x.

Learn how to apply the trapezoidal rule to a given integral, construct the x-values, compute sine, log, and e^x values, and verify results using a scientific calculator.

Apply Simpson's one third rule to approximate definite integrals. Use h/3 times the sum of first and last y-values, plus 4 times odd ordinates and 2 times even ordinates.

Apply Simpson's one-third rule with four subintervals to approximate the integral of e^x from 0 to 4, then compare with e^4 minus e^0 and compute the error.

Explore the Simpson's 3/8 rule as a quadrature method for integrating from x0 to x0 plus N by dx, derived from the general quadrature formula.

Compute the integral from 0 to 1 of 1/(1+x^2) using Simpson's 3/8 rule with six subintervals, obtaining 0.78539 and approximating pi as 3.14156.

Apply Weddell's rule for numerical integration to compute quadrature with n=6, comparing its accuracy to Simpson's rule, for functions with y as a polynomial in x.

Apply Bayes rule to approximate the integral of 1/(1+x^2) from 0 to 6 using six equal parts and the 3/10 rule, yielding 1.373448.

Explore numerical solutions of algebraic and transcendental equations, and learn what defines a polynomial, a root, and a transcendental function. See how a polynomial of order n has n roots and contrast algebraic equations with examples involving logarithmic, exponential, and trigonometric terms.

Explore synthetic division, using x minus alpha to obtain a quotient and remainder, and learn how to compute roots and derivatives of polynomials with a step-by-step table method.

Use synthetic division to divide a fourth-order polynomial by 3x-2, with alpha equals 2/3, obtaining a quotient of degree three and remainder 1/81.

Learn to differentiate a polynomial using synthetic division by building a division table with alpha and the coefficients.

Learn to compute P(2), P'(2), P''(2), and P'''(2) for p(x)=2x^3-6x+13 using synthetic division with alpha=2. The method yields P(2)=17, P'(2)=18, P''(2)=24, and P'''(2)=12.

Apply synthetic division to a cubic p(x) = x^3 - 3x^2 + 4x - 5, obtaining P(2) = -1, P(-2) = 4, P''(2) = 6, P'''(2) = 6.

Explore graphical methods to locate roots of equations by plotting f(x) and finding intersections; use successive approximations and initial estimates for algebraic and transcendental cases.

Compare the graphs of sin x and x minus one to locate the approximate root, finding the intersection near 1.9.

Find the approximate root of e^{-x} - sin x = 0 by graphical intersection of y = e^{-x} and y = sin x near x ≈ 0.6, for the initial approximation.

Apply the bisection method to find roots of continuous functions by checking sign changes, iteratively halving the interval, and refining the root to the desired accuracy.

Locate the root of the equation x cubed minus nine x plus one between 2 and 4 using the bisection method, iterating midpoints to converge on approximately 2.94.

Explore the regula falsi (false position) method for root finding, using small-interval straight-line approximations and line-intersection updates to locate f(x)=0.

Learn the iteration method for solving f(x) equals zero by rewriting it as x equals five x and generating successive approximations from an initial guess.

Apply the newton-raphson method to approximate roots of f(x)=0 by Taylor expanding around an initial guess and using x_{n+1}=x_n - f(x_n)/f'(x_n).

Learn the basics of solving systems of linear equations with iterative methods, including the Jacobi iterative method and Gauss-siedel method, using simultaneous and successive replacement approaches.

demonstrates solving a linear system using the Jacobi iterative method, showing initial estimates x≈3.14, y≈4.80, z≈2.03 and refined results x≈2.40, y≈3.54, z≈1.95.

Learn to solve a linear system with the Gauss-Seidel method, performing iterative refinements of x, y, and z from zeros to x ≈ 2.42, y ≈ 3.57, z ≈ 1.92.