tools for math1070_2019

tools, some simple MATLAB tools for MATH1070, "Numerical Mathematical Analysis", Instructor: Catalin Trenchea, University of Pittsburgh, fall 2019, MWF 12:00-12:50, Room PUBHL A522.

Location: http://people.sc.fsu.edu/~jburkardt/classes/math1070_2019/tools/tools.html

approx: Find a simple function that approximates a function everywhere in [a,b]

• approx_bernstein.m, approximates a function f(x) over the interval [a,b] using a Bernstein polynomial.
• approx_cheby.m, approximates a function f(x) over the interval [a,b] by the interpolating polynomial at n Chebyshev points, which often is close to the minmax polynomial.
• approx_leastsquares.m, approximates a function f(x) over the interval [a,b] using an m-degree polynomial which minimizes the RMS norm difference to n sample values of f(x).

diff: Estimate the derivative of a function at a point

• diff_center.m, uses centered differences to estimate the derivative of a function f(x) at a point x0 using a stepsize h.
• diff_forward.m, uses forward differences to estimate the derivative of a function f(x) at a point x0 using a stepsize h.
• diff2_center.m, uses centered differences to estimate the second derivative of a function f(x) at a point x0 using a stepsize h.

interp: find a simple function that matches a function f(x) at selected points (x,f(x)), or data (x,y)

• interp_cheby.m, interpolate f(x) over [a,b] by constructing a polynomial based on n Chebyshev spaced points;
• interp_equal.m, interpolate f(x) over [a,b] by constructing a polynomial based on n equally spaced points;
• interp_ncs.m, interpolate a function f(x) with a natural cubic spline (NCS), which uses the 'zero second derivative' condition at the endpoints.
• interp_spline.m, interpolate a function f(x) with a cubic spline, which uses the 'not-a-knot' condition at the endpoints.
• interp_spline_data.m, interpolate (x,y) data with a cubic spline, using the 'not-a-knot' condition at the endpoints.
• interp_trig.m, interpolate a function f(x) at n equally spaced points over the interval [a,b], using a trigonometric polynomial.

nonlin: Solve a single nonlinear equation f(x)=0

• nonlin_bisect.m, uses bisection to seek a root of f(x) in the change-of-sign interval [a,b];
• nonlin_fixed_point.m, uses fixed point iteration, x=g(x), to seek a root of f(x), given a starting point x0 and a number of iterations it.
• nonlin_newton.m, uses Newton's method to seek a zero of a function given formulas for f(x), f'(x), and a starting point a.
• nonlin_regula.m, uses the regula falsi method to seek a root of f(x) in the change-of-sign interval [a,b];
• nonlin_secant.m, uses the secant method to seek a root of f(x) given two estimates a and b.

norm: function norms and dot products

• dot_l2.m, estimates the L2 dot product of f(x) and g(x) over the interval [a,b];
• norm_l1.m, estimates the L1 norm of f(x) over the interval [a,b];
• norm_l2.m, estimates the L2 norm of f(x) over the interval [a,b];
• norm_loo.m, estimates the L-infinity norm of f(x) over the interval [a,b] by sampling n equally spaced values.
• norm_rms.m, estimates the root mean square (RMS) norm of f(x) over the interval [a,b];

ode: Estimate the solution of a differential equation

• ode_euler.m, applies the Euler method to estimate the solution of an ordinary differential equation y'=f(x,y), over the interval [a,b], with initial condition y(a)=ya, using n steps.
• ode_euler_system.m, applies the Euler method to estimate the solution of a system of ordinary differential equations y'=f(x,y), over the interval [a,b], with initial condition vector y(a)=ya, using n steps.
• ode_euler_backward.m, applies the backward Euler method to estimate the solution of an ordinary differential equation y'=f(x,y), over the interval [a,b], with initial condition y(a)=ya, using n steps.
• ode_midpoint.m, applies the midpoint method to estimate the solution of an ordinary differential equation y'=f(x,y), over the interval [a,b], with initial condition y(a)=ya, using n steps.
• ode_midpoint_system.m, applies the midpoint method to estimate the solution of a system of ordinary differential equations y'=f(x,y), over the interval [a,b], with initial condition vector y(a)=ya, using n steps.
• ode_rk4.m, applies a fourth order Runge-Kutta method to estimate the solution of an ordinary differential equation y'=f(x,y), over the interval [a,b], with initial condition y(a)=ya, using n steps.
• ode_trapezoidal.m, applies the trapezoidal method to estimate the solution of an ordinary differential equation y'=f(x,y), over the interval [a,b], with initial condition y(a)=ya, using n steps.

opt: Find the minimum or maximum of a function

• opt_golden.m, estimates the location of a local minimum of a function f(x), using a formula for f(x), an interval[a,b] over which the function is unimodal ("U-shaped"), a maximum number of iterations, and a tolerance for the interval width.
• opt_gradient_descent.m, estimates the location of a local minimum of a function f(x), using a formula for f'(x), a starting point x0, and a stepsize factor gamma.
• opt_quadratic.m, interactively uses quadratic interpolation to estimate a critical point of a function f(x) given three starting points, an iteration limit n, and tolerances for x and y.
• opt_sample.m, estimates the minimum and maximum of a function f(x) over the interval [a,b] using n random sample points. (Use a large value for n!)

quad: Estimate the integral of a function

• quad_gauss.m, uses an n-point Gauss quadrature rule to estimate the integral of a function f(x) over the interval [a,b].
• quad_monte_carlo.m, uses n random samples to estimate the integral of a function f(x) over the interval [a,b].
• quad_trap.m, uses an n-interval trapezoidal quadrature rule to estimate the integral of a function f(x) over the interval [a,b].

util: utilities

• iplot.m, plots f(x) over [a,b];
• timestamp.m, prints the YMDHMS date as a timestamp;