Interpolation

An interpolant is a POLYINTERP data structure created with either the PolynomialInterpolation or the CubicSpline command.

PolynomialInterpolation

Creates the POLYINTERP data structure, from which can be extracted the interpolating polynomial and its properties.

CubicSpline

Constructs a cubic spline for numeric data points in the form $\left[x\,y\right]$.

The following commands work on an interpolant, a POLYINTERP data structure.

AddPoint

Recomputes an interpolant with an additional point, provided the interpolant was created with the PolynomialInterpolation or the CubicSpline command.

ApproximateExactUpperBound

For an interpolant created with the PolynomialInterpolation command, and for each indicated point, returns the value of the interpolating polynomial, the value of the interpolated function, and the upper bound of the remainder term.

ApproximateValue

For specified points, returns the value(s) of an interpolating polynomial created with either the PolynomialInterpolation or CubicSpline command.

BasisFunctions

For interpolants constructed with the PolynomialInterpolation command using the Lagrange, Newton, or Hermite method, returns the method's basis functions.

DataPoints

Retrieves the data points (interpolated points) from an interpolant constructed by either the PolynomialInterpolation or CubicSpline command.

DividedDifferenceTable

Constructs a divided-difference table from an interpolant created with the PolynomialInterpolation command using either the Hermite or Newton method.

Draw

For an interpolant constructed by either the PolynomialInterpolation or CubicSpline command, draws a graph of one or more of the following: ApproximateValue, BasisFunctions, DataPoints, ExactValue, Function, Interpolant.

ExactValue

At specified points, for an interpolant constructed by either the PolynomialInterpolation or CubicSpline command, returns the exact values of the interpolated function.

Function

For an interpolant constructed by either the PolynomialInterpolation or CubicSpline command, returns the interpolated function.

Interpolant

Extracts the interpolating polynomial from an interpolant constructed by either the PolynomialInterpolation or CubicSpline command.

InterpolantRemainderTerm

For an interpolant constructed by the PolynomialInterpolation command, returns the interpolating polynomial and the remainder term.

LinearSystem

For an interpolant created by the CubicSpline command, returns, in the form of a matrix and vector, the linear equations whose solution determines the spline.

NevilleTable

For interpolating polynomials constructed by the PolynomialInterpolation command using  Neville's method, returns the Neville table.

RemainderTerm

For interpolating polynomials constructed by the PolynomialInterpolation command, returns the remainder (error) term.

UpperBoundOfRemainderTerm

For an interpolant constructed by either the PolynomialInterpolation command or CubicSpline command (clamped endpoint conditions), returns the upper bound of the absolute value of the remainder term.

Quadrature

AdaptiveQuadrature

A scaled-down version of the Quadrature command, tailored to just those methods of numeric integration that support an adaptive implementation.

Quadrature

Numeric integration by various techniques, including adaptive methods. For adaptive methods, a table showing the subinterval selections can be returned.

Root Finding

Bisection

Numeric root-finding for the function $f\left(x\right)$ by the bisection method. Possible returns include the value of the root, a sequence of bracketing subintervals, a table of these subintervals along with function values and errors, a graph showing the subintervals, and an animation of the convergence of the subintervals to the root.

FalsePosition

Numeric root-finding for the function $f\left(x\right)$ by the method of false position. Possible returns include the value of the root, a sequence of bracketing subintervals, a table of these subintervals along with function values and errors, a graph showing the approximations, and an animation of the convergence of the approximations to the root.

FixedPointIteration

Fixed-point (Picard, linear) iteration is used to find a root of the function $f\left(x\right)$ by converting it to $g\left(x\right)\=x-f\left(x\right)$. Possible returns include the value of the root, a sequence of iterates, a table of iterates and associated errors, a graph showing the iterates and a cobweb diagram, and an animation of the convergence of the iterates to the root.

ModifiedNewton

Roots of the function $f\left(x\right)$ are found by the classic Newton's method for roots of multiplicity 1, and by a modified algorithm for roots of multiplicity $m\>1$. Possible returns include the approximate root, a sequence of iterates, a table of iterates and their errors, a graph showing the iterates and an animation showing the convergence of the iterates to the root.

Newton

Roots of the function $f\left(x\right)$ are found by the classic Newton's method for roots of multiplicity 1; the method fails for roots of multiplicity $m\>1$. Possible returns include the approximate root, a sequence of iterates, a table of iterates and their errors, a graph showing the iterates and an animation showing the convergence of the iterates to the root.

Roots

The parent command for iterative root-finding, incorporates each of the separate commands in this Root-Finding section.

Secant

Roots of the function $f\left(x\right)$ are found by the secant method. Possible returns include the approximate root, a sequence of iterates, a table of iterates and their errors, a graph showing secants and iterates, and an animation showing the convergence of the secants and iterates to the root.

Steffensen

Roots of the function $f\left(x\right)$ are found by fixed-point iteration accelerated by a version of Aitken's ${\mathrm{\Delta }}^2$ technique. Possible returns include the approximate root, the sequence of accelerated iterates, a table of all iterates and their errors, a graph showing the iterates, and an animation showing the conversion of the iteration.

Numerical Linear Algebra

BackSubstitution

Back substitution applied to the system $A\mathbf{x}\=\mathbf{b}$, where $A$ is upper triangular.

Distance

Computes a norm of the difference between two vectors.

ForwardSubstitution

Forward substitution applied to the system $A\mathbf{x}\=\mathbf{b}$, where $A$ is lower triangular.

IsConvergent

Determines whether or not the Jacobi, Gauss-Seidel, or SOR methods for the solution of $A\mathbf{x}\=\mathbf{b}$ converge.

IsMatrixShape

Determines if a matrix $A$ is diagonal, strictly diagonally dominant, diagonally dominant, Hermitian, positive definite, symmetric, triangular[upper], triangular[lower], or tridiagonal.

IterativeApproximate

Obtain an approximate solution of $A\mathbf{x}\=\mathbf{b}$ by Jacobi, Gauss-Seidel, or SOR iteration. Possible returns include the approximate solution, a sequence of iterates, a list of errors of the iterates, a column graph of the errors of the iterates, and if $n\=3$, a graph of the path taken in ${\mathrm{\ℝ}}^3$ by the iterations.

IterativeFormula

Determines the matrix $T$ and vector c that express the Jacobi, Gauss-Seidel, and SOR iterations in the form $\mathbf{x}\=T\mathbf{x}\+\mathbf{c}$. Possible returns include one or more of L, U, D, T, c, the spectral radius, or a list of iterates.

IterativeFormulaTutor

Interactive implementation of the IterativeFormula command.

LeadingPrincipalSubmatrix

Returns the $n$th leading principal submatrix of the matrix $A$.

LinearSolve

Provides a numerical solution to the linear system $\mathrm{Ax}\=b$

MatrixConvergence

For the square matrix $A$, determines if the spectral radius is strictly less than 1 so that $\underset{k\to \infty }{lim}{\left(A^k\right)}_{i\,j}\=0$ for each $i\,j\=1..n$, where $n$ is the dimension of $A$.

MatrixDecomposition

Returns, among others, the following decompositions for the matrix $A$:

LU, PLU, LU[tridiagonal], PLU[scaled], LDU, LDLt, Cholesky

MatrixDecompositionTutor

Interactive implementation of the MatrixDecomposition command.

SpectralRadius

For the square matrix $A$, determines the spectral radius, that is, the maximal absolute value of the eigenvalues.

VectorLimit

For a vector $\mathbf{V}\left(n\right)$, returns $\underset{n\to \infty }{lim}\mathbf{V}\left(n\right)$, the vector of limits of the components of $\mathbf{V}$. Essentially, it maps the limit operator onto the components of V.

Initial-Value Problem

AdamsBashforth

AdamsBashforthMoulton

AdamsMoulton

Euler

RungeKutta

Taylor

EulerTutor

Interactive implementation of the Euler command.

InitialValueProblem

For the initial value problem $y\prime \=f\left(t\,y\right)\,y\left(t_0\right)\=y_0$, this "parent" command can be instantiated to implement any one of the six methods listed above.

InitialValueProblemTutor

Interactive implementation of the InitialValueProblem command.

General

AbsoluteError

Given an exact and an approximate value, returns the absolute error in the approximate value.

NumberOfSignificantDigits

Given an exact and an approximate value, returns, according to the usage in the Burden/Faires reference, the number of significant digits in the approximate value.

RateOfConvergence

Indicates, by means of the Landau big "O" notation, the rate of convergence of a sequence described by its $n$th term.

RelativeError

Given an exact and an approximate value, returns the relative error in the approximate value.

TaylorPolynomial

Constructs a Taylor polynomial, and can provide its remainder term. If a point is given, the return includes the exact and approximate values at that point, and the error-bound for the approximation.