Student/NumericalAnalysis/Glossary - Maple Help

Student[NumericalAnalysis] Glossary of Commands

The purpose of this help page is to provide a quick reference to the commands in the Student[NumericalAnalysis] subpackage.

The following table lists all of the commands of the Student[NumericalAnalysis] subpackage. The commands are categorized by subject area.

For a comprehensive description of this subpackage, see Student[NumericalAnalysis].

 Interpolation An interpolant is a POLYINTERP data structure created with either the PolynomialInterpolation or the CubicSpline command. Creates the POLYINTERP data structure, from which can be extracted the interpolating polynomial and its properties. 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. Recomputes an interpolant with an additional point, provided the interpolant was created with the PolynomialInterpolation or the CubicSpline command. 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. For specified points, returns the value(s) of an interpolating polynomial created with either the PolynomialInterpolation or CubicSpline command. For interpolants constructed with the PolynomialInterpolation command using the Lagrange, Newton, or Hermite method, returns the method's basis functions. Retrieves the data points (interpolated points) from an interpolant constructed by either the PolynomialInterpolation or CubicSpline command. Constructs a divided-difference table from an interpolant created with the PolynomialInterpolation command using either the Hermite or Newton method. 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. At specified points, for an interpolant constructed by either the PolynomialInterpolation or CubicSpline command, returns the exact values of the interpolated function. For an interpolant constructed by either the PolynomialInterpolation or CubicSpline command, returns the interpolated function. Extracts the interpolating polynomial from an interpolant constructed by either the PolynomialInterpolation or CubicSpline command. For an interpolant constructed by the PolynomialInterpolation command, returns the interpolating polynomial and the remainder term. 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. For interpolating polynomials constructed by the PolynomialInterpolation command using  Neville's method, returns the Neville table. For interpolating polynomials constructed by the PolynomialInterpolation command, returns the remainder (error) term. 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 A scaled-down version of the Quadrature command, tailored to just those methods of numeric integration that support an adaptive implementation. Numeric integration by various techniques, including adaptive methods. For adaptive methods, a table showing the subinterval selections can be returned.



 Root Finding 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. 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. 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. 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. 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. The parent command for iterative root-finding, incorporates each of the separate commands in this Root-Finding section. 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. 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 Back substitution applied to the system $A\mathbf{x}=\mathbf{b}$, where $A$ is upper triangular. Computes a norm of the difference between two vectors. Forward substitution applied to the system $A\mathbf{x}=\mathbf{b}$, where $A$ is lower triangular. Determines whether or not the Jacobi, Gauss-Seidel, or SOR methods for the solution of $A\mathbf{x}=\mathbf{b}$ converge. Determines if a matrix $A$ is diagonal, strictly diagonally dominant, diagonally dominant, Hermitian, positive definite, symmetric, triangular[upper], triangular[lower], or tridiagonal. 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. 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. Interactive implementation of the IterativeFormula command. Returns the $n$th leading principal submatrix of the matrix $A$. Provides a numerical solution to the linear system $\mathrm{Ax}=b$ For the square matrix $A$, determines if the spectral radius is strictly less than 1 so that $\underset{k\to \infty }{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\left({A}^{k}\right)}_{i,j}=0$ for each $i,j=1..n$, where $n$ is the dimension of $A$. Returns, among others, the following decompositions for the matrix $A$: LU, PLU, LU[tridiagonal], PLU[scaled], LDU, LDLt, Cholesky Interactive implementation of the MatrixDecomposition command. For the square matrix $A$, determines the spectral radius, that is, the maximal absolute value of the eigenvalues. 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

 • For the initial value problem $y\prime =f\left(t,y\right),y\left({t}_{0}\right)={y}_{0}$, returns one of: the computed value $y\left(b\right),b>{t}_{0}$, the absolute error in $y\left(b\right)$, a graph of the numeric solution along with a graph of the solution computed by one of Maple's best numeric solvers, or a table of computed values and the absolute value of their errors.
 • The RungeKutta command implements one of the following methods: Midpoint, Third-Order, Fourth-Order, Runge-Kutta-Fehlberg, Heun, Modified Euler.
 • The Taylor command defaults to a third-degree Taylor polynomial, but this can be modified with the order option.

Interactive implementation of the Euler command.

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.

Interactive implementation of the InitialValueProblem command.



 General Given an exact and an approximate value, returns the absolute error in the approximate value. 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. Indicates, by means of the Landau big "O" notation, the rate of convergence of a sequence described by its $n$th term. Given an exact and an approximate value, returns the relative error in the approximate value. 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.