Documentation

This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

linalg::vandermondeSolve

Solve a linear Vandermonde system

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

Syntax

linalg::vandermondeSolve(v, y, <Transposed>)

Description

linalg::vandermondeSolve(v, y) returns the solution of the linear Vandermonde system with i = 1, …, n.

linalg::vandermondeSolve uses O(n2) elementary operations to solve the Vandermonde system. It is faster than the general solver solve and the linear solvers linsolve, numeric::linsolve, linalg::matlinsolve and numeric::matlinsolve.

The solution returned by linalg::vandermondeSolve([v[i] $ i=1..n], [y[i] $ i=1..n]) yields the coefficients of the polynomial p(v) = x1 + x2v + ··· + xnvn - 1 interpolating the data table (v1, y1), …, (vn, yn), i.e.,

.

See Example 1.

Examples

Example 1

The Vandermonde points v and the right hand side y of the linear system are entered as vectors:

delete y0, y1, y2:
v := matrix([[0, 1, 2]]); 
y := matrix([[y0, y1, y2]])

The solution vector is:

x := linalg::vandermondeSolve(v, y)

The solution yields the coefficients of the interpolating polynomial:

P := v -> _plus(x[i+1]*v^i $ i=0..2):

through the points (0, y0), (1, y1), (2, y2):

P(v[1]), P(v[2]), P(v[3])

With the optional argument Transposed, the linear system with the transposed Vandermonde matrix corresponding to v is solved:

linalg::vandermondeSolve(v, y, Transposed)

delete v, y, x, P:

Example 2

The Vandermonde points v and the right hand side y of the linear system are entered as 2×1 matrices:

Mat := Dom::Matrix(Dom::ExpressionField(normal)):
delete v1, v2, y1, y2:
v := Mat([v1, v2]): 
y := Mat([y1, y2]):

We define the vectors over the domain Dom::ExpressionField(normal) in order to simplify intermediate computations.

Next, we compute the solution of the corresponding Vandermonde system:

x := linalg::vandermondeSolve(v, y)

We construct the Vandermonde matrix V and verify the result:

V := Mat([[1, v[1]], [1, v[2]]])

V * x

delete Mat, v, y, x, V:

Example 3

We solve a Vandermonde system over the field 7 (the integers modulo 7) represented by the domain Dom::IntegerMod(7):

MatZ7 := Dom::Matrix(Dom::IntegerMod(7)):
v := MatZ7([1, 2, 3]): y := MatZ7([0, 1, 2]):
linalg::vandermondeSolve(v, y)

delete MatZ7, v, y:

Parameters

v

A vector with distinct elements (a vector is an n×1 or 1 ×n matrix of category Cat::Matrix). Alternatively, a list with n distinct elements is also accepted.

y

A vector of the same dimension and domain type as v. Alternatively, a list with n elements is also accepted.

Options

Transposed

Returns the solution of the transposed system with i = 1, …, n.

Return Values

Vector of the same domain type as y.

Algorithms

The Vandermonde matrix

generated by v = [v1, …, vn] is invertible if and only if the vi are distinct.

The vector returned by linalg::vandermondeSolve(x, y) is the unique solution of .

The vector x returned by linalg::vandermondeSolve(x, y, Transposed) is the unique solution of .

Was this topic helpful?