Construct a Vandermonde matrix with symbolic entries:

Show the elements:

Get the determinant:

Construct a Vandermonde matrix with numerical entries:

Get the ordinary representation of the matrix:

Solve a linear system with the matrix (a dual Vandermonde problem):

The solution gives the coefficients of an interpolating polynomial approximating the cosine:

A rectangular Vandermonde matrix:

Show the elements:

A confluent Vandermonde matrix:

Show the elements:

A rectangular confluent Vandermonde matrix:

Show the elements:

Represent a non-confluent Vandermonde matrix as a structured array:

Show the elements:

The structured representation typically uses much less memory:

VandermondeMatrix objects include properties that give information about the array:

The "Nodes" property gives the vector of nodes that generates the Vandermonde matrix:

The "Permutation" property gives a permutation vector associated with the Vandermonde matrix:

The "Multiplicities" property gives the multiplicities of each unique node:

The "Confluent" property gives True if the Vandermonde matrix is confluent, and False otherwise:

The "Transposed" property gives True if the Vandermonde matrix is transposed, and False otherwise:

The "Summary" property gives a brief summary of information about the array:

The "StructuredAlgorithms" property gives a list of functions that have algorithms that use the structure of the representation:

Structured algorithms are typically faster and/or more accurate:

The exact determinant can be computed:

The value computed from VandermondeMatrix is much more accurate:

The structured algorithms may avoid problems from ill-conditioning:

Solve the linear system using VandermondeMatrix:

Solve the linear system using the normal matrix:

The residual is smaller with VandermondeMatrix:

When appropriate, structured algorithms return another structured array:

Evaluating its conjugate transpose gives a structured result:

The product with the original matrix is no longer a Vandermonde matrix:

Generate a set of n+1 distinct samples from a function:

Use VandermondeMatrix to find a polynomial of degree n that interpolates the function at those samples:

Define the polynomial:

Compare the original function and the interpolating polynomial:

Determine the coefficients of a polynomial that satisfies the interpolation conditions and the derivative conditions :

Define the polynomial:

Verify that the interpolation and derivative conditions are satisfied:

Define a set of 4 distinct points:

Fit a cubic Vandermonde polynomial to the set of points:

Columns of the inverse of the VandermondeMatrix define a set of polynomials of degree n, called Lagrange polynomials:

Lagrange polynomials satisfy the identity :

Represent the Vandermonde polynomial in terms of Lagrange polynomials:

Plot and compare:

Define a polynomial with roots :

Construct a VandermondeMatrix from the roots:

The Frobenius companion matrix of a polynomial with known roots is similar to a diagonal matrix containing the roots, with the Vandermonde matrix acting as the similarity transformation:

The last column of the Frobenius companion matrix contains the negative of coefficients of the polynomial up to degree n-1:

These are also the same as the elementary symmetric polynomials, up to sign:

Solve for the coefficients of an n-order finite difference formula:

Assemble the finite difference formula:

This is equivalent to the result of DifferenceQuotient:

Define a function for computing the nodes and weights of a closed Newton–Cotes rule [MathWorld]:

The trapezoidal rule is the order-1 Newton–Cotes formula:

Simpson's rule is the order-2 Newton–Cotes formula:

Boole's rule is the order-4 Newton–Cotes formula:

An n-term sum of exponential functions:

Sample the exponential function at points:

Use Prony's method [Wikipedia] to recover the sum of exponentials from the data:

A confluent Vandermonde matrix can be obtained as the limit of a non-confluent matrix:

Take the successive divided differences of each row:

The limit as is the confluent Vandermonde matrix:

FourierMatrix can be represented as a VandermondeMatrix:

The discriminant of a polynomial can be expressed as the square of Det of a VandermondeMatrix:

Solve a confluent Vandermonde system:

These are the coefficients of InterpolatingPolynomial with derivative values specified:

Demonstrate a factorization of the inverse of a Vandermonde matrix in terms of diagonal, Vandermonde and Hankel matrices:

Express a Cauchy matrix as a product of diagonal, Vandermonde and Hankel matrices:

A matrix and vector:

Define a function for computing the Krylov matrix from a given matrix and vector:

Compute the eigenvalues of the matrix:

The action of a matrix exponential on a vector can be expressed in terms of a Krylov matrix and the inverse of a Vandermonde matrix:

Verify that the same result can be obtained from MatrixExp:

Verify that the same result can be obtained from DSolveValue:

Linear Caputo differential equations can also be solved in terms of a Krylov matrix and the inverse of a Vandermonde matrix, along with the Mittag–Leffler function MittagLefflerE:

Verify that the same result can be obtained from DSolveValue:

The Butcher tableau of an implicit Runge–Kutta method can be obtained by solving a primal Vandermonde system constructed from the roots of a Legendre polynomial:

Define a function for computing the implicit Runge–Kutta–Gauss coefficients to a given precision:

Use these coefficients in NDSolveValue to solve a stiff differential equation modeling flame propagation, due to Shampine:

Compare the result with the exact solution: