# vandermonde

vandermonde, a MATLAB code which implements the Bjork-Pereyra algorithm for accurate solution of linear systems involving the Vandermonde matrix.

A univariate NxN Vandermonde matrix is defined by a parameter vector ALPHA of N distinct real values, and has the form:

```        1             1            ... 1
alpha1        alpha2       ... alphan
alpha1^2      alpha2^2     ... alphan^2
...           ...          ... ...
alpha1^(n-1)  alpha2^(n-1) ... alphan^(n-1)
```

If p(x) is a polynomial of degree N-1, which is required to interpolate a function f(x) at N distinct points ALPHA, then the coefficients C of the polynomial can be found from the interpolation equations, which can be written as a linear system involving a transposed Vandermonde matrix:

```        1    alpha1  alpha1^2  ... alpha1^(n-1)    c0           f(alpha1)
1    alpha2  alpha2^2  ... alpha2^(n-1)    c1           f(alpha2)
...  ...     ...       ... ...             ...       =  ...
1    alphan  alphan^2  ... alphan^(n-1)    c(n-1)       f(alphan)
```

### Languages:

vandermonde is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

quadrature_weights_vandermonde, a MATLAB code which computes the weights of a quadrature rule using the Vandermonde matrix, assuming that the points have been specified.

test_mat, a MATLAB code which defines test matrices for which some of the determinant, eigenvalues, inverse, null vectors, P*L*U factorization or linear system solution are already known, including the Vandermonde matrix.

vandermonde_approx_1d, a MATLAB code which finds a polynomial approximant to data of a 1D argument by setting up and solving an overdetermined linear system for the polynomial coefficients, involving the Vandermonde matrix.

vandermonde_approx_2d, a MATLAB code which finds a polynomial approximant p(x,y) to data z(x,y) of a 2D argument by setting up and solving an overdetermined linear system for the polynomial coefficients involving the Vandermonde matrix.

vandermonde_interp_1d, a MATLAB code which finds a polynomial interpolant to a function of 1D data by setting up and solving a linear system for the polynomial coefficients, involving the Vandermonde matrix.

vandermonde_interp_2d, a MATLAB code which finds a polynomial interpolant to data z(x,y) of a 2D argument by setting up and solving a linear system for the polynomial coefficients, involving the Vandermonde matrix.

### Reference:

1. Ake Bjorck, Victor Pereyra,
Solution of Vandermonde Systems of Equations,
Mathematics of Computation,
Volume 24, Number 112, October 1970, pages 893-903.
2. Gustavo Galimberti, Victor Pereyra,
Numerical Differentiation and the Solution of Multidimensional Vandermonde Systems,
Mathematics of Computation,
Volume 24, Number 110, April 1970, pages 357-364.
3. Nicholas Higham,
Stability analysis of algorithms for solving confluent Vandermonde-like systems,
SIAM Journal on Matrix Analysis and Applications,
Volume 11, 1990, pages 23-41.

### Source Code:

• bivand1.m, defines a bivariate Vandermonde matrix associated with polynomials of total degree less than N.
• bivand2.m, defines a bivariate Vandermonde matrix associated with polynomials of maximum degree less than N.
• dvand.m, solves a transposed Vandermonde linear system.
• dvandprg.m, solves a transposed Vandermonde linear system "progressively", that is, using N successive calls, adding one more ALPHA each time.
• pvand.m, solves a Vandermonde linear system.
• pvandprg.m, solves a Vandermonde linear system "progressively", that is, using N successive calls, adding one more ALPHA each time.
• r8mat_print.m, prints an R8MAT;
• r8mat_print_some.m, prints some of an R8MAT;
• vand1.m, defines a univariate Vandermonde matrix.