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Polynomial Interpolation
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
But polynomial interpolation is usually a poor way to approximate a function. The Vandermonde matrix is frequently ill-conditioned. Even if we solve it accurately, however, if n is large and x is large then error in evaluating the polynomial p(x)=anxn+an−1xn−1+⋯+a1x+a0
The last chapter
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
PROOF Consider the equations expressing Ak′ in terms of the ∑jjiAj for k < s. These are s linear equations for s unknowns. The coefficient matrix is a Vandermonde matrix. As we know from Chapter 4, Vandermonde matrices are invertible over any field. It follows that the solution is uniquely determined.
Numerical Differentiation and Integration
Published in Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer, Classical and Modern Numerical Analysis, 2009
Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer
for the interval [0, 1]. Note that A is the transpose of the Vandermonde matrix that arises as the matrix for the system of equations for the coefficients of the interpolating polynomial at the points xi. (See page 211.) Thus, the matrix A is nonsingular, because xj ≠ xℓ for j ≠ ℓ.
Parallel model order reduction based on block discrete Fourier transform and Krylov subspace for parametric systems
Published in International Journal of Systems Science, 2023
Before introducing the DFT and the block DFT, some notations are firstly given. Let be the imaginary unit such that . For an integer , let us define . The complex number is an hth root of unity, i.e. . The Vandermonde matrix formed with the complex number is called the discrete Fourier transform matrix of order h. It is non-singular and its inverse is , where denotes the complex conjugate of the matrix .
A local meshless procedure to determine the unknown control parameter in the multi-dimensional inverse problems
Published in Inverse Problems in Science and Engineering, 2021
Mehdi Dehghan, Nasim Shafieeabyaneh, Mostafa Abbaszadeh
FD formulas for a regular mesh are generally derived from 1-D case. To obtain FD formulas in high dimension in Cartesian grids, the 1-D FD can be separately employed in each spatial direction. Now, the main goal is to find a polynomial-based FD-formula. There are various approaches for detecting the weights in 1-D FD formulas. The simplest of them is that the formulas must be precise for the monomials up to the highest degree possible [40, 41]. For instance, we approximate the linear differential operator at the point with points over its stencil. Hence, the unknowns -related are determined by solving the following linear system [40] The Vandermonde matrix is nonsingular if the points are distinct. This approach is not applicable to scattered points and multivariate functions because the unisolvency feature does not resolve in this case [40].
Lyapunov-based boundary feedback design for parabolic PDEs
Published in International Journal of Control, 2021
We are now ready to turn to the proof of the lemma. It suffices to show that the single-input, linear, time-invariant system is controllable. The Kalman rank controllability test for system (79) gives the square matrix Since for , it follows that the matrix is invertible with non-zero determinant. Since the matrix is a Vandermonde matrix and since , it follows that the matrix has a non-zero determinant. It follows from (80) that the determinant of the matrix is non-zero. Thus and system (79) is controllable. The proof is complete.