Solution of Vandermonde systems of equations

Authors:
Ake Björck and Victor Pereyra

Journal:
Math. Comp. **24** (1970), 893-903

MSC:
Primary 65.35

MathSciNet review:
0290541

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Abstract: We obtain in this paper a considerable improvement over a method developed earlier by Ballester and Pereyra for the solution of systems of linear equations with Vandermonde matrices of coefficients. This is achieved by observing that a part of the earlier algorithm is equivalent to Newton's interpolation method. This allows also to produce a progressive algorithm which is significantly more efficient than previous available methods. Algol-60 programs and numerical results are included. Confluent Vandermonde systems are also briefly discussed.

**[1 C]**C. Ballester and V. Pereyra,*On the construction of discrete approximations to linear differential expressions*, Math. Comp.**21**(1967), 297–302. MR**0228167**, 10.1090/S0025-5718-1967-0228167-8**[2]**Walter Gautschi,*On inverses of Vandermonde and confluent Vandermonde matrices*, Numer. Math.**4**(1962), 117–123. MR**0139627****[3]**J. W. Jerome & L. L. Schumaker,*A Note on Obtaining Natural Spline Functions by the Abstract Approach of Laurent*, MRC Technical Report #776, University of Wisconsin, Madison, Wis., 1967.**[4]**J. N. Lyness and C. B. Moler,*Van der Monde systems and numerical differentiation*, Numer. Math.**8**(1966), 458–464. MR**0201071****[5]**J. F. Traub,*Associated polynomials and uniform methods for the solution of linear problems*, SIAM Rev.**8**(1966), 277–301. MR**0207238****[6]**J. F. Traub,*Iterative methods for the solution of equations*, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0169356****[7]**G. Galimberti and V. Pereyra,*Numerical differentiation and the solution of multidimensional Vandermonde systems*, Math. Comp.**24**(1970), 357–364. MR**0275668**, 10.1090/S0025-5718-1970-0275668-2**[8]**G. Galimberti and V. Pereyra,*Solving confluent Vandermonde systems of Hermite type*, Numer. Math.**18**(1971/72), 44–60. MR**0300417****[9]**S.-Å. Gustafson,*Rapid Computation of Interpolation Formulae and Mechanical Quadrature Rules*, Technical Report CS #70-152, Stanford University, Stanford, Calif., 1970.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1970-0290541-1

Keywords:
Vandermonde systems,
confluent Vandermonde systems

Article copyright:
© Copyright 1970
American Mathematical Society