The Experts below are selected from a list of 306 Experts worldwide ranked by ideXlab platform
John P Boyd - One of the best experts on this subject based on the ideXlab platform.
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a fourier Companion Matrix multiplication Matrix with real valued elements finding the roots of a trigonometric polynomial by Matrix eigensolving
Numerical Mathematics-theory Methods and Applications, 2013Co-Authors: John P BoydAbstract:We show that the zeros of a trigonometric polynomial of degree N with the usual (2 N + 1) terms can be calculated by computing the eigenvalues of a Matrix of dimension 2 N with real-valued elements M jk . This Matrix is a multiplication Matrix in the sense that, after first defining a vector whose elements are the first 2 N basis functions, . This relationship is the eigenproblem; the zeros t k are the arccosine function of λ k /2 where the λ k are the eigenvalues of . We dub this the “Fourier Division Companion Matrix”, or FDCM for short, because it is derived using trigonometric polynomial division. We show through examples that the algorithm computes both real and complex-valued roots, even double roots, to near machine precision accuracy.
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a comparison of Companion Matrix methods to find roots of a trigonometric polynomial
Journal of Computational Physics, 2013Co-Authors: John P BoydAbstract:Abstract A trigonometric polynomial is a truncated Fourier series of the form f N ( t ) ≡ ∑ j = 0 N a j cos ( jt ) + ∑ j = 1 N b j sin ( jt ) . It has been previously shown by the author that zeros of such a polynomial can be computed as the eigenvalues of a Companion Matrix with elements which are complex valued combinations of the Fourier coefficients, the “CCM” method. However, previous work provided no examples, so one goal of this new work is to experimentally test the CCM method. A second goal is introduce a new alternative, the elimination/Chebyshev algorithm, and experimentally compare it with the CCM scheme. The elimination/Chebyshev Matrix (ECM) algorithm yields a Companion Matrix with real-valued elements, albeit at the price of usefulness only for real roots. The new elimination scheme first converts the trigonometric rootfinding problem to a pair of polynomial equations in the variables ( c , s ) where c ≡ cos ( t ) and s ≡ sin ( t ) . The elimination method next reduces the system to a single univariate polynomial P ( c ) . We show that this same polynomial is the resultant of the system and is also a generator of the Groebner basis with lexicographic ordering for the system. Both methods give very high numerical accuracy for real-valued roots, typically at least 11 decimal places in Matlab/IEEE 754 16 digit floating point arithmetic. The CCM algorithm is typically one or two decimal places more accurate, though these differences disappear if the roots are “Newton-polished” by a single Newton’s iteration. The complex-valued Matrix is accurate for complex-valued roots, too, though accuracy decreases with the magnitude of the imaginary part of the root. The cost of both methods scales as O ( N 3 ) floating point operations. In spite of intimate connections of the elimination/Chebyshev scheme to two well-established technologies for solving systems of equations, resultants and Groebner bases, and the advantages of using only real-valued arithmetic to obtain a Companion Matrix with real-valued elements, the ECM algorithm is noticeably inferior to the complex-valued Companion Matrix in simplicity, ease of programming, and accuracy.
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computing the real roots of a fourier series plus linear polynomial a chebyshev Companion Matrix approach
Applied Mathematics and Computation, 2012Co-Authors: John P Boyd, Burhan SadiqAbstract:Abstract Fourier series often need to be generalized by appending a linear polynomial to the usual series of sines and cosines. The integral of a trigonometric polynomial is one example; another is a time series of climate data where the periodic oscillations of the diurnal and annual cycles are accompanied by a non-periodic trend (global warming). Stock market averages fluctuate about a generally upward trend. Such non-periodic variations with time are commonly called “secular trends”. We borrow “secular” to label a truncated Fourier series plus a linear trend as a “linear”, secular trigonometric polynomial. Standard Fourier rootfinding methods are wrecked by the extra, nonperiodic term. Therefore, we introduce a new algorithm for computing the zeros of a Fourier polynomial-with-secular-trend. First, we expand the linear secular trigonometric polynomial f N ( t ) as a truncated Chebyshev series. Because of the special structure, it is easy to calculate a problem-dependent truncation M such that the error of the truncated Chebyshev series is guaranteed to be less than a user-specified tolerance. We then find the roots of the truncated Chebyshev series as the eigenvalues of the Chebyshev Companion Matrix. This computes all roots, but we explain why the method is not reliable for complex-valued roots unless these are close to the real axis. No a priori information is required of the user except the coefficients of the linear secular trigonometric polynomial. Numerical examples show that 13 decimal place accuracy for real roots is typical.
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numerical experiments on the accuracy of the chebyshev frobenius Companion Matrix method for finding the zeros of a truncated series of chebyshev polynomials
Journal of Computational and Applied Mathematics, 2007Co-Authors: John P Boyd, Daniel H GallyAbstract:For a function f(x) that is smooth on the interval [email protected]?[a,b] but otherwise arbitrary, the real-valued roots on the interval can always be found by the following two-part procedure. First, expand f(x) as a Chebyshev polynomial series on the interval and truncate for sufficiently large N. Second, find the zeros of the truncated Chebyshev series. The roots of an arbitrary polynomial of degree N, when written in the form of a truncated Chebyshev series, are the eigenvalues of an NxN Matrix whose elements are simple, explicit functions of the coefficients of the Chebyshev series. This Matrix is a generalization of the Frobenius Companion Matrix. We show by experimenting with random polynomials, Wilkinson's notoriously ill-conditioned polynomial, and polynomials with high-order roots that the Chebyshev Companion Matrix method is remarkably accurate for finding zeros on the target interval, yielding roots close to full machine precision. We also show that it is easy and cheap to apply Newton's iteration directly to the Chebyshev series so as to refine the roots to full machine precision, using the Companion Matrix eigenvalues as the starting point. Lastly, we derive a couple of theorems. The first shows that simple roots are stable under small perturbations of magnitude @e to a Chebyshev coefficient: the shift in the root x"* is bounded by @e/df/dx(x"*)+O(@e^2) for sufficiently small @e. Second, we show that polynomials with definite parity (only even or only odd powers of x) can be solved by a Companion Matrix whose size is one less than the number of nonzero coefficients, a vast cost-saving.
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Computing the zeros of a Fourier series or a Chebyshev series or general orthogonal polynomial series with parity symmetries
Computers & Mathematics With Applications, 2007Co-Authors: John P BoydAbstract:In recent years, good algorithms have been developed for finding the zeros of trigonometric polynomials and of ordinary polynomials when written in the form of a truncated Chebyshev polynomial or Legendre polynomial series. In each case, the roots can be found from the eigenvalues of a generalized Frobenius Companion Matrix whose elements are trivial functions of the Fourier coefficients or Chebyshev coefficients. However, the QR method for computing the Companion Matrix eigenvalues has a cost that grows proportionally to N^3 where N is the polynomial degree. (By exploiting the special structure of the Companion matrices, the cost can be reduced to O(N^2), but only for large N.) Here, we show that if the polynomial has definite parity, such as a trigonometric polynomial composed only of cosines or a polynomial that is a sum only of Chebyshev polynomials of odd degree, one can exploit these symmetries to halve the size of the problem. This reduces costs in the Companion Matrix method by a factor ranging between four and eight. For trigonometric polynomials, we give transformations that dramatically reduce costs even if the roots are found by an algorithm other than the Companion Matrix procedure. We further give reductions for trigonometric polynomials with double parity symmetries which save a factor of sixteen to a factor of sixty-four in the Companion Matrix algorithm. Special functions such as spherical harmonics, Mathieu functions, prolate spheroidal wavefunctions and Hough functions, all represented by truncated Fourier series with double parity, are a rich source of applications.
Miguel Vacahern - One of the best experts on this subject based on the ideXlab platform.
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Roots of the Underwood’s equations in short-cut distillation from a Companion Matrix eigenvalues
Chemical Engineering Science, 2012Co-Authors: Rosendo Monroy-loperena, Miguel VacahernAbstract:Abstract The design of distillation columns requires the calculation of the minimum reflux. For ideal mixtures, the well-known Underwood’s equations, which can also be applied to complex columns (e.g., several feeds and side products and side-stream strippers and enrichers) are used to calculate it. When distributed components other than the light and the heavy key are considered in the separation, the knowledge of all the roots of the Underwood’s feed equation is essential. However, the discontinuous form of the Underwood´s feed equation; makes the search of all roots a hard task. Using the fact that the Underwood’s equation can be transform into a polynomial form, in this work a Companion Matrix of the polynomial is presented, permitting the solution of the equation as a Generalized Eigenvalue Problem, arriving to a reliable and efficient method for the calculation of all roots of the Underwood’s equations. One main feature of the proposed approach is that not initial guesses are required to find all the roots of the Underwood’s equation. Open, robust and reliable software such as EISPACK can be used to routinely calculate the minimum reflux ratio by Underwood’s equations.
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roots of the underwood s equations in short cut distillation from a Companion Matrix eigenvalues
Chemical Engineering Science, 2012Co-Authors: Rosendo Monroyloperena, Miguel VacahernAbstract:Abstract The design of distillation columns requires the calculation of the minimum reflux. For ideal mixtures, the well-known Underwood’s equations, which can also be applied to complex columns (e.g., several feeds and side products and side-stream strippers and enrichers) are used to calculate it. When distributed components other than the light and the heavy key are considered in the separation, the knowledge of all the roots of the Underwood’s feed equation is essential. However, the discontinuous form of the Underwood´s feed equation; makes the search of all roots a hard task. Using the fact that the Underwood’s equation can be transform into a polynomial form, in this work a Companion Matrix of the polynomial is presented, permitting the solution of the equation as a Generalized Eigenvalue Problem, arriving to a reliable and efficient method for the calculation of all roots of the Underwood’s equations. One main feature of the proposed approach is that not initial guesses are required to find all the roots of the Underwood’s equation. Open, robust and reliable software such as EISPACK can be used to routinely calculate the minimum reflux ratio by Underwood’s equations.
David S Watkins - One of the best experts on this subject based on the ideXlab platform.
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fast and backward stable computation of roots of polynomials part ii backward error analysis Companion Matrix and Companion pencil
SIAM Journal on Matrix Analysis and Applications, 2018Co-Authors: Jared L Aurentz, Raf Vandebril, Thomas Mach, Leonardo Robol, David S WatkinsAbstract:This work is a continuation of work by [J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, J. Matrix Anal. Appl., 36 (2015), pp. 942--973]. In that paper we introduced a Companion QR algorithm that finds the roots of a polynomial by computing the eigenvalues of the Companion Matrix in $O(n^{2})$ time using $O(n)$ memory. We proved that the method is backward stable. Here we introduce, as an alternative, a Companion QZ algorithm that solves a generalized eigenvalue problem for a Companion pencil. More importantly, we provide an improved backward error analysis that takes advantage of the special structure of the problem. The improvement is also due, in part, to an improvement in the accuracy (in both theory and practice) of the turnover operation, which is the key component of our algorithms. We prove that for the Companion QR algorithm, the backward error on the polynomial coefficients varies linearly with the norm of the polynomial's vector of coefficients. Thus, the Companion QR algorithm has a sm...
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fast and backward stable computation of roots of polynomials part ii backward error analysis Companion Matrix and Companion pencil
arXiv: Numerical Analysis, 2016Co-Authors: Jared L Aurentz, Raf Vandebril, Thomas Mach, Leonardo Robol, David S WatkinsAbstract:This work is a continuation of "Fast and backward stable computation of roots of polynomials" by J.L. Aurentz, T. Mach, R. Vandebril, and D.S. Watkins, SIAM Journal on Matrix Analysis and Applications, 36(3): 942--973, 2015. In that paper we introduced a Companion QR algorithm that finds the roots of a polynomial by computing the eigenvalues of the Companion Matrix in $O(n^{2})$ time using $O(n)$ memory. We proved that the method is backward stable. Here we introduce, as an alternative, a Companion QZ algorithm that solves a generalized eigenvalue problem for a Companion pencil. More importantly, we provide an improved backward error analysis that takes advantage of the special structure of the problem. The improvement is also due, in part, to an improvement in the accuracy (in both theory and practice) of the turnover operation, which is the key component of our algorithms. We prove that for the Companion QR algorithm, the backward error on the polynomial coefficients varies linearly with the norm of the polynomial's vector of coefficients. Thus the Companion QR algorithm has a smaller backward error than the unstructured QR algorithm (used by MATLAB's \texttt{roots} command, for example), for which the backward error on the polynomial coefficients grows quadratically with the norm of the coefficient vector. The Companion QZ algorithm has the same favorable backward error as Companion QR, provided that the polynomial coefficients are properly scaled.
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fast computation of the zeros of a polynomial via factorization of the Companion Matrix
SIAM Journal on Scientific Computing, 2013Co-Authors: Jared L Aurentz, Raf Vandebril, David S WatkinsAbstract:A new fast algorithm for computing the zeros of a polynomial in $O(n^{2})$ time using $O(n)$ memory is developed. The eigenvalues of the Frobenius Companion Matrix are computed by applying a nonunitary analogue of Francis's implicitly shifted $QR$ algorithm to a factored form of the Matrix. The algorithm achieves high speed and low memory use by preserving the factored form. It also provides a residual and an error estimate for each root. Numerical tests confirm the high speed of the algorithm.
Jared L Aurentz - One of the best experts on this subject based on the ideXlab platform.
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fast and backward stable computation of roots of polynomials part ii backward error analysis Companion Matrix and Companion pencil
SIAM Journal on Matrix Analysis and Applications, 2018Co-Authors: Jared L Aurentz, Raf Vandebril, Thomas Mach, Leonardo Robol, David S WatkinsAbstract:This work is a continuation of work by [J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, J. Matrix Anal. Appl., 36 (2015), pp. 942--973]. In that paper we introduced a Companion QR algorithm that finds the roots of a polynomial by computing the eigenvalues of the Companion Matrix in $O(n^{2})$ time using $O(n)$ memory. We proved that the method is backward stable. Here we introduce, as an alternative, a Companion QZ algorithm that solves a generalized eigenvalue problem for a Companion pencil. More importantly, we provide an improved backward error analysis that takes advantage of the special structure of the problem. The improvement is also due, in part, to an improvement in the accuracy (in both theory and practice) of the turnover operation, which is the key component of our algorithms. We prove that for the Companion QR algorithm, the backward error on the polynomial coefficients varies linearly with the norm of the polynomial's vector of coefficients. Thus, the Companion QR algorithm has a sm...
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fast and backward stable computation of roots of polynomials part ii backward error analysis Companion Matrix and Companion pencil
arXiv: Numerical Analysis, 2016Co-Authors: Jared L Aurentz, Raf Vandebril, Thomas Mach, Leonardo Robol, David S WatkinsAbstract:This work is a continuation of "Fast and backward stable computation of roots of polynomials" by J.L. Aurentz, T. Mach, R. Vandebril, and D.S. Watkins, SIAM Journal on Matrix Analysis and Applications, 36(3): 942--973, 2015. In that paper we introduced a Companion QR algorithm that finds the roots of a polynomial by computing the eigenvalues of the Companion Matrix in $O(n^{2})$ time using $O(n)$ memory. We proved that the method is backward stable. Here we introduce, as an alternative, a Companion QZ algorithm that solves a generalized eigenvalue problem for a Companion pencil. More importantly, we provide an improved backward error analysis that takes advantage of the special structure of the problem. The improvement is also due, in part, to an improvement in the accuracy (in both theory and practice) of the turnover operation, which is the key component of our algorithms. We prove that for the Companion QR algorithm, the backward error on the polynomial coefficients varies linearly with the norm of the polynomial's vector of coefficients. Thus the Companion QR algorithm has a smaller backward error than the unstructured QR algorithm (used by MATLAB's \texttt{roots} command, for example), for which the backward error on the polynomial coefficients grows quadratically with the norm of the coefficient vector. The Companion QZ algorithm has the same favorable backward error as Companion QR, provided that the polynomial coefficients are properly scaled.
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fast computation of the zeros of a polynomial via factorization of the Companion Matrix
SIAM Journal on Scientific Computing, 2013Co-Authors: Jared L Aurentz, Raf Vandebril, David S WatkinsAbstract:A new fast algorithm for computing the zeros of a polynomial in $O(n^{2})$ time using $O(n)$ memory is developed. The eigenvalues of the Frobenius Companion Matrix are computed by applying a nonunitary analogue of Francis's implicitly shifted $QR$ algorithm to a factored form of the Matrix. The algorithm achieves high speed and low memory use by preserving the factored form. It also provides a residual and an error estimate for each root. Numerical tests confirm the high speed of the algorithm.
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a factorization of the inverse of the shifted Companion Matrix
Electronic Journal of Linear Algebra, 2013Co-Authors: Jared L AurentzAbstract:A common method for computing the zeros of an n-th degree polynomial is to compute the eigenvalues of its Companion Matrix C. A method for factoring the shifted Companion Matrix CI is presented. The factorization is a product of 2n 1 essentially 2 × 2 matrices and requires O(n) storage. Once the factorization is computed it immediately yields factorizations of both (CI) 1 and (CI) � . The cost of multiplying a vector by (CI) 1 is shown to be O(n), and therefore, any shift and invert eigenvalue method can be implemented efficiently. This factorization is generalized to arbitrary forms of the Companion Matrix.
Rosendo Monroyloperena - One of the best experts on this subject based on the ideXlab platform.
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roots of the underwood s equations in short cut distillation from a Companion Matrix eigenvalues
Chemical Engineering Science, 2012Co-Authors: Rosendo Monroyloperena, Miguel VacahernAbstract:Abstract The design of distillation columns requires the calculation of the minimum reflux. For ideal mixtures, the well-known Underwood’s equations, which can also be applied to complex columns (e.g., several feeds and side products and side-stream strippers and enrichers) are used to calculate it. When distributed components other than the light and the heavy key are considered in the separation, the knowledge of all the roots of the Underwood’s feed equation is essential. However, the discontinuous form of the Underwood´s feed equation; makes the search of all roots a hard task. Using the fact that the Underwood’s equation can be transform into a polynomial form, in this work a Companion Matrix of the polynomial is presented, permitting the solution of the equation as a Generalized Eigenvalue Problem, arriving to a reliable and efficient method for the calculation of all roots of the Underwood’s equations. One main feature of the proposed approach is that not initial guesses are required to find all the roots of the Underwood’s equation. Open, robust and reliable software such as EISPACK can be used to routinely calculate the minimum reflux ratio by Underwood’s equations.