메뉴 건너뛰기




Volumn 55, Issue 2, 2013, Pages 375-396

Finding the zeros of a univariate equation: Proxy rootfinders, chebyshev interpolation, and the companion matrix

Author keywords

Chebyshev polynomials; Rootfinding; Transcendental equations

Indexed keywords

ALGEBRAIC GEOMETRY; CHEBYSHEV INTERPOLATION; CHEBYSHEV POLYNOMIALS; COMPANION MATRIX; INFINITE INTERVAL; ROOTFINDING; TRANSCENDENTAL EQUATIONS; TRIVIAL FUNCTIONS;

EID: 84881296663     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/110838297     Document Type: Article
Times cited : (41)

References (56)
  • 1
    • 0016598313 scopus 로고
    • Companion matrix analog for orthogonal polynomials
    • S. Barnett, Companion matrix analog for orthogonal polynomials, Linear Algebra Appl., 12 (1975), pp. 197-208.
    • (1975) Linear Algebra Appl. , vol.12 , pp. 197-208
    • Barnett, S.1
  • 2
    • 6344220010 scopus 로고    scopus 로고
    • An extension of MATLAB to continuous functions and operators
    • Z. Battles and L. N. Trefethen, An extension of MATLAB to continuous functions and operators, SIAM J. Sci. Comput., 25 (2004), pp. 1743-1770.
    • (2004) SIAM J. Sci. Comput. , vol.25 , pp. 1743-1770
    • Battles, Z.1    Trefethen, L.N.2
  • 3
    • 0002042339 scopus 로고
    • Quelques remarques sur linterpolation
    • S. N. Bernstein, Quelques remarques sur linterpolation, J. Math. Ann., 79 (1918), pp. 1-12.
    • (1918) J. Math. Ann. , vol.79 , pp. 1-12
    • Bernstein, S.N.1
  • 4
    • 0001234290 scopus 로고
    • Complex coordinate methods for hydrodynamic instabilities and Sturm-Liouville problems with an interior singularity
    • J. P. Boyd, Complex coordinate methods for hydrodynamic instabilities and Sturm-Liouville problems with an interior singularity, J. Comput. Phys., 57 (1985), pp. 454-471.
    • (1985) J. Comput. Phys. , vol.57 , pp. 454-471
    • Boyd, J.P.1
  • 5
    • 0001098372 scopus 로고
    • Polynomial series versus sinc expansions for functions with corner or endpoint singularities
    • J. P. Boyd, Polynomial series versus sinc expansions for functions with corner or endpoint singularities, J. Comput. Phys., 64 (1986), pp. 266-269.
    • (1986) J. Comput. Phys. , vol.64 , pp. 266-269
    • Boyd, J.P.1
  • 6
    • 0002717130 scopus 로고
    • Orthogonal rational functions on a semi-infinite interval
    • J. P. Boyd, Orthogonal rational functions on a semi-infinite interval, J. Comput. Phys., 70 (1987), pp. 63-88.
    • (1987) J. Comput. Phys. , vol.70 , pp. 63-88
    • Boyd, J.P.1
  • 7
    • 0001718791 scopus 로고
    • Spectral methods using rational basis functions on an infinite interval
    • J. P. Boyd, Spectral methods using rational basis functions on an infinite interval, J. Comput. Phys., 69 (1987), pp. 112-142.
    • (1987) J. Comput. Phys. , vol.69 , pp. 112-142
    • Boyd, J.P.1
  • 8
    • 0345708608 scopus 로고
    • A Chebyshev polynomial interval-searching method ("Lanczos economization") for solving a nonlinear equation with application to the nonlinear eigenvalue problem
    • J. P. Boyd, A Chebyshev polynomial interval-searching method ("Lanczos economization") for solving a nonlinear equation with application to the nonlinear eigenvalue problem, J. Comput. Phys., 118 (1995), pp. 1-8.
    • (1995) J. Comput. Phys. , vol.118 , pp. 1-8
    • Boyd, J.P.1
  • 10
    • 0344033726 scopus 로고    scopus 로고
    • Computing zeros on a real interval through Chebyshev expansion and polynomial rootfinding
    • J. P. Boyd, Computing zeros on a real interval through Chebyshev expansion and polynomial rootfinding, SIAM J. Numer. Anal., 40 (2002), pp. 1666-1682.
    • (2002) SIAM J. Numer. Anal. , vol.40 , pp. 1666-1682
    • Boyd, J.P.1
  • 11
    • 33744519285 scopus 로고    scopus 로고
    • Computing real roots of a polynomial in Chebyshev series form through subdivision
    • J. P. Boyd, Computing real roots of a polynomial in Chebyshev series form through subdivision, Appl. Numer. Math., 56 (2006), pp. 1077-1091.
    • (2006) Appl. Numer. Math. , vol.56 , pp. 1077-1091
    • Boyd, J.P.1
  • 12
    • 33644594466 scopus 로고    scopus 로고
    • Computing real roots of a polynomial in Chebyshev series form through subdivision with linear testing and cubic solves
    • J. P. Boyd, Computing real roots of a polynomial in Chebyshev series form through subdivision with linear testing and cubic solves, Appl. Math. Comput., 174 (2006), pp. 1642-1648.
    • (2006) Appl. Math. Comput. , vol.174 , pp. 1642-1648
    • Boyd, J.P.1
  • 13
    • 33846670221 scopus 로고    scopus 로고
    • Computing the zeros, maxima and inflection points of Chebyshev, Legendre and Fourier series: Solving transcendental equations by spectral interpolation and polynomial rootfinding
    • J. P. Boyd, Computing the zeros, maxima and inflection points of Chebyshev, Legendre and Fourier series: Solving transcendental equations by spectral interpolation and polynomial rootfinding, J. Engrg. Math., 56 (2006), pp. 203-219.
    • (2006) J. Engrg. Math. , vol.56 , pp. 203-219
    • Boyd, J.P.1
  • 14
    • 34250793396 scopus 로고    scopus 로고
    • Computing the zeros of a Fourier series or a Chebyshev series or general orthogonal polynomial series with parity symmetries
    • J. P. Boyd, Computing the zeros of a Fourier series or a Chebyshev series or general orthogonal polynomial series with parity symmetries, Comput. Math. Appl., 54 (2007), pp. 336- 349.
    • (2007) Comput. Math. Appl , vol.54 , pp. 336-349
    • Boyd, J.P.1
  • 15
    • 33750522035 scopus 로고    scopus 로고
    • Polynomialization of Kepler's equation through Chebyshev polynomial expansion of the sine: A non-iterative rootfinder
    • J. P. Boyd, Polynomialization of Kepler's equation through Chebyshev polynomial expansion of the sine: A non-iterative rootfinder, Appl. Numer. Math., 57 (2007), pp. 12-18.
    • (2007) Appl. Numer. Math. , vol.57 , pp. 12-18
    • Boyd, J.P.1
  • 16
    • 62149111517 scopus 로고    scopus 로고
    • Large-degree asymptotics and exponential asymptotics for Fourier coefficients and transforms, Chebyshev and other spectral coefficients
    • J. P. Boyd, Large-degree asymptotics and exponential asymptotics for Fourier coefficients and transforms, Chebyshev and other spectral coefficients, J. Engrg. Math., 63 (2009), pp. 355- 399.
    • (2009) J. Engrg. Math. , vol.63 , pp. 355-399
    • Boyd, J.P.1
  • 17
    • 84881295100 scopus 로고    scopus 로고
    • A comparison of companion matrix methods to find roots of a trigonometric polynomial
    • to appear
    • J. P. Boyd, A comparison of companion matrix methods to find roots of a trigonometric polynomial, J. Comput. Phys., to appear.
    • J. Comput. Phys
    • Boyd, J.P.1
  • 18
    • 84856953597 scopus 로고    scopus 로고
    • Numerical perturbative and Chebyshev inversion of the incomplete elliptic integral of the second kin
    • J. P. Boyd, Numerical, perturbative and Chebyshev inversion of the incomplete elliptic integral of the second kin, Appl. Math. Comput., 218 (2012), pp. 7005-7013.
    • (2012) Appl. Math. Comput. , vol.218 , pp. 7005-7013
    • Boyd, J.P.1
  • 19
    • 34247520925 scopus 로고    scopus 로고
    • Numerical experiments on the accuracy of the Chebyshev- Frobenius companion matrix method for finding the zeros of a truncated series of Chebyshev polynomials
    • J. P. Boyd and D. H. Gally, Numerical experiments on the accuracy of the Chebyshev- Frobenius companion matrix method for finding the zeros of a truncated series of Chebyshev polynomials, J. Comput. Appl. Math., 205 (2007), pp. 281-295.
    • (2007) J. Comput. Appl. Math. , vol.205 , pp. 281-295
    • Boyd, J.P.1    Gally, D.H.2
  • 20
    • 78651277394 scopus 로고    scopus 로고
    • Exponentially-convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions, Part II: Multi-interval schemes
    • J. P. Boyd and J. R. Ong, Exponentially-convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions, Part II: Multi-interval schemes, Appl. Numer. Math., 61 (2011), pp. 460-472.
    • (2011) Appl. Numer. Math. , vol.61 , pp. 460-472
    • Boyd, J.P.1    Ong, J.R.2
  • 21
    • 84867328299 scopus 로고    scopus 로고
    • Computing the real roots of a Fourier series-plus-linearpolynomial: A Chebyshev companion matrix approach
    • J. P. Boyd and B. A. Sadiq, Computing the real roots of a Fourier series-plus-linearpolynomial: A Chebyshev companion matrix approach, Appl. Math. 219 (2012), pp. 819- 826.
    • (2012) Appl. Math , vol.219 , pp. 819-826
    • Boyd, J.P.1    Sadiq, B.A.2
  • 23
    • 0000624238 scopus 로고
    • A method for numerical integration on an automatic computer
    • C. W. Clenshaw and A. R. Curtis, A method for numerical integration on an automatic computer, Numer. Math., 2 (1960), pp. 197-205.
    • (1960) Numer. Math. , vol.2 , pp. 197-205
    • Clenshaw, C.W.1    Curtis, A.R.2
  • 24
    • 84968470212 scopus 로고
    • An algorithm for the machine calculation of complex Fourier series
    • J. W. Cooley and J. W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp., 19 (1965), pp. 297-301.
    • (1965) Math. Comp. , vol.19 , pp. 297-301
    • Cooley, J.W.1    Tukey, J.W.2
  • 26
    • 84975687800 scopus 로고    scopus 로고
    • On a generalized companion matrix pencil for matrix polynomials expressed in the Lagrange basis
    • D. Wang and L.-H. Zhi, eds., Trends in Mathematics, Birkhäuser, Basel
    • R. M. Corless, On a generalized companion matrix pencil for matrix polynomials expressed in the Lagrange basis, in Symbolic-Numeric Computation, D. Wang and L.-H. Zhi, eds., Trends in Mathematics, Birkhäuser, Basel, 2007, pp. 1-15.
    • (2007) Symbolic-Numeric Computation , pp. 1-15
    • Corless, R.M.1
  • 28
    • 0001030604 scopus 로고
    • Faber polynomials and the Faber series
    • J. H. Curtiss, Faber polynomials and the Faber series, Amer. Math. Monthly, 78 (1971), pp. 577-596.
    • (1971) Amer. Math. Monthly , vol.78 , pp. 577-596
    • Curtiss, J.H.1
  • 31
    • 33750187718 scopus 로고    scopus 로고
    • Roots of polynomials expressed in terms of orthogonal polynomials
    • D. Day and L. Romero, Roots of polynomials expressed in terms of orthogonal polynomials, SIAM J. Numer. Anal., 43 (2005), pp. 1969-1987.
    • (2005) SIAM J. Numer. Anal. , vol.43 , pp. 1969-1987
    • Day, D.1    Romero, L.2
  • 32
    • 77953612515 scopus 로고    scopus 로고
    • Automatic spectral collocation for integral, integro-differential, and integrally reformulated differential equations
    • T. A. Driscoll, Automatic spectral collocation for integral, integro-differential, and integrally reformulated differential equations, J. Comput. Phys., 229 (2010), pp. 5980-5998.
    • (2010) J. Comput. Phys. , vol.229 , pp. 5980-5998
    • Driscoll, T.A.1
  • 33
    • 0030516403 scopus 로고    scopus 로고
    • On the optimal stability of the Bernstein basis
    • R. T. Farouki and T. N. T. Goodman, On the optimal stability of the Bernstein basis, Math. Comput., 65 (1996), pp. 1553-1566.
    • (1996) Math. Comput. , vol.65 , pp. 1553-1566
    • Farouki, R.T.1    Goodman, T.N.T.2
  • 35
    • 84968486951 scopus 로고
    • The condition of polynomials in power form
    • W. Gautschi, The condition of polynomials in power form, Math. Comput., 33 (1979), pp. 343- 352.
    • (1979) Math. Comput , vol.33 , pp. 343-352
    • Gautschi, W.1
  • 36
    • 79955463557 scopus 로고
    • Chebyshev nodes for interpolation on a class of ellipses
    • A. G. Law and B. N. Sahney, eds., Academic Press, New York
    • K. O. Geddes, Chebyshev nodes for interpolation on a class of ellipses, in Theory of Approximations with Applications, A. G. Law and B. N. Sahney, eds., Academic Press, New York, 1976, pp. 155-170.
    • (1976) Theory of Approximations with Applications , pp. 155-170
    • Geddes, K.O.1
  • 37
    • 0016485297 scopus 로고
    • Polynomial approximation by projections on the unit circle
    • K. O. Geddes and J. C. Mason, Polynomial approximation by projections on the unit circle, SIAM J. Numer. Anal., 12 (1975), pp. 111-120.
    • (1975) SIAM J. Numer. Anal. , vol.12 , pp. 111-120
    • Geddes, K.O.1    Mason, J.C.2
  • 39
    • 84881272973 scopus 로고
    • Translated from vestnik leningrad
    • Translated from Vestnik Leningrad Univ. Mat. Mekh. Astr., 7 (1973), pp. 18-24.
    • (1973) Univ. Mat. Mekh. Astr , vol.7 , pp. 18-24
  • 40
    • 0347344020 scopus 로고
    • The colleague matrix, a Chebyshev analogue of the companion matrix, Quart
    • I. J. Good, The colleague matrix, a Chebyshev analogue of the companion matrix, Quart. J. Math., 12 (1961), pp. 61-68.
    • (1961) J. Math. , vol.12 , pp. 61-68
    • Good, I.J.1
  • 42
    • 0347171230 scopus 로고
    • A variable order pseudospectral method for two-point boundary value problems
    • S. J. Jacobs, A variable order pseudospectral method for two-point boundary value problems, J. Comput. Phys., 88 (1990), pp. 169-182.
    • (1990) J. Comput. Phys. , vol.88 , pp. 169-182
    • Jacobs, S.J.1
  • 44
    • 0346003632 scopus 로고
    • An algebraic algorithm for calculating the roots of a general quintic equation from its coefficients
    • R. B. King and E. R. Canfield, An algebraic algorithm for calculating the roots of a general quintic equation from its coefficients, J. Math. Phys., 32 (1991), pp. 823-825.
    • (1991) J. Math. Phys. , vol.32 , pp. 823-825
    • King, R.B.1    Canfield, E.R.2
  • 45
    • 0038419092 scopus 로고
    • Icosahedral symmetry and the quintic equation
    • R. B. King and E. R. Canfield, Icosahedral symmetry and the quintic equation, Comput. Math. Appl., 24 (1992), pp. 13-28.
    • (1992) Comput. Math. Appl. , vol.24 , pp. 13-28
    • King, R.B.1    Canfield, E.R.2
  • 46
    • 85162699057 scopus 로고
    • Trigonometric interpolation of empirical and analytical functions
    • C. Lanczos, Trigonometric interpolation of empirical and analytical functions, J.Math. Phys., 17 (1938), pp. 123-199.
    • (1938) J.Math. Phys , vol.17 , pp. 123-199
    • Lanczos, C.1
  • 47
    • 84881273630 scopus 로고    scopus 로고
    • Reprinted in cornelius lanczos
    • North Carolina State University, Raleigh, NC, to 3-297
    • Reprinted in Cornelius Lanczos: Collected Papers with Commentaries, Vol. 3, W. R. Davis et al., eds., North Carolina State University, Raleigh, NC, 1997, pp. 3-221 to 3-297.
    • (1997) Collected Papers with Commentaries , vol.3 , pp. 3-221
    • Davis, W.R.1
  • 50
    • 0016047525 scopus 로고
    • On quadratic approximation
    • R. E. Shafer, On quadratic approximation, SIAM J. Numer. Anal., 11 (1974), pp. 447-460.
    • (1974) SIAM J. Numer. Anal. , vol.11 , pp. 447-460
    • Shafer, R.E.1
  • 51
    • 84979317428 scopus 로고
    • Die Lage der Nullstellen eines Polynoms III
    • W. Specht, Die Lage der Nullstellen eines Polynoms III, Math. Nach., 16 (1957), pp. 369-389.
    • (1957) Math. Nach. , vol.16 , pp. 369-389
    • Specht, W.1
  • 52
    • 84979312784 scopus 로고
    • Die Lage der Nullstellen eines Polynoms IV
    • W. Specht, Die Lage der Nullstellen eines Polynoms IV, Math. Nach., 21 (1960), pp. 201-222.
    • (1960) Math. Nach. , vol.21 , pp. 201-222
    • Specht, W.1
  • 55
    • 84856954407 scopus 로고    scopus 로고
    • The Chebfun Development Team
    • L. N. Trefethen et al., Chebfun Version 4.0, The Chebfun Development Team, 2011, http://www.maths.ox.ac.uk/chebfun/.
    • (2011) Chebfun Version 4.0
    • Trefethen, L.N.1
  • 56
    • 0030095821 scopus 로고    scopus 로고
    • Symbolic computation on complex polynomial solution of differential equations
    • J. Zhang, Symbolic computation on complex polynomial solution of differential equations, J. Symbolic Comput., 22 (1996), pp. 345-354.
    • (1996) J. Symbolic Comput. , vol.22 , pp. 345-354
    • Zhang, J.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.