Difference between revisions of "Budan-Fourier theorem"
From Encyclopedia of Mathematics
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The number of roots of an algebraic equation | The number of roots of an algebraic equation | ||
| − | + | $$f(x)=0$$ | |
| + | comprised in an interval $(a,b), a<b$, is equal to or is smaller, by an even number, than $\tau=t_1-t_2$, where $t_1$ is the number of changes in sign in the series of derivatives of the polynomial $f(x)$ at the point $a$, i.e. in the series | ||
| − | + | $$f(a),f'(a),\dots,f^{(n)}(a),$$ | |
| + | while $t_2$ is the number of changes in sign in this series at the point $b$. Each multiple root is counted according to its multiplicity. Established by F. Budan (1807) {{Cite|Bu}} and J. Fourier (1820). | ||
| − | + | ====Comments==== | |
| − | + | An application of the Budan–Fourier theorem in numerical analysis may be found in | |
| − | + | {{Cite|BoSc}}, where it is used in the interpolation by spline functions. | |
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====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|AlMaCh}}||valign="top"| Alexandroff, P.S., Markuschewitsch, A.I., Chintschin, A.J., "Encyclopaedia of elementary mathematics", '''2. Algebra''', Moscow-Leningrad (1951) pp. 331 (In Russian) {{MR|0080060}} {{ZBL|0365.00003}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Bu}}||valign="top"| Budan, F. D., " Nouvelle méthode pour la résolution des équations numériques", Paris: Courcier (1807) [http://books.google.com/books/about/Nouvelle_m%C3%A9thode_pour_la_r%C3%A9solution_de.html?id=VyMOAAAAQAAJ&redir_esc=y google books] | ||
| + | |- | ||
| + | |valign="top"|{{Ref|BoSc}}||valign="top"| C. de Boor, I.J. Schoenberg, "Cardinal interpolation and spline functions VIII. The Budan–Fourier theorem for splines and applications." K. Bohmer (ed.) G. Meinardus (ed.) W. Schempp (ed.), ''Spline functions'', ''Lect. notes in math.'', '''501''', Springer (1976) {{MR|0493059}} {{ZBL|0319.41010}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ho}}||valign="top"| A.S. Householder, "Unitary triangularization of a nonsymmetric matrix" ''J. Assoc. Comp. Mach.'', '''5''' (1958) pp. 339–342 {{MR|0111128}} {{ZBL|0121.33802}} | ||
| + | |- | ||
| + | |} | ||
Revision as of 12:17, 16 April 2012
2020 Mathematics Subject Classification: Primary: 12Y Secondary: 65T [MSN][ZBL]
The number of roots of an algebraic equation
$$f(x)=0$$ comprised in an interval $(a,b), a<b$, is equal to or is smaller, by an even number, than $\tau=t_1-t_2$, where $t_1$ is the number of changes in sign in the series of derivatives of the polynomial $f(x)$ at the point $a$, i.e. in the series
$$f(a),f'(a),\dots,f^{(n)}(a),$$ while $t_2$ is the number of changes in sign in this series at the point $b$. Each multiple root is counted according to its multiplicity. Established by F. Budan (1807) [Bu] and J. Fourier (1820).
Comments
An application of the Budan–Fourier theorem in numerical analysis may be found in [BoSc], where it is used in the interpolation by spline functions.
References
| [AlMaCh] | Alexandroff, P.S., Markuschewitsch, A.I., Chintschin, A.J., "Encyclopaedia of elementary mathematics", 2. Algebra, Moscow-Leningrad (1951) pp. 331 (In Russian) MR0080060 Zbl 0365.00003 |
| [Bu] | Budan, F. D., " Nouvelle méthode pour la résolution des équations numériques", Paris: Courcier (1807) google books |
| [BoSc] | C. de Boor, I.J. Schoenberg, "Cardinal interpolation and spline functions VIII. The Budan–Fourier theorem for splines and applications." K. Bohmer (ed.) G. Meinardus (ed.) W. Schempp (ed.), Spline functions, Lect. notes in math., 501, Springer (1976) MR0493059 Zbl 0319.41010 |
| [Ho] | A.S. Householder, "Unitary triangularization of a nonsymmetric matrix" J. Assoc. Comp. Mach., 5 (1958) pp. 339–342 MR0111128 Zbl 0121.33802 |
How to Cite This Entry:
Budan-Fourier theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Budan-Fourier_theorem&oldid=22213
Budan-Fourier theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Budan-Fourier_theorem&oldid=22213
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article