A practical error formula for multivariate rational interpolation and approximation
We consider exact and approximate multivariate interpolation of a function f ( x 1 , . . . , x d ) by a rational function p n , m / q n , m ( x 1 , . . . , x d ) and develop an error formula for the difference f − p n , m / q n , m . The similarity with a well-known univariate formula for the...
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| Published in | Numerical algorithms Vol. 55; no. 2-3; pp. 233 - 243 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Boston
Springer US
01.11.2010
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1017-1398 1572-9265 |
| DOI | 10.1007/s11075-010-9380-2 |
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| Abstract | We consider exact and approximate multivariate interpolation of a function
f
(
x
1
, . . . ,
x
d
) by a rational function
p
n
,
m
/
q
n
,
m
(
x
1
, . . . ,
x
d
) and develop an error formula for the difference
f
−
p
n
,
m
/
q
n
,
m
. The similarity with a well-known univariate formula for the error in rational interpolation is striking. Exact interpolation is through point values for
f
and approximate interpolation is through intervals bounding
f
. The latter allows for some measurement error on the function values, which is controlled and limited by the nature of the interval data. To achieve this result we make use of an error formula obtained for multivariate polynomial interpolation, which we first present in a more general form. The practical usefulness of the error formula in multivariate rational interpolation is illustrated by means of a 4-dimensional example, which is only one of the several problems we tested it on. |
|---|---|
| AbstractList | We consider exact and approximate multivariate interpolation of a function f(x 1,...,x d ) by a rational function p n,m /q n,m (x 1,...,x d ) and develop an error formula for the difference f-p n,m /q n,m . The similarity with a well-known univariate formula for the error in rational interpolation is striking. Exact interpolation is through point values for f and approximate interpolation is through intervals bounding f. The latter allows for some measurement error on the function values, which is controlled and limited by the nature of the interval data. To achieve this result we make use of an error formula obtained for multivariate polynomial interpolation, which we first present in a more general form. The practical usefulness of the error formula in multivariate rational interpolation is illustrated by means of a 4-dimensional example, which is only one of the several problems we tested it on. We consider exact and approximate multivariate interpolation of a function f(x1 , . . . , xd) by a rational function pn,m/qn,m(x1 , . . . , xd) and develop an error formula for the difference f − pn,m/qn,m. The similarity with a well-known univariate formula for the error in rational interpolation is striking. Exact interpolation is through point values for f and approximate interpolation is through intervals bounding f. The latter allows for some measurement error on the function values, which is controlled and limited by the nature of the interval data. To achieve this result we make use of an error formula obtained for multivariate polynomial interpolation, which we first present in a more general form. The practical usefulness of the error formula in multivariate rational interpolation is illustrated by means of a 4-dimensional example, which is only one of the several problems we tested it on. We consider exact and approximate multivariate interpolation of a function f ( x 1 , . . . , x d ) by a rational function p n , m / q n , m ( x 1 , . . . , x d ) and develop an error formula for the difference f − p n , m / q n , m . The similarity with a well-known univariate formula for the error in rational interpolation is striking. Exact interpolation is through point values for f and approximate interpolation is through intervals bounding f . The latter allows for some measurement error on the function values, which is controlled and limited by the nature of the interval data. To achieve this result we make use of an error formula obtained for multivariate polynomial interpolation, which we first present in a more general form. The practical usefulness of the error formula in multivariate rational interpolation is illustrated by means of a 4-dimensional example, which is only one of the several problems we tested it on. |
| Author | Yang, Xianglan Cuyt, Annie |
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| Cites_doi | 10.1017/CBO9780511530074 10.1016/j.ejor.2006.12.028 10.1007/BF02432000 10.1017/CBO9780511800948 10.1090/S0025-5718-1995-1297477-5 10.1007/978-3-540-24588-9_22 10.1109/TMTT.2006.873637 10.1023/A:1018981505752 10.1063/1.2990963 10.1007/s11075-007-9077-3 |
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| DOI | 10.1007/s11075-010-9380-2 |
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| Keywords | Multivariate interpolation Rational interpolation Interpolation error |
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| References | Higham, D.J.: An Introduction to Financial Option Valuation: Mathematics, Stochastics, and Computation. Cambridge University Press (2004) Salazar CelisOCuytAVerdonkBRational approximation of vertical segmentsNumer. Algorithms2007453753881151.4101110.1007/s11075-007-9077-32355995 SauerTXuYOn multivariate Lagrange interpolationMath. Comput.199564211114711700823.410021297477 Becuwe, S., Cuyt, A., Verdonk, B.: Multivariate rational interpolation of scattered data. In: Lirkov, I., Margenov, S., Waśniewski, J., Yalamov, P. (eds.) LNCS, vol. 2907, pp. 204–213 (2004) Salazar Celis, O., Cuyt, A., Van Deun, J.: Symbolic and interval rational interpolation: the problem of unattainable data. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) International Conference on Numerical Analysis and Applied Mathematics. AIP Conference Proceedings, vol. 1048, pp. 466–469 (2008) LiMApproximate inversion of the Black-Scholes formula using rational functionsEur. J. Oper. Res.200818527437591137.9145910.1016/j.ejor.2006.12.028 CuytALeninRBBecuweSVerdonkBAdaptive multivariate rational data fitting with applications in electromagneticsIEEE Trans. Microwave Theor. Tech.2006542265227410.1109/TMTT.2006.873637 Baker, G.A., Jr., Graves-Morris, P.: Padé Approximants, 2nd edn. Cambridge University Press (1996) GascaMSauerTPolynomial interpolation in several variablesAdv. Comput. Math.20001243774100943.4100110.1023/A:10189815057521768957 SauerTComputational aspects of multivariate polynomial interpolationAdv. Comput. Math.1995332192370831.6500610.1007/BF024320001325032 CuytAWuytackLNonlinear Methods in Numerical Analysis1987AmsterdamNorth-Holland0609.65001 T Sauer (9380_CR11) 1995; 64 A Cuyt (9380_CR3) 2006; 54 O Salazar Celis (9380_CR9) 2007; 45 9380_CR1 9380_CR2 M Li (9380_CR7) 2008; 185 T Sauer (9380_CR10) 1995; 3 9380_CR6 A Cuyt (9380_CR4) 1987 9380_CR8 M Gasca (9380_CR5) 2000; 12 |
| References_xml | – reference: Higham, D.J.: An Introduction to Financial Option Valuation: Mathematics, Stochastics, and Computation. Cambridge University Press (2004) – reference: Salazar Celis, O., Cuyt, A., Van Deun, J.: Symbolic and interval rational interpolation: the problem of unattainable data. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) International Conference on Numerical Analysis and Applied Mathematics. AIP Conference Proceedings, vol. 1048, pp. 466–469 (2008) – reference: SauerTXuYOn multivariate Lagrange interpolationMath. Comput.199564211114711700823.410021297477 – reference: Baker, G.A., Jr., Graves-Morris, P.: Padé Approximants, 2nd edn. Cambridge University Press (1996) – reference: SauerTComputational aspects of multivariate polynomial interpolationAdv. Comput. Math.1995332192370831.6500610.1007/BF024320001325032 – reference: CuytALeninRBBecuweSVerdonkBAdaptive multivariate rational data fitting with applications in electromagneticsIEEE Trans. Microwave Theor. Tech.2006542265227410.1109/TMTT.2006.873637 – reference: Becuwe, S., Cuyt, A., Verdonk, B.: Multivariate rational interpolation of scattered data. In: Lirkov, I., Margenov, S., Waśniewski, J., Yalamov, P. (eds.) LNCS, vol. 2907, pp. 204–213 (2004) – reference: LiMApproximate inversion of the Black-Scholes formula using rational functionsEur. J. Oper. Res.200818527437591137.9145910.1016/j.ejor.2006.12.028 – reference: CuytAWuytackLNonlinear Methods in Numerical Analysis1987AmsterdamNorth-Holland0609.65001 – reference: GascaMSauerTPolynomial interpolation in several variablesAdv. Comput. Math.20001243774100943.4100110.1023/A:10189815057521768957 – reference: Salazar CelisOCuytAVerdonkBRational approximation of vertical segmentsNumer. Algorithms2007453753881151.4101110.1007/s11075-007-9077-32355995 – ident: 9380_CR1 doi: 10.1017/CBO9780511530074 – volume: 185 start-page: 743 issue: 2 year: 2008 ident: 9380_CR7 publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2006.12.028 – volume: 3 start-page: 219 issue: 3 year: 1995 ident: 9380_CR10 publication-title: Adv. Comput. Math. doi: 10.1007/BF02432000 – ident: 9380_CR6 doi: 10.1017/CBO9780511800948 – volume: 64 start-page: 1147 issue: 211 year: 1995 ident: 9380_CR11 publication-title: Math. Comput. doi: 10.1090/S0025-5718-1995-1297477-5 – volume-title: Nonlinear Methods in Numerical Analysis year: 1987 ident: 9380_CR4 – ident: 9380_CR2 doi: 10.1007/978-3-540-24588-9_22 – volume: 54 start-page: 2265 year: 2006 ident: 9380_CR3 publication-title: IEEE Trans. Microwave Theor. Tech. doi: 10.1109/TMTT.2006.873637 – volume: 12 start-page: 377 issue: 4 year: 2000 ident: 9380_CR5 publication-title: Adv. Comput. Math. doi: 10.1023/A:1018981505752 – ident: 9380_CR8 doi: 10.1063/1.2990963 – volume: 45 start-page: 375 year: 2007 ident: 9380_CR9 publication-title: Numer. Algorithms doi: 10.1007/s11075-007-9077-3 |
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| Snippet | We consider exact and approximate multivariate interpolation of a function
f
(
x
1
, . . . ,
x
d
) by a rational function
p
n
,
m
/
q
n
,
m
(
x
1
, . . . , ... We consider exact and approximate multivariate interpolation of a function f(x1 , . . . , xd) by a rational function pn,m/qn,m(x1 , . . . , xd) and develop an... We consider exact and approximate multivariate interpolation of a function f(x 1,...,x d ) by a rational function p n,m /q n,m (x 1,...,x d ) and develop an... |
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| SubjectTerms | Algebra Algorithms Approximation Computer Science Error analysis Errors Interpolation Intervals Mathematical analysis Mathematical models Multivariate analysis Numeric Computing Numerical Analysis Original Paper Permissible error Polynomials Rational functions Theory of Computation |
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| Title | A practical error formula for multivariate rational interpolation and approximation |
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