{"id":260,"date":"2018-07-12T06:42:29","date_gmt":"2018-07-12T06:42:29","guid":{"rendered":"http:\/\/itp15.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=260"},"modified":"2019-05-15T08:56:24","modified_gmt":"2019-05-15T08:56:24","slug":"function-approximation","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/chapter\/function-approximation\/","title":{"rendered":"Function Approximation"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n \r\n

1.\u00a0\u00a0 Introduction:<\/strong><\/p>\r\n \r\n

An evaluation of an arbitrary function f on a digital computer involves approximating this function by a polynomial function P. This is because computational capabilities of digital computers are restricted to arithmetic operations and evaluation of a polynomial function involves only basic arithmetic operations.<\/p>\r\n \r\n

Interpolation and curve fitting techniques can be used for function approximation. However, both these techniques use finite number of data points obtained either by evaluating a function to be approximated at finite number of arguments, or obtained experimentally. Such approximations are completely dependent on these finite data values and do not use any other properties of the function to be approximated. Also, the interpolating polynomial is constrained to pass through the given data points where as least square polynomial minimizes the total square error. And hence, the approximating polynomial obtained by Interpolation or least square curve fitting techniques may not have much control over the error |f(x) \u2013 P(x)| at some points in the domain of approximation. Thus, though these techniques are efficient for estimating a functional relationship from the available finite data points, they are not adequate for function approximation.<\/p>\r\n \r\n

The \u00a0main \u00a0focus \u00a0in \u00a0function \u00a0approximation \u00a0is \u00a0to \u00a0find \u00a0a \u00a0polynomial \u00a0P \u00a0that \u00a0minimizes {Maximum |f(x) \u2013 P(x)|: x in the domain of approximation}. For this purpose, two techniques namely, Taylor\u2019s \/ Maclaurin\u2019s series approximation and approximation by Chebyshev polynomials, are discussed in this module.<\/p>\r\n \r\n

2.\u00a0\u00a0 Taylor\u2019s Polynomial Approximation:<\/strong><\/p>\r\n \r\n

This approximation is applicable to the functions which are sufficiently smooth and based on well-known Taylor\u2019s theorem as stated below.<\/p>\r\n \r\n

Taylor\u2019s theorem: <\/strong>Suppose that function f has (n + 1) continuous derivatives on the interval [a, b] and suppose that x0<\/sub> \u03f5 [a, b]. Then for every x \u03f5 [a, b] there exists a number c(x) is in between x0 and x, such that f(x) = Pn<\/sub>(x) + Rn<\/sub>(x) where<\/p>\r\n \r\n\r\n\"\" \"\" \"\" \"\" \"\" \"\" \"\" \"\" \"\"\r\n\r\nAdditional Reading Material<\/strong>\r\n\r\n \r\n

The following are some of the references on Interpolation. Detailed discussion on Newton\u2019s divided differences and Newton\u2019s divided differences interpolation is given in reference 3 and 4. Computational aspects are discussed in reference 1. And MATLAB programs can be found in reference 5.<\/p>\r\n\r\n\r\n\r\n
you can view video on Function Approximation<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

References<\/strong>:<\/p>\r\n\r\n

    \r\n \t
  1. Steven C Chapra & Raymond Canale: N. Methods for Engineers, 5th Ed. TMH, 2009<\/li>\r\n \t
  2. Conte Samuel D. & Carl de Boor: Elementary N A \u2013 An Algorithmic approach, 3rd Ed., TMH, 2005.<\/li>\r\n \t
  3. Alfio Quarteroni, Recardo Sacco, Fausto Saleri: N. Mathematics (Text in Applied Math.) 2nd Ed. SV, 2007<\/li>\r\n \t
  4. Richard L. Burden & J. Douglas Faires: N A \u2013Theory & Applications, Cengage Learning, 2005.<\/li>\r\n \t
  5. Won Y. Yang, Wenwu Cao, Tae-Sang Chung, & John Morris: Applied NM Using Matlab, WSE, 2005<\/li>\r\n \t
  6. Kendall Atkinson & Weimin Han: Elimentary Numerical Analysis, Third edition, Wiley Dreamtech India(P) Ltd., 2004.<\/li>\r\n \t
  7. Matheus Grasselli & Dimitry Pelinovsky: Numerical Mathematics, Narosa Publishing House, 2009.<\/li>\r\n \t
  8. s. s. ssatry: Introductory Methods of Numerical Analysis, PHI, 1989.<\/li>\r\n \t
  9. Polynomial interpolation, From Wikipedia, the free encyclopaedia<\/li>\r\n<\/ol>","rendered":"
    \"epgp<\/a>
    \n<\/span><\/div>\n

     <\/p>\n

    1.\u00a0\u00a0 Introduction:<\/strong><\/p>\n

     <\/p>\n

    An evaluation of an arbitrary function f on a digital computer involves approximating this function by a polynomial function P. This is because computational capabilities of digital computers are restricted to arithmetic operations and evaluation of a polynomial function involves only basic arithmetic operations.<\/p>\n

     <\/p>\n

    Interpolation and curve fitting techniques can be used for function approximation. However, both these techniques use finite number of data points obtained either by evaluating a function to be approximated at finite number of arguments, or obtained experimentally. Such approximations are completely dependent on these finite data values and do not use any other properties of the function to be approximated. Also, the interpolating polynomial is constrained to pass through the given data points where as least square polynomial minimizes the total square error. And hence, the approximating polynomial obtained by Interpolation or least square curve fitting techniques may not have much control over the error |f(x) \u2013 P(x)| at some points in the domain of approximation. Thus, though these techniques are efficient for estimating a functional relationship from the available finite data points, they are not adequate for function approximation.<\/p>\n

     <\/p>\n

    The \u00a0main \u00a0focus \u00a0in \u00a0function \u00a0approximation \u00a0is \u00a0to \u00a0find \u00a0a \u00a0polynomial \u00a0P \u00a0that \u00a0minimizes {Maximum |f(x) \u2013 P(x)|: x in the domain of approximation}. For this purpose, two techniques namely, Taylor\u2019s \/ Maclaurin\u2019s series approximation and approximation by Chebyshev polynomials, are discussed in this module.<\/p>\n

     <\/p>\n

    2.\u00a0\u00a0 Taylor\u2019s Polynomial Approximation:<\/strong><\/p>\n

     <\/p>\n

    This approximation is applicable to the functions which are sufficiently smooth and based on well-known Taylor\u2019s theorem as stated below.<\/p>\n

     <\/p>\n

    Taylor\u2019s theorem: <\/strong>Suppose that function f has (n + 1) continuous derivatives on the interval [a, b] and suppose that x0<\/sub> \u03f5 [a, b]. Then for every x \u03f5 [a, b] there exists a number c(x) is in between x0 and x, such that f(x) = Pn<\/sub>(x) + Rn<\/sub>(x) where<\/p>\n

     <\/p>\n

    \"\" \"\" \"\" \"\" \"\" \"\" \"\" \"\" \"\"<\/p>\n

    Additional Reading Material<\/strong><\/p>\n

     <\/p>\n

    The following are some of the references on Interpolation. Detailed discussion on Newton\u2019s divided differences and Newton\u2019s divided differences interpolation is given in reference 3 and 4. Computational aspects are discussed in reference 1. And MATLAB programs can be found in reference 5.<\/p>\n\n\n\n
    you can view video on Function Approximation<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    References<\/strong>:<\/p>\n

      \n
    1. Steven C Chapra & Raymond Canale: N. Methods for Engineers, 5th Ed. TMH, 2009<\/li>\n
    2. Conte Samuel D. & Carl de Boor: Elementary N A \u2013 An Algorithmic approach, 3rd Ed., TMH, 2005.<\/li>\n
    3. Alfio Quarteroni, Recardo Sacco, Fausto Saleri: N. Mathematics (Text in Applied Math.) 2nd Ed. SV, 2007<\/li>\n
    4. Richard L. Burden & J. Douglas Faires: N A \u2013Theory & Applications, Cengage Learning, 2005.<\/li>\n
    5. Won Y. Yang, Wenwu Cao, Tae-Sang Chung, & John Morris: Applied NM Using Matlab, WSE, 2005<\/li>\n
    6. Kendall Atkinson & Weimin Han: Elimentary Numerical Analysis, Third edition, Wiley Dreamtech India(P) Ltd., 2004.<\/li>\n
    7. Matheus Grasselli & Dimitry Pelinovsky: Numerical Mathematics, Narosa Publishing House, 2009.<\/li>\n
    8. s. s. ssatry: Introductory Methods of Numerical Analysis, PHI, 1989.<\/li>\n
    9. Polynomial interpolation, From Wikipedia, the free encyclopaedia<\/li>\n<\/ol>\n","protected":false},"author":4,"menu_order":20,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["dr-v-d-pathak"],"pb_section_license":""},"chapter-type":[],"contributor":[59],"license":[],"class_list":["post-260","chapter","type-chapter","status-publish","hentry","contributor-dr-v-d-pathak"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/pressbooks\/v2\/chapters\/260","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/wp\/v2\/users\/4"}],"version-history":[{"count":4,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/pressbooks\/v2\/chapters\/260\/revisions"}],"predecessor-version":[{"id":602,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/pressbooks\/v2\/chapters\/260\/revisions\/602"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/pressbooks\/v2\/chapters\/260\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/wp\/v2\/media?parent=260"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/pressbooks\/v2\/chapter-type?post=260"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/wp\/v2\/contributor?post=260"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp15\/wp-json\/wp\/v2\/license?post=260"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}