Global polynomial interpolants suffer from the Runge Phenomenon if the data sites (nodes) are not chosen correctly.

Transcription

1 Piecewise polynomial interpolation Global polynomial interpolants suffer from the Runge Phenomenon if the data sites (nodes) are not chosen correctly. In many applications, one does not have the freedom to choose the data sites. In these situations it may be better to interpolate the data using a piecewise polynomial interpolant. Additionally, if one wishes to impose further constraints on the interpolant, such as maximum/minimum preserving or that it have a certain shape, then piecewise polynomials are the right tool.

2 Piecewise linear interpolation The simplest piecewise polynomial interpolant is piecewise linear, or the kindergartener s interpolant

3 Piecewise linear interpolation <latexit sha_base64="idwfly9rfqjeelsswhnxhmdeya=">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</latexit> <latexit sha_base64="idwfly9rfqjeelsswhnxhmdeya=">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</latexit> <latexit sha_base64="idwfly9rfqjeelsswhnxhmdeya=">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</latexit> <latexit sha_base64="idwfly9rfqjeelsswhnxhmdeya=">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</latexit> <latexit sha_base64="idwfly9rfqjeelsswhnxhmdeya=">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</latexit> <latexit sha_base64="8mhyvikpjisjbny7ytsollltua=">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</latexit> <latexit sha_base64="8mhyvikpjisjbny7ytsollltua=">aaacqxicbvffb9mwehccjfh+dfccxitzhdrpucutgvcanikxhngyilkkpooc59jaczzivoxevr4bx5cvwsfabsorzxk+xfu/p5fpduuhgmgt+ef+vnb7+/cg9x88fpr4epdkuylrzwhksnqwcimskfgigilzconregkxct5xzv/cqnaifj9w7acrcgwsmscmshepjrc5ypmyp6gwzxto9pvksudkizbxtl7rjzbppjshvt+m4vdm4p4ttlaptsdyabcxqkawjzdfxhxk6+sjnwtt/wvzletdfbjngyqvhdakt/p4wnulplxbsjkkhkzd4mkf5zpffxcn4hqaxxjovvcehfcjalu5gtoy9djkvzqdrsdfrqxwwfcareiyc4yrc5nbb//hzwvmisufwnopiuvzliivd74cmqgnhtraubbur5svmjmlaygkqgrrxelyqgvwputfhnk4coa5ntjde9knnhlhtzwfrk8n4sfnkzojvpjdwnz8khgzoqvcvn5bm5jpcyr/vmffco/rf+v/9mf9jm+p7fctcs8/heswdmd</latexit> <latexit sha_base64="8mhyvikpjisjbny7ytsollltua=">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</latexit> <latexit sha_base64="8mhyvikpjisjbny7ytsollltua=">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</latexit> The simplest piecewise polynomial interpolant is piecewise linear, or the kindergartener s interpolant For each interval [x k,x k+ ], k =,...,n, fit a linear polynomial to the data L k (x) =f k + f k+ x k+ f k x k {z } k (x x k ) {z } s = f k + k s

4 Piecewise linear interpolation The simplest piecewise polynomial interpolant is piecewise linear, or the kindergartener s interpolant Issues with this approach: Creates corners in the data at each data site. Derivative of the curve will not be continuous If data values come from a function then the accuracy in approximating the function will be quite low.

5 Piecewise Hermite interpolation Suppose that in addition to function values at the data sites we also know what the derivative of the interpolant should be. Red lines indicate the slope (or derivative) the interpolant should have at the points For each interval [x k,x k+ ], k =,...,n, we want a polynomial H k (x) that interpolates the given data and has a derivative that matches values given: <latexit sha_base64="plksqzgllspfaity4uouchlw=">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</latexit> () H k (x k )=f k () H k (x k+ )=f k+ () H k (x k)=f k (4) H k (x k+) =f k+

6 <latexit sha_base64="wxmwopw6ndp4p6gbdoxphahsee=">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</latexit> <latexit sha_base64="wxmwopw6ndp4p6gbdoxphahsee=">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</latexit> <latexit sha_base64="wxmwopw6ndp4p6gbdoxphahsee=">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</latexit> <latexit sha_base64="wxmwopw6ndp4p6gbdoxphahsee=">aaadvnicbvjlb9naelyt+ic8ujhygdegpbsnkibequkqxkunvksgvopdtf6p7cxrxcu7thnz/i8gq5w7q+bcrpkh8xh/cmt/p4vh4mhbh9/oxban5bw9/yvn968pdr4yftradfjc5yjioupc7pfgzqcoujk6zesyxhlvost/ky5/nwfuhfyndphhjgwreqhgzfjouuueeeolfacycbila5gyoygq6gpxkhazeybxcoilvokjcdg9jbni99wwafmzylpvpbjogqaoxyrlbxfxz54qu+u4fqgizh9lwyz4jzgc+nyd58wia7grsed/yfiyekpkukxpvty4ouxno/ab/gnsne8tfefra9gf/s8dkhtcuetgngl5qikrmp/et7t8jyuigbwetgwpaqydktzgw6vm9ug+atwcwpyiznbwo8jklvt7u9/plg7tgsalbzsqosetl9c8selrlpkx4e/s5os5vzwivxlkwxmjccswjfbxvikipcwuvkristq8/atl9pqnkqxglfkfmdrubg7n6ud/cupchgetuqisskj4zaowkpxe9yobqotirvwqydyn5tnwmowmk4juysc9orxzupqszu5ckhp+abxf+rgrpd7wf9wlvjucfunua7lrdt4ptw+hkwknefoc6frdjyzqfz5bw7i4e79f7k/v+oi8bu5ty4pdbcznzgrtv8a+vsla==</latexit> <latexit sha_base64="zoivgyuzuhhhg4rhuphfmptv4=">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</latexit> <latexit sha_base64="zoivgyuzuhhhg4rhuphfmptv4=">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</latexit> <latexit sha_base64="zoivgyuzuhhhg4rhuphfmptv4=">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</latexit> <latexit sha_base64="zoivgyuzuhhhg4rhuphfmptv4=">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</latexit> Piecewise cubic Hermite interpolation For each interval [x k,x k+ ], k =,...,n, we want a polynomial H k (x) that interpolates the given data and has a derivative that matches values given: <latexit sha_base64="plksqzgllspfaity4uouchlw=">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</latexit> () H k (x k )=f k () H k (x k+ )=f k+ () H k (x k)=f k (4) H k (x k+) =f k+ This gives four conditions, so we look for a polynomial of degree three (cubic) in each interval [x k,x k+ ]: H k (x) =b k (x x k ) + c k (x x k ) + d k (x x k )+e k Now we just need to determine b k, c k, d k, e k to match the conditions. After some intensive algebra, we can write the solution for the coe cients as follows: e k = f k d k = f k c k = k d k d k+ h k b k = d k k + d k+ h k where h k = x k+ x k and k =(f k+ f k )/h k. <latexit sha_base64="5phxebpuzjcv4uym9klwggc=">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</latexit>

7 Piecewise Hermite interpolation Suppose that in addition to function values at the data sites we also know what the derivative of the interpolant should be. Red lines indicate the slope (or derivative) the interpolant should have at the points For each interval [x k,x k+ ], k =,...,n, we want a polynomial H k (x) that interpolates the given data and has a derivative that matches values given: <latexit sha_base64="plksqzgllspfaity4uouchlw=">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</latexit> () H k (x k )=f k () H k (x k+ )=f k+ () H k (x k)=f k (4) H k (x k+) =f k+