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The Interpolation Problem

The Interpolation Problem. The general interpolation problem We have a set of abscissas and a function Target: to find a polynomial P ( x ) such that , .

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The Interpolation Problem

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  1. The Interpolation Problem • The general interpolation problem • We have a set of abscissas and a function • Target: to find a polynomial P(x) such that , . • Observation: the function is not “critical”, one can substitute it by the set of values , . )c) Gershon Elber, Technion

  2. Specific Interpolation Problems • We have a set of abscissas and a function • Target: to find a polynomial P(x) such that , • , . This is the classical Hermite interpolation. • Target: to find P(x) such that in each abscissa there is a certain degree of contact between f(x) and P(x). This is the general Hermite interpolation. • Observation: once again the function f(x) is not “critical”. )c) Gershon Elber, Technion

  3. B-spline Functions Interpolation Problems • We have a set of knots • and a function • Requirements: • , s+1 constraints • , s-1 constraints • , s-1 constraints • Two additional requirements: and )c) Gershon Elber, Technion

  4. Complete Cubic Interpolation with B-splines • B-spline interpolants are polynomials by parts • There are discontinuities at knots • Requirements: • Degree 3 • Continuity C(2) • The abscissas are knots • First and immediate diagnosis: the internal knots have multiplicity 1 )c) Gershon Elber, Technion

  5. Complete Cubic Interpolation with B-splines • The B-spline curve: • We have to interpolate s+1 values to interpolate • We have s+3 degrees of freedom Pi )c) Gershon Elber, Technion

  6. Complete Cubic Interpolation with B-splines • In order to interpolate and we choose open end conditions • The end points are: and • In general: • Two additional constraints: )c) Gershon Elber, Technion

  7. Complete Cubic Interpolation with B-splines • At only and • Moreover, • Therefore, • Similarly, )c) Gershon Elber, Technion

  8. Complete Cubic Interpolation with B-splines • Matriceal system of equations )c) Gershon Elber, Technion

  9. Complete Cubic Interpolation with B-splines )c) Gershon Elber, Technion

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