Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

- Interpolation with unequal interval by Dr. Nirav Vyas 13300 views
- Interpolation Generalized by Mohammad Tawfik 1491 views
- Newton divided difference interpola... by VISHAL DONGA 689 views
- interpolation by 8laddu8 19858 views
- Newton’s Forward & backward interp... by Meet Patel 7018 views
- Interpolation Methods by Mohammad Tawfik 3653 views

1,180 views

Published on

No Downloads

Total views

1,180

On SlideShare

0

From Embeds

0

Number of Embeds

2

Shares

0

Downloads

105

Comments

0

Likes

3

No embeds

No notes for slide

- 1. Part 5b: INTERPOLATION – – – – Newton’s Divided-Differences Lagrange Polynomial Interpolation Inverse Interpolation Spline Interpolation
- 2. Introduction: Interpolation is the process of estimating intermediate values between precisely defined data points. For n+1 scattered data points there is a unique n-th order polynomial fit function. n+1=2 n+1=3 n+1=4 Polynomial interpolation is the process of determining the unique nth-order polynomial that fits (n+1) data points. One can define the n-th order polynomial in different formats, e.g., Newton polynomials These formats are well-suited for computational implementations Lagrange polynomials
- 3. Newton’s Divided Differences One of the most popular interpolating functions Linear interpolation: Simplest form of interpolation: connect two data points by a underlying straight line, and estimate the intermediate value. f(x) f1(x) fit function Similarity of the triangles: f1 ( x) f ( x0 ) x x0 f(x1) f1(x) f(x0) f ( x1 ) f ( x0 ) x1 x0 finite divided difference of first derivative x0 x x1 f1 ( x) f ( x0 ) f ( x1 ) f ( x0 ) ( x x0 ) x1 x0 for data points f1 ( xi ) function f ( xi ) represents the first order interpolation linear interpolation formula
- 4. Quadratic interpolation: If you have three data points, you can introduce some curvature for a better fitting. A second-order polynomial (quadratic polynomial) of the form f 2 ( x) b0 b1 ( x x0 ) b2 ( x x0 )( x x1 ) To determine the values of the coefficients; x x x x0 x1 x2 b0 f ( x0 ) b1 f ( x1 ) f ( x0 ) x1 x0 b2 If you expand the terms, this is nothing different a general polynomial Linear interpolation formula f ( x2 ) f ( x1 ) f ( x1 ) f ( x0 ) x2 x1 x1 x0 ( x2 x0 ) Quadratic interpolation formula finite divided difference of second derivative
- 5. General form of Newton’s interpolating polynomials: In general, to fit an n-the order Newton’s polynomial to (n+1) data points: f n ( x) b0 b1 ( x x0 ) b2 ( x x0 )( x x1 ) .. bn ( x x0 )( x x1 )..( x xn 1 ) where the coefficients: b0 f [ x0 ] b1 f [ x1 , x0 ] b2 f [ x2 , x1 , x0 ] data points n-th finite divided difference: … bn brackets represent the function evaluations for finite divided-differences f [ xn , xn 1 ,.., x1 , x0 ] f [ xn , xn 1 ,.., x1 , x0 ] f [ xn , xn 1 ,.., x1 ] f [ xn 1 ,.., x1 , x0 ] ( xn x0 )
- 6. These differences can be evaluated for the coefficients and substituted into the fitting function. f n ( x) f ( x0 ) ( x x0 ) f [ x1 , x0 ] ( x x0 )(x x1 ) f [ x2 , x1 , x0 ] .. ( x x0 )(x x1 )..(x xn 1 ) f [ xn , xn 1 ,.., x0 ] x values are not need to be equally spaced. x values are not necessarily in order. Newton’s divideddifference interpolating polynomial
- 7. Error for Newton’s interpolating polynomials: Newton’s divided difference formula is similar to Taylor expansion formula, adding higher order derivatives of the underlying function. A truncation error can be defined as in the case of Taylor series approximation: Rn f ( n 1) ( ) ( x x0 )( x x1 )..( x xn ) (n 1)! where is somewhere in the interval containing the unknown and the data. For Taylor series approximation error Rn f ( n 1) ( ) ( xi (n 1)! 1 xi ) n Above formulation requires prior knowledge of the underlying function and its derivative, so cannot be evaluated. 1
- 8. An alternative formulation that does not require prior knowledge of the underlying function: Rn f [ x, xn , xn 1 ,..., x0 ]( x x0 )( x x1 )..( x xn ) (n+1)th finite divided difference One more data point (xn+1) is needed to evaluate the equation. Rn f [ xn 1 , xn , xn 1 ,..., x0 ]( x x0 )( x x1 )..( x xn ) This relationship is equivalent to Rn f n 1 ( x) f n 1 ( x) f n ( x) f n ( x) Rn (next estimate) - (current estimate) increment added to the (n)th order case to calculate (n+1)th order case is equal to the error for the n-th order case.
- 9. Lagrange Polynomial Interpolation A Lagrange polynomial can be stated concisely as f n ( x) Li ( x) f ( xi ) Li ( x) j 0 j i i 0 For example: n 1 n 2 x xj n n x x0 f ( x1 ) x1 x0 xi xj In fact, Lagrange polynomials is just a different formulation of Newton’s polynomials f1 ( x) x x1 f ( x0 ) x0 x1 f1 ( x) ( x x0 )(x x2 ) ( x x1 )(x x2 ) f ( x0 ) f ( x1 ) ( x0 x1 )(x0 x2 ) ( x1 x0 )(x1 x2 ) ( x x0 )(x x1 ) f ( x2 ) ( x2 x0 )(x2 x1 )
- 10. In the formula, each term Li (x) will be equal to 1 for x=xi , and zero for all other data points. Thus, each product Li (x) fi(x) takes on the value of fi(x) at the data point.
- 11. Error is defined same as before Rn f [ x, xn , xn 1 ,..., x0 ]( x x0 )( x x1 )..( x xn ) (An additional point (xn+1) is needed for evaluation) In summary: Newton’s method is preferable for exploratory computations (n is not known a priori). > Newton method has advantages because of the insight for the behavior between different orders (consider Taylor series). > Error estimate in Newton method can easily be implemented as it employs a finite difference. Lagrange method is preferable when only one interpolation is performed (order n is known a priori), > It is easier for computational implementation.
- 12. Polynomial coefficients: Newton and Lagrange methods do not provide the coefficients of the conventional form f ( x) a0 a1 x a2 x 2 ... an x n With (n+1) data points, all the (n+1) coefficients can be determined by using elimination techniques. For example for n=2: 2 f ( x) a0 a1 x a2 x satisfies the following linear equations f ( x0 ) a0 a1 x0 2 a2 x0 f ( x1 ) a0 a1 x1 a2 x12 f ( x2 ) a0 a1 x2 2 a 2 x2 The process is notoriously illconditioned and susceptible to round-off errors: keep the order (n) small. use Lagrange or Newton interpolation.
- 13. Inverse Interpolation dependant variable independent variable Values of x are usually evenly spaced. Normally interpolation concerns finding an approximate f (x) for a given intermediate value of x. What if reverse is needed, that is value of f(x) is given and need to find the corresponding x value (inverse interpolation). Two possible solutions: Switch x by f(x) and apply Lagrange/Newton interpolation. (this method is not suitable because there is no guarantee that the new abscissa values will be evenly distributed,-in fact, usually highly uneven.) Apply normal interpolation, and find the x value that satisfies the given f(x) value a root finding problem. f (x) x
- 14. Equally spaced data: Newton/Lagrange methods are compatible for arbitrarily spaced data Before the computer era, equally spaced data had to be used, but computer implementation of these methods do not require it anymore. Evenly spaced data is required for other applications too, e.g., numerical differentiation and integration. Extrapolation: The process of f(x) for a point outside of the range of x values. If the extrapolated x value is not near the evaluation points, the error can be very large. So, extreme caution is fit curve required during extrapolation. true curve extrapolation
- 15. Spline Interpolation Sometimes fitting higher order polynomials results in erroneous results, especially at sharp changes. Spline interpolation provide s smoother transition between data points. Apply a different lower order polynomial to each interval of the data points. Continuity is maintained by constraining the derivatives at the knots. Here the polynomial interpolation overshoots between data points. Spline offer a smoother and a meaningful transition knot interval interval A different spline function is defined for each interval
- 16. Linear Splines: Each interval is connected by a straight line. For each interval f ( x) f ( xi ) mi ( x xi ) where mi f ( xi 1 ) f ( xi ) ( xi 1 xi ) Linear splines is identical to the first order polynomial fit. Linear spline function is discontinuous at the knots. So, we need to use higher order polynomials to maintain continuity. In general, for m-th derivative to be continuous, an order (m+1) spline fit must be used.
- 17. Quadratic Splines: In quadratic splines each interval is represented by a different quadratic polynomial. f ( x) ai x 2 bi x ci For n+1 data points there are n intervals. This makes a total of 3n unknowns to be solved. Conditions: 1. At the interior knots adjacent functions must meet the data: 2n-2 equations 2. First and last function must pass through end points: 2 equations 3. First derivatives at the interior knots must be equal: n-1 equations 4. The final constrain is chosen arbitrarily: 1 equation Note that continuity of the second derivative is not ensured at the knots. These simultaneous linear equations are solved to obtain all the coefficients.
- 18. Cubic Splines: For cubic splines a different third order (cubic) polynomial is defined for each interval. f ( x) ai x 3 bi x 2 ci x d i For n+1 data points there are n intervals: 4n unknowns to be solved. Fifth condition is chosen arbitrarily (called natural spline) Conditions: 1. The function values must meet interior knots: 2n-2 equations 2. The first and last function must pass through end points: 2 equations 3. First derivatives at the knots must be equal: n-1 equations 4. Second derivatives at the interior knots must be equal: n-1 equations 5. The second derivatives at the end knots are zero: 2 equations

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment