Es272 ch5b


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Es272 ch5b

  1. 1. Part 5b: INTERPOLATION – – – – Newton’s Divided-Differences Lagrange Polynomial Interpolation Inverse Interpolation Spline Interpolation
  2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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