1 / 33

Halfway Down

Halfway Down Halfway down the stairs Is a stair Where I sit. There isn't any Other stair Quite like It. I'm not at the bottom, I'm not at the top; So this is the stair Where I always Stop.

basil
Download Presentation

Halfway Down

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Halfway Down Halfway down the stairsIs a stair Where I sit.There isn't any Other stairQuite likeIt.I'm not at the bottom,I'm not at the top;So this is the stairWhere I alwaysStop. Halfway up the stairs Isn't up,And isn't down.It isn't in the nursery,It isn't in the town.And all sorts of funny thoughtsRun round my head:"It isn't really Anywhere!It's somewhere else instead!"

  2. Halfway Down Halfway down the stairsIs a stair Where I sit.There isn't any Other stairQuite likeIt.I'm not at the bottom,I'm not at the top;So this is the stairWhere I alwaysStop. Halfway up the stairs Isn't up,And isn't down.It isn't in the nursery,It isn't in the town.And all sorts of funny thoughtsRun round my head:"It isn't really Anywhere!It's somewhere else instead!" A.A. Milne

  3. Interpolation – the ups and downs

  4. Interpolation and Fitting • Discrete datum points obtained by some process • Provide basis for a description of behaviour at any point • Description usually in some function form: • Function can have some physical form based on theoretical behaviour or can be a more general form • Use interpolation every day, e.g. measurements, weights, solving ODEs etc. or

  5. Interpolation and Fitting • Interpolation vs Fitting – in general: • Interpolation reproduces • datum values at the datum • locations • Fitting seeks to provide the • best overall description • Interpolation vs Extrapolation • -Interpolation: unknown value bracketed by known values • -Extrapolation: unknown value has known value on one side only y x y x

  6. Polynomial Interpolation • Determine the coefficients of a polynomial one order less than the number of datum points: • Coefficients found by solving linear system of equations: • Significant problems with this kind of interpolation, especially as the number of datum points becomes large: • “expensive” to solve for the coefficients • System of equations (Vandermonde matrix) becomes ill-conditioned • High order polynomials oscillate between points

  7. Polynomial (and other) Fitting • Polynomials useful for small number of points – at least 3 for a quadratic • Can be useful for fitting rather than datum interpolation • Given polynomial to “reasonable” order, find coefficients such that function is a good representation of data, e.g. minimise the sum of the squared difference between function and datum values (called Least Squares Fit): • If we have N data points and our polynomial is of order m, how can we solve? The problem is overdetermined. don’t know know

  8. Polynomial (and other) Fitting • Problem looks like: • Two options: • Solve the normal equations: ATAa = ATb. Solved by standard LU decomposition, but tend to be badly conditioned and prone to roundoff error. • Do a singular value decomposition on A. Naturally finds the solution which is the best least squares approximation. A= a= b= Aa=b

  9. Up or Down? Halfway up the stairs Isn't up,And isn't down.It isn't in the nursery,It isn't in the town. Up How much up and how much down? Down

  10. Lagrangian Interpolation entire range y sub-interval x • Used for large number of datum points • Break whole interpolation range into smaller intervals • Use low order polynomial to interpolate over sub-interval only • Overcomes problems with high-order polynomial interpolation

  11. Linear Lagrange Interpolation sub-interval y (xj+1,yj+1) (xj,yj) x yj(x) • Put a straight line between each pair of points What form do Lj and Lj+1 take?

  12. Linear Lagrange Interpolation sub-interval y (xj+1,yj+1) (xj,yj) x yj(x) • Put a straight line between each pair of points Lj+1 1 Important!! L act as weighting functions: some of yj and some of yj+1 xj+1 x xj

  13. Quadratic Lagrange Interpolation sub-interval y (xj+1,yj+1) (xj-1,yj-1) (xj,yj) x yj(x) • Put a quadratic between each triplet of points

  14. Quadratic Lagrange Interpolation Lj-1 Lj Lj+1 1 1 1 xj-1 xj xj+1 xj-1 xj xj+1 xj-1 xj xj+1

  15. Quadratic Lagrange Interpolation Still true? Check! L continue to act as weighting functions: some of yj-1, some of yj, and some of yj+1

  16. Lagrange Interpolation of any Order • An mth order Lagrange interpolation using m+1 datum points is represented by: • The weighting functions, or Lagrange polynomials, are described by: multiply

  17. Lagrange Interpolation of any Order • Always true that: • A problem with Lagrange interpolation is the lack of slope continuity between the sub-interval (element) boundaries. sub-interval y (xj+1,yj+1) (xj-1,yj-1) (xj,yj) x slope discontinuous Not always important

  18. Next time…Function continuity and Cubic Splines

  19. The story continues…Function continuity and Cubic Splines

  20. Function Continuity

  21. Function Continuity • C0 continuity – value continuous • C1 continuity – gradient or slope continuous • C2 continuity – curvature (2nd derivative) continuous

  22. Interpolation with Slope Continuity sub-interval: j y • Definitions: kj kj+1 x yj(x) xj xj+1 • Four “degrees of freedom” over interval: pj(xj), pj(xj+1), p’j(xj)=kj and p’j(xj+1 )=kj+1 • Four dof can be fitted by a cubic polynomial. • Gradients may be known at x0 and xN. • May be reasonable to set p’’=0 at x0 and xN. p’’=0 p’’ 0

  23. Cubic Spline Polynomial • General form: • Spline – from thin rods used by engineers to fit smooth curves through a number of points. • pj(x) must satisfy: pj(xj)=yj, pj(xj+1)=yj+1, p’j(xj)=kj and p’j(xj+1 )=kj+1 • Two steps to make pj(x) useful. (1) Determine a0 to a3. (2) Determine k values. Two sets of equations necessary.

  24. Step 1: Spline Coefficients • To satisfy: pj(xj)=yj, p’j(xj)=kj , pj(xj+1)=yj+1 and p’j(xj+1 )=kj+1

  25. …some algebra later…

  26. Step 1: Spline Coefficients • Solving for a2 and a3: • Once the kj values are known, the spline function is determined.

  27. Step 2: Unknown Gradients sub-interval: j sub-interval: j-1 • For cubic in each interval: • Evaluating the second derivatives: • Equating the second derivatives: y’’ p’’j-1(x) p’’j(x) xj x

  28. Step 2: Unknown Gradients k0 y’’N-1(xN)=0 y • Known or • assumed • information: • If k0 and kN are known, there are N-1 points, N-1 unknown k values and N-1 equations. • Equal number of equations and unknowns. y’’0(x0)=0 N-1 internalpoints kN x x0 xN

  29. Step 2: Unknown Gradients • If k0 and kN are not known, there are N+1 points, N+1 unknown k values and N-1 equations of the type: • and two equations: • from the condition of zero curvature (p’’=0) at the end-points. • Still equal number of equations and unknowns.

  30. Step 2: Unknown Gradients • In both cases, equations are put together into a matrix of linear equations in the unknown k values. • The system of equations is solved. • Each row has only 3 non-zero terms – one on either side of the diagonal. Very efficient to solve. = k

  31. Summary • Gradient continuous interpolations can be produced using cubic splines. • Gradients must be known at the extremum points – or assumption of zero curvature can be used. • System of linear equations solved to give gradients at each discrete point. (Step 2) • Gradients used to determine coefficents of cubic spline interpolation for each sub-interval. (Step 1) • Cubic splines can be used to interpolate to x points lying between known discrete points.

  32. The end…

More Related