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Файл:Derive_v5_05 / Derive / SPLINE
.docSPLINE_S.DOC - by Steven Schonefeld The file SPLINE_S.MTH contains procedures for generating a "natural" cubic spline (in parametric form) for a matrix m_ of points. The function SPLINE(m_, t) expects the x-coordinates of the points contained in m_ to be in increasing order. When you [E]xpand SPLINE(m_, t), you get a matrix SS. Each row of SS is a vector representing parametric equations for a segment of the desired spline. These parametric equations should be [P]lotted using 0 (zero) for the lower limit and 1 (one) for the upper limit of the parameter t. Once the limits (0 and 1) have been set, you may plot all the parametric equations at once with: [Ctrl Enter]. In case you are interpolating a specific function F(x), you may use the function DIFFERENCE(SS) to generate a matrix of parametric equations for the difference between the spline SS and the function F(x). The function PARA(m_, t) does not require the x-coordinates of m_ to be in increasing order. It will give a matrix of parametric equations [P(t), Q(t)] that will smoothly connect the points in matrix m_. In all cases, the matrices of parametric equations should be plotted using 0 and 1 for the limits on the parameter. In the following examples, #29 was [E]xpanded to give the matrix #30 of parametric equations for the natural cubic spline that will interpolate the matrix of points: [[1, 1], [3, 2], [4, 2], [5, 3]]. Line #32 was [E]xpanded to give the spline matrix #33 that will interpolate the function F(x) at the points having x-coordinates 0, 0.5, 1, .., 6.5, 7. We next, [A]uthored and expanded DIFFERENCE(#33) to get the spline matrix #35. We may [P]lot #35 to see how close the spline comes to the function F(x). Finally, #36 was [E]xpanded to give the matrix #37 of parametric equations that smoothly connect the closed rectangle: [[0, 0], [3, 0], [3, 2], [0, 2], [0, 0]]. #29: SPLINE([[1, 1], [3, 2], [4, 2], [5, 3]], t) #30: [[2 t + 1, -0.521739 t^3 + 1.52173 t + 1], [t + 3 , 0.413043 t^3 - 0.391304 t^2 - 0.0217391 t + 2], [t + 4 , -0.282608 t^3 + 0.847826 t^2 + 0.434782 t + 2]] #31: F(x) := SIN(x) + COS(x/2) #32: SPLINE(VECTOR([x, F(x)], x, 0, 7, 0.5), t) #33: [[0.5 t, -0.0333467 t^3 + 0.481684 t + 1], .. (rest of matrix omitted) #34: DIFFERENCE(#33) #35: [[0.5 t, -COS(0.25 t) - SIN(0.5 t) - 0.0333467 t^3 + 0.481684 t + 1], .. (rest of matrix omitted) #36: PARA([[0, 0], [3, 0], [3, 2], [0, 2], [0, 0]], t) #37: [[3.53571 t - 0.535714 t^3, 0.642857 t^3 - 0.642857 t], .. (rest of matrix omitted) For more information on cubic splines see the book: NUMERICAL ANALYSIS via DERIVE, by Steven Schonefeld, MathWare, 604 E Mumford Dr., Urbana, IL 61801 1-800-255-2468.