Demo 5. 1 Degree Reduction and Raising Matrices for Degree 5

Transcription

1 Demo This demo contains the calculation procedures to generate the results shown in Appendix A of the lab report. Degree Reduction and Raising Matrices for Degree M,4 = M 4, = M,3 = M 3, = M, = M, = M, = M, = M, = M, = ( )

2 QuadClip Applied to a Polynomial of th Degree with Two Roots X 3X 4 X 3 + 4X X + Called QuadClip with input polynomial on interval [, ]: p = X 3X 4 X 3 + 4X X Recursion Branch for Input Interval [, ] Normalized monomial und Bézier representations and the Bézier polygon: p = X 3X 4 X 3 + 4X X + = B, (X) B, (X) B, (X) +.B 3, (X) + B 4, (X) + B, (X) Best approximation polynomials of degree,,, 3 and 4: q =. =.B, q =.374X.487 =.487B, +.987B, q =.48X.7874X.748 =.748B,.448B, +.87B, q 3 =.X X.9X =.7349B,3.7B, B, B 3,3 q 4 = 7.X 4 7.X X 7.97X +.99 =.99B, B,4 +.B, B 3, B 4,4

3 q q q q 3 q 4 p Degree reduction and raising matrices: M, = M, = Degree reduction and raising: q =.48X.7874X.748 =.748B,.448B, +.87B, q =.877 X X X X.7874X.748 =.748B,.87B,.749B, +.B 3, +.874B 4, +.87B, p q q The maximum difference of the Bézier coefficients is δ =.4. Bounding polynomials M and m: Root of M and m: Intersection intervals: M =.48X.7874X m =.48X.7874X.4743 N(M) = {} N(m) = {.93,.78} 3

4 [, ] Longest intersection interval: = Bisection: first half [,.] und second half [., ]. Recursion Branch on the First Half [,.] Normalized monomial und Bézier representations and the Bézier polygon: p =.78X.87X 4.87X 3 + X 7.X + = B, (X).B, (X) B, (X).87B 3, (X).87B 4, (X) +.87B, (X) Best approximation polynomials of degree,,, 3 and 4: q =.978 =.978B, q =.37944X.979 =.979B,.3844B, q = 4.839X.X +.84 =.84B,.837B, +.499B, q 3 = 4.798X X 7.997X +.4 =.4B,3.488B,3.4494B, B 3,3 q 4 =.3437X 4 3.X 3 +.X 7.93X +.3 =.3B,4.89B,4.83B,4.89B 3,4 +.B 4,4 4

5 q q q q 3 q 4 p Degree reduction and raising matrices: M, = M, = Degree reduction and raising: q = 4.839X.X +.84 =.84B,.837B, +.499B, q =.7 X 3.37 X X X.X +.84 =.84B,.399B,.7997B,.87B 3,.4734B 4, +.499B, p q q The maximum difference of the Bézier coefficients is δ =.834. Bounding polynomials M and m: Root of M and m: Intersection intervals: M = 4.839X.X +.89 m = 4.839X.X +.3 N(M) = {.883,.79787} N(m) = {.84,.978}

6 [.84,.883], [.79787,.978] Longest intersection interval:.93 = Selective recursion: interval : [.9,.448], interval : [ ,.48339],.3 Recursion Branch in Interval : [.9,.448] Normalized monomial und Bézier representations and the Bézier polygon: p =.78X.49X 4.338X X.933X +.83 =.83B, (X) +.799B, (X).88B, (X).89B 3, (X).89B 4, (X).373B, (X) Best approximation polynomials of degree,,, 3 and 4: q =.849 =.849B, q =.387X +.43 =.43B,.4733B, q =.4X X =.874B,.4473B,.374B, q 3 =.87X X.9397X =.8338B,3.7787B,3.3783B,3.3733B 3,3 q 4 =.97X X X.93X +.83 =.83B,4 +.74B,4.438B,4.8B 3,4.373B 4,4

7 q q q q 3 q 4 p Degree reduction and raising matrices: M, = M, = Degree reduction and raising: q =.4X X =.874B,.4473B,.374B, q = X.88 X X 3 +.4X X =.874B, +.388B,.947B,.87B 3,.884B 4,.374B, p q q The maximum difference of the Bézier coefficients is δ =.8. Bounding polynomials M and m: Root of M and m: Intersection intervals: M =.4X X m =.4X X N(M) = {.339, 3.38} N(m) = { , 3.39} 7

8 [ ,.339] Longest intersection interval:.398 = Selective recursion: interval : [.88,.88],.4 Recursion Branch in Interval : [.88,.88] Found root in interval [.88,.88] at recursion depth 4!. Recursion Branch in Interval : [ ,.48339] Normalized monomial und Bézier representations and the Bézier polygon: p =.78X X 4.878X X +.347X.47 =.47B, (X).3B, (X).83B, (X) +.744B 3, (X) B 4, (X) +.3B, (X) Best approximation polynomials of degree,,, 3 and 4: q =.877 =.877B, q =.373X.44 =.44B, +.9B, q =.8X +.37X.8 =.8B,.39B, +.4B, q 3 =.94X X +.348X.48 =.48B,3.93B, B,3 +.98B 3,3 q 4 =.743X X X +.349X.4 =.4B,4.97B,4.9B, B 3,4 +.B 4,4 8

9 q q q q 3 q 4 p Degree reduction and raising matrices: M, = M, = Degree reduction and raising: q =.8X +.37X.8 =.8B,.39B, +.4B, q = X X X 3 +.8X +.37X.8 =.8B,.33B,.47B, B 3, B 4, +.4B, p q q The maximum difference of the Bézier coefficients is δ =.773. Bounding polynomials M and m: Root of M and m: Intersection intervals: M =.8X +.37X.43 m =.8X +.37X.7 N(M) = { 3.79,.779} N(m) = { 3.737,.838} 9

10 [.779,.838] Longest intersection interval:.89 = Selective recursion: interval : [.44788,.4489],. Recursion Branch in Interval : [.44788,.4489] Found root in interval [.44788,.4489] at recursion depth 4!.7 Recursion Branch on the Second Half [., ] Normalized monomial und Bézier representations and the Bézier polygon: p =.78X +.787X 4.8X 3.937X +.3X +.87 =.87B, (X) +.B, (X) +.937B, (X) +.87B 3, (X) +.B 4, (X) + B, (X) Best approximation polynomials of degree,,, 3 and 4: q =.978 =.978B, q =.493X =.44494B, B, q =.8473X +.37X +.39 =.39B, +.948B, B, q 3 =.794X 3.744X X =.939B,3 +.8B, B, B 3,3 q 4 = 3.787X X 3.848X X +.8 =.8B,4 +.74B, B,4 +.39B 3, B 4,4

11 q q q q 3 q 4 p Degree reduction and raising matrices: M, = M, = Degree reduction and raising: q =.8473X +.37X +.39 =.39B, +.948B, B, q = 3.79 X X 4.89 X X +.37X +.39 =.39B, +.7B, +.778B, B 3, +.887B 4, B, p q q The maximum difference of the Bézier coefficients is δ =.387. Bounding polynomials M and m: Root of M and m: Intersection intervals: M =.8473X +.37X m =.8473X +.37X.937 N(M) = {.4774,.999} N(m) = {.49,.}

12 [,.49] Longest intersection interval:.49 = Selective recursion: interval : [.,.333],.8 Recursion Branch in Interval : [.,.333] Normalized monomial und Bézier representations and the Bézier polygon: p = X X 4.888X 3.49X X +.87 =.87B, (X) B, (X) +.4B, (X) B 3, (X) +.93B 4, (X) +.38B, (X) Best approximation polynomials of degree,,, 3 and 4: q =.398 =.398B, q =.948X +.94 =.94B, +.3B, q =.44X X =.873B, +.7B, +.34B, q 3 =.998X 3.8X X +.87 =.87B,3 +.3B,3 +.8B,3 +.38B 3,3 q 4 = 8.89 X 4.89X 3.477X X +.87 =.87B, B,4 +.88B, B 3,4 +.38B 4,4

13 q q q q 3 q 4 p Degree reduction and raising matrices: M, = M, = Degree reduction and raising: q =.44X X =.873B, +.7B, +.34B, q =.733 X X 4.4 X 3.44X X =.873B, B, +.47B, B 3, +.933B 4, +.34B, p q q The maximum difference of the Bézier coefficients is δ =.94. Bounding polynomials M and m: Root of M and m: Intersection intervals: M =.44X X +.87 m =.44X X N(M) = {.8, 4.838} N(m) = {.8, 4.838} 3

14 No intersection intervals with the x axis. 4

15 .9 Result: Root Intervals Input Polynomial on Interval [, ] p = X 3X 4 X 3 + 4X X Result: Root Intervals [.88,.88], [.44788,.4489] with precision ε =..