Wednesday, December 13, 2017

HP Prime: Area by Quadratic Splines


HP Prime:  Area by Quadratic Splines

Introduction

The program QUADSUM calculates the area under the curve described by the set of points (x_n, y_n).  The points are connected, in groups of three, by quadratic splines.  Thus, points (x1, y1), (x2, y2), and (x3, y3) are connected by a quadratic spline, (x3, y3), (x4, y4), (x5, y5) are connected by another quadratic spline, and so on.

Screen shot by HP Prime, labels I added using MS Paint (the old one)

  
The number of points for QUADSUM must be odd. 

HP Prime Program QUADSUM

EXPORT QUADSUM(LX,LY)
BEGIN
// EWS 2017-12-10
// Area by connecting
// points using quadratic
// curves
// number of points must be odd
LOCAL A,S,T; // A=0
S:=SIZE(LX);
IF FP(S/2)==0 THEN
RETURN "Invalid: Number of
points must be odd";
KILL;
END;
LOCAL T,M,MA,MB,MC;
FOR T FROM 1 TO S-2 STEP 2 DO
M:=CAS.LSQ([[1,LX(T),LX(T)^2],
[1,LX(T+1),LX(T+1)^2],
[1,LX(T+2),LX(T+2)^2]],
[[LY(T)],[LY(T+1)],[LY(T+2)]]);
MA:=M(3,1);
MB:=M(2,1);
MC:=M(1,1);
A:=A+
(MA*LX(T+2)^3/3+MB*LX(T+2)^2/2+
MC*LX(T+2))-
(MA*LX(T)^3/3+MB*LX(T)^2/2+
MC*LX(T));
END;
RETURN A;
END;

Example

Find the area under the curve with these points connected by quadratic splines:
(0,2), (1,1), (2,2), (3,6), (4,4)



Note that the point (2,2) ends the first spline and starts the second.

QUADSUM({0,1,2,3,4}, {2,1,2,6,4}) returns 12.6666666667

FYI:  The polynomial described would be the piecewise equation:
y = { x^2 -2x + 2 for 0 < x ≤ 2,  -3x^2 + 19x – 24 for 2 < x ≤ 4

Eddie


This blog is property of Edward Shore, 2017

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