Area by Quadratic Splines
12-13-2017, 01:42 PM
Post: #1
 Eddie W. Shore Senior Member Posts: 1,358 Joined: Dec 2013
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.

The number of points for QUADSUM must be odd.

HP Prime Program QUADSUM
Code:
 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
12-14-2017, 06:57 AM
Post: #2
 AlexFekken Member Posts: 151 Joined: May 2016
RE: Area by Quadratic Splines
Doesn't this mean you are basically doing a repeated Simpson's rule, but with more work?

https://en.wikipedia.org/wiki/Simpson%27s_rule
12-20-2017, 05:03 AM
Post: #3
 AlexFekken Member Posts: 151 Joined: May 2016
RE: Area by Quadratic Splines
(12-14-2017 06:57 AM)AlexFekken Wrote:  Doesn't this mean you are basically doing a repeated Simpson's rule, but with more work?

Actually no, not if the points aren't equidistant of course...
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