Lagrangian Interpolation

[Eddie W. Shore]

The program LARG estimates a point (x, y) using the Lagrangian interpolating polynomial below:

y =
y0 * (x - x1) * (x - x2) ÷ ((x0 - x1) * (x0 - x2))
+ y1 * (x - x0) * (x - x2) ÷ ((x1 - x0) * (x1 - x2))
+ y2 * (x - x0) * (x - x1) ÷ ((x2 - x0) * (x2 - x1))

The polynomial has three set points that defined points (x0, y0), (x1, y1), and (x2, y2).

This program is based off the Interpolations program for the HP 67 and HP 97 calculators.

Source

HP 67/97 Math Pac I. Hewlett Packard. Corvallis, OR. 1976

Code:

01  LBL ᵀ LARG
02  SF 27
03  CF 00
04  STOP

05  LBL A
06  STO 01
07  x<>y
08  STO 04
09  RTN

10  LBL B
11  STO 02 
12  x<>y
13  STO 05
14  RTN

15  LBL C
16  STO 03
17  x<>y
18  STO 06
19  CF 00
20  RTN

21  LBL D
22  STO 00
23  FS? 00
24  GTO 15
25  RCL 04
26  RCL 05
27  -
28  RCL 04
29  RCL 06
30  -
31  *
32  ST/ 01  
33  RCL 05
34  RCL 04
35  -
36  RCL 05
37  RCL 06
38  -
39  *
40  ST/ 02
41  RCL 06
42  RCL 04
43  -
44  RCL 06
45  RCL 05
46  - 
47  *
48  ST/ 03
49  SF 00

50  LBL 15
51  RCL 00
52  RCL 05
53  -
54  STO 07
55  RCL 00
56  RCL 06
57  -
58  STO 08
59  *
60  RCL 01
61  *
62  RCL 00
63  RCL 04
64  -
65  STO 09
66  RCL 08
67  *
68  RCL 02
69  *
70  +
71  RCL 09
72  RCL 07
73  *
74  RCL 03
75  *
76  +
77  STOP

78  LBL E
79  CF 27
80  CF 00
81  END

Memory Registers Used:

R01 = y0
R02 = y1
R03 = y2

R04 = x0
R05 = x1
R06 = x2

R07, R08, R09

Instructions

The program LARG is set up to use the USER keyboard as follows:

Key A ([Σ+]): Enter the point (x0, y0). Type x0, press [ENTER], type y0, press [Σ+].

Key B ([1/x]): Enter the point (x1, y1). Type x1, press [ENTER], type y1, press [1/x].

Key C ([√]): Enter the point (x2, y2). Type x2, press [ENTER], type y2, press [√].

Key D ([LOG]): Calculate y.

Key E ([LN]): Exits the program and turns off the user keyboard.



Example


An estimated polynomial runs through the points (1, 7), (3, 9), and (5, 4). Estimate the point at x = 2 and x = 6.

XEQ LARG

1 ENTER 7 [Σ+] (A)
3 ENTER 9 [1/x] (B)
5 ENTER 4 [√] (C)

2 [LOG] (D) returns about 8.8750

6 [LOG] (D) returns about -1.1250

When done, press [LN] (E) to exit or turn off the user keyboard.

[Thomas Klemm]

Somewhat related:

• Lagrangian Interpolation
• (42S) Newton Polynomial Interpolation