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+--- Thread: Lagrangian Interpolation (/thread-151.html)
Pages: 1 2
RE: Lagrangian Interpolation - PedroLeiva - 07-09-2024 12:58 PM
Yes, you are right. It was in March 2019, I had forgotten, Sorry
RE: Lagrangian Interpolation - Albert Chan - 07-09-2024 08:43 PM
(03-14-2019 07:22 PM)PedroLeiva Wrote: For 2nd order polynomials we can use your program, just loading three points;
so we cannot use it for 3rd. or 4th. order, considering we will need 4 or 5 points. Am I right?
Yes, you can!
It does not matter how many points. Example, for 5 points:
Code:
x1 y1
x2 y2
x3 y3 y3'
x4 y4 y4'
x5 y5 y5' y5''(x1,y1),(x2,y2),(x3,y3) --> (x3,y3')
(x1,y1),(x2,y2),(x4,y4) --> (x4,y4')
(x1,y1),(x2,y2),(x5,y5) --> (x5,y5')
(x3,y3'),(x4,y4'),(x5,y5') --> (x5,y5'')
This work with simple secant line too.
(03-07-2015 11:49 PM)bshoring Wrote: A program for the HP-25 gave this example:
X1=-11 Y1=121 X2=3 Y2=9 X3=2 Y3=4
X=2.5 Y=6.25 etc.
Code:
X Y Interpolate @ X=2.5
2 4
3 9 6.5
-11 121 -0.5 6.25(2,4), (3,9) --> (2.5, 6.5)
(2,4), (-11,121) --> (2.5, -0.5)
(3,6.5), (-11,-0.5) --> (2.5, 6.25)
RE: Lagrangian Interpolation - Thomas Klemm - 07-10-2024 05:09 AM
(07-09-2024 08:43 PM)Albert Chan Wrote: Yes, you can!
That's nice.
But because of the memory limitation of the HP-25 to only 8 registers, the HP memory extension™ must be used. (See picture)
In addition, the method is a bit tedious if the function is to be interpolated at several points.
RE: Lagrangian Interpolation - Albert Chan - 07-10-2024 11:23 AM
(07-10-2024 05:09 AM)Thomas Klemm Wrote: But because of the memory limitation of the HP-25 to only 8 registers,
the HP memory extension™ must be used. (See picture)
Can scrap paper work too?
Quote:In addition, the method is a bit tedious if the function is to be interpolated at several points.
We can use Acton Forman's method for polynomial coefficients too.
Instead of interpolating for a value, do divided difference (i.e. slope)
Code:
X Y D D^2
2 4
3 9 5
-11 121 -9 1f(x) = 4 + (x-2)*(5 + (x-3)*1) = 4 + (x-2)*(x+2) = x^2
RE: Lagrangian Interpolation - rprosperi - 07-10-2024 11:34 AM
(07-10-2024 11:23 AM)Albert Chan Wrote:(07-10-2024 05:09 AM)Thomas Klemm Wrote: But because of the memory limitation of the HP-25 to only 8 registers,
the HP memory extension™ must be used. (See picture)
Can scrap paper work too?
It can. but it's not anywhere near as collectible...
RE: Lagrangian Interpolation - Thomas Klemm - 07-11-2024 04:19 AM
(07-10-2024 11:23 AM)Albert Chan Wrote:Code:
X Y D D^2
2 4
3 9 5
-11 121 -9 1
I assume that -9 should rather be -8:
\(
\frac{121 - 9}{-11 - 3} = \frac{112}{-14} = -8
\)
And then:
\(
\frac{-8 - 5}{-11 - 2} = \frac{-13}{-13} = 1
\)
RE: Lagrangian Interpolation - Albert Chan - 07-11-2024 10:05 AM
Hi, Thomas Klemm
Your way works too, but I prefer Acton's Forman style.
I like to do row-by-row, and this lined up numbers to use.
Acton Forman style
Code:
f D D^2 D^3
1 1
2 8 7/1=7
5 125 124/4=31 24/3=8
7 343 342/6=57 50/5=10 2/2=1Conventional Divided Difference, we need to trace the diagonal for x's to use.
Code:
f D D^2 D^3
1 1
7/1=7
2 8 32/4=8
117/3=39 6/6=1
5 125 70/5=14
218/2=109
7 343Both ways get the same [1,7,8,1] digaonal
f = 1 + (x-1)*(7 + (x-2)*(8 + (x-5)*1)) = x^3
RE: Lagrangian Interpolation - Thomas Klemm - 07-12-2024 08:28 AM
(03-14-2019 07:58 PM)Thomas Klemm Wrote: Given the restrictions of the HP-25 I'm afraid we can't go further than 3 points.
(07-09-2024 08:43 PM)Albert Chan Wrote: Yes, you can!
Here we go:
Code:
01: 24 00 : RCL 0
02: 41 : -
03: 23 06 : STO 6
04: 24 04 : RCL 4
05: 41 : -
06: 24 07 : RCL 7
07: 61 : *
08: 24 05 : RCL 5
09: 51 : +
10: 24 06 : RCL 6
11: 24 02 : RCL 2
12: 41 : -
13: 61 : *
14: 24 03 : RCL 3
15: 51 : +
16: 24 06 : RCL 6
17: 61 : *
18: 24 01 : RCL 1
19: 51 : +
20: 13 00 : GTO 00
21: 24 00 : RCL 0
22: 23 41 02 : STO - 2
23: 23 41 04 : STO - 4
24: 23 41 06 : STO - 6
25: 24 01 : RCL 1
26: 23 41 03 : STO - 3
27: 23 41 05 : STO - 5
28: 23 41 07 : STO - 7
29: 24 04 : RCL 4
30: 23 71 05 : STO / 5
31: 24 06 : RCL 6
32: 23 71 07 : STO / 7
33: 24 02 : RCL 2
34: 23 71 03 : STO / 3
35: 24 03 : RCL 3
36: 23 41 05 : STO - 5
37: 23 41 07 : STO - 7
38: 22 : Rv
39: 41 : -
40: 23 71 07 : STO / 7
41: 21 : x<->y
42: 14 73 : f LASTx
43: 41 : -
44: 23 71 05 : STO / 5
45: 24 05 : RCL 5
46: 23 41 07 : STO - 7
47: 22 : Rv
48: 41 : -
49: 23 71 07 : STO / 7Example
Interpolation of the \(sin\) function using well known values:
\(
\begin{align}
(30&, 0.5) \\
(45&, \sqrt{0.5}) \\
(60&, \sqrt{0.75}) \\
(90&, 1) \\
\end{align}
\)
Enter the data
30 STO 0
.5 STO 1
45 STO 2
.5 \(\sqrt{x}\) STO 3
60 STO 4
.75 \(\sqrt{x}\) STO 5
90 STO 5
1 STO 6
Calculation of coefficients
GTO 21
R/S
Interpolation of values
37 R/S
0.6020
37 sin
0.6018
49 R/S
0.7546
49 sin
0.7547
73 R/S
0.9572
73 sin
0.9563