Lagrangian Interpolation
07-09-2024, 12:58 PM
Post: #21
 PedroLeiva Member Posts: 226 Joined: Jun 2014
RE: Lagrangian Interpolation
Yes, you are right. It was in March 2019, I had forgotten, Sorry
07-09-2024, 08:43 PM
Post: #22
 Albert Chan Senior Member Posts: 2,680 Joined: Jul 2018
RE: Lagrangian Interpolation
(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)
07-10-2024, 05:09 AM
Post: #23
 Thomas Klemm Senior Member Posts: 2,105 Joined: Dec 2013
RE: Lagrangian Interpolation
(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 extensionâ„˘ must be used. (See picture)
In addition, the method is a bit tedious if the function is to be interpolated at several points.

Attached File(s) Thumbnail(s)

07-10-2024, 11:23 AM
Post: #24
 Albert Chan Senior Member Posts: 2,680 Joined: Jul 2018
RE: Lagrangian Interpolation
(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 extensionâ„˘ 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   1

f(x) = 4 + (x-2)*(5 + (x-3)*1) = 4 + (x-2)*(x+2) = x^2
07-10-2024, 11:34 AM
Post: #25
 rprosperi Super Moderator Posts: 6,357 Joined: Dec 2013
RE: Lagrangian Interpolation
(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 extensionâ„˘ must be used. (See picture)

Can scrap paper work too?

It can. but it's not anywhere near as collectible...

--Bob Prosperi
07-11-2024, 04:19 AM
Post: #26
 Thomas Klemm Senior Member Posts: 2,105 Joined: Dec 2013
RE: Lagrangian Interpolation
(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:

$$\frac{-8 - 5}{-11 - 2} = \frac{-13}{-13} = 1$$
07-11-2024, 10:05 AM
Post: #27
 Albert Chan Senior Member Posts: 2,680 Joined: Jul 2018
RE: Lagrangian Interpolation
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=1

Conventional 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   343

Both ways get the same [1,7,8,1] digaonal

f = 1 + (x-1)*(7 + (x-2)*(8 + (x-5)*1)) = x^3
07-12-2024, 08:28 AM
Post: #28
 Thomas Klemm Senior Member Posts: 2,105 Joined: Dec 2013
RE: Lagrangian Interpolation
(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 / 7

Example

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
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