Interpolation Equation

[MNH]

The following is from my online Hydraulics and Hydrology class.

The required elevation for a volume of 0.46 acre-ft. is somewhere between 74 ft. and 75 ft. The
accumulated volumes are therefore between 0.39 acre-ft. and 0.65 acre-ft. Use the interpolation
equation to figure it out.

I never heard of the term interpolation equation. Is it the same as linear regression? I found
y - y1 = [(y2 - y1) / (x2 - x1)] (x - x1) on Wikipedia. I could easily write something for my HP 48G to crunch these numbers, but I figure there's already something on the HP 48G to do the same thing. There always seems to be a semantics issue involved when trying to figure out how to do something like this.

My class learning module gives an example of calculating this using Microsoft Excel, but since I can't read the blurry text in the formula bar, I'm resorting to doing the math on my calculator.

[Namir]

You already have the formula, and should be easy to write in in RPL. you need to input x1,y1,x2,y2,and x from the stack and sore them in named variables. The you write an rPL code for the equation using your named variables.

Simple!

Namir

[SlideRule]

   

BEST!
SlideRule

[Thomas Klemm]

You could use linear regression:

[[.39 74]
 [.65 75]]
STOΣ
.46
PREDY

74.2692307692

[KeithB]

There are a number of interpolation formulas depending on the number of points you use to interpolate with:
Check out section 25.2 of Handbook of Mathematical functions by Abramowitz and Stegun.
https://personal.math.ubc.ca/~cbm/aands/...stegun.pdf

[Namir]

(07-08-2024 02:09 PM)Thomas Klemm Wrote:  You could use linear regression:

[[.39 74]
 [.65 75]]
STOΣ
.46
PREDY

74.2692307692

Cool!

[Gil]

You could use this program INTERP, 
that works for n (n not too big) points :
it finds the exact Polynom that goes through the n points.

Code:


\<< "1) Create 
once
 Mat 
XY

 w/ the n given points
 [[x1 y1]
  [x2 y2]
  [   ]
  [xn yn]] in Stack 2
 2)
Arg:
value 
xo INTERP

 3)Repeat step 2)
   for other values xo
" DROP XY \-> xo XY
  \<< XY SIZE OBJ\-> 1 ==
    IF
    THEN DROP XY OBJ\-> 1 SWAP + \->ARRY 'XY' STO
    ELSE DROP2
    END XY SIZE 1 GET 1 ==
    IF
    THEN XY DUP 2 * 2 ROW+ 'XY' STO
    END XY 1 COL- DUP SIZE OBJ\-> DROP 1 - \-> x size
    \<< 1 size 1 +
      FOR i 0 size
        FOR j x i GET j ^
        NEXT
      NEXT size 1 + DUP 2 \->LIST \->ARRY / "a0\183x^0"
      IF size 1 >
      THEN "+" +
      END "+a" + size \->STR DUP SIZE 1 - 1 SWAP SUB + "\183x^" + size \->STR DUP SIZE 1 - 1 SWAP SUB + \->TAG DUP TRN 0 size
      FOR i xo i ^
      NEXT size 1 + 1 2 \->LIST \->ARRY * OBJ\-> DROP "y(" xo + ")" + \->TAG
    \>>
  \>>
\>>

[Thomas Klemm]

(07-08-2024 02:09 PM)Thomas Klemm Wrote:  STOΣ

For those unfamiliar:

(08-02-2022 11:54 AM)Thomas Klemm Wrote:  I was pleasantly surprised by the following:

[LeftShift] ΣDAT → STOΣ
[RightShift] ΣDAT → RCLΣ

Both make perfect sense once you understand how those shift keys work with variables.

[Gil]

It works for any variable :

[LeftShift] XXxxxx→ 'XXxxxx' STO
[RightShift] XXxxxx → 'XXxxxx' RCL

[Thomas Klemm]

(07-08-2024 07:46 PM)Gil Wrote:  It works for any variable

There’s a difference between [ΣDAT] in the VAR menu and [ΣDAT] in the STAT DATA menu.
You will notice it if you press the [LeftShift] key within a program:

Code:

« 'ΣDAT' STO »
« STOΣ »

[Thomas Klemm]

(07-08-2024 01:23 AM)MNH Wrote:  Is it the same as linear regression?

(02-10-2024 06:26 PM)Thomas Klemm Wrote:  For just two points \(P_1 = (x_1, y_1)\) and \(P_2 = (x_2, y_2)\) we want to solve the following linear system of equations:

\(
\begin{bmatrix}
x_1 & 1 \\
x_2 & 1 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
a \\
b \\
\end{bmatrix}
=
\begin{bmatrix}
y_1 \\
y_2 \\
\end{bmatrix}
\)

This leads to:

\(
\begin{align}
a &= \frac{y_1 - y_2}{x_1 - x_2} \\
\\
b &= \frac{x_1 y_2 - x_2 y_1}{x_1 - x_2} \\
\end{align}
\)

However we solve instead the following system of equations:

\(
\begin{bmatrix}
\sum x^2 & \sum x \\
\sum x & n \\
\end{bmatrix}
\cdot
\begin{bmatrix}
a \\
b \\
\end{bmatrix}
=
\begin{bmatrix}
\sum xy \\
\sum y \\
\end{bmatrix}
\)

In case of \(n = 2\) the solutions are:

\(
\begin{align}
a
&= \frac{2(x_1 y_1 + x_2 y_2) - (x_1 + x_2)(y_1 + y_2)}{2 (x_1^2 + x_2^2) - (x_1 + x_2)^2} \\
&= \frac{(x_1 - x_2)(y_1 - y_2)}{(x_1 - x_2)^2} \\
&= \frac{y_1 - y_2}{x_1 - x_2} \\
\\
b
&= \frac{(x_1^2 + x_2^2)(y_1 + y_2) - (x_1 + x_2)(x_1 y_1 + x_2 y_2)}{2 (x_1^2 + x_2^2) - (x_1 + x_2)^2} \\
&= \frac{(x_1 - x_2)(x_1 y_2 - x_2 y_1)}{(x_1 - x_2)^2} \\
&= \frac{x_1 y_2 - x_2 y_1}{x_1 - x_2} \\
\end{align}
\)

Thus, we end up with the same result.

[PedroLeiva]

I would like to check my comprehension and test the formulas, could you please give me an example? Thank you in advance. Pedro

[Albert Chan]

Linear regression line minimize Mean Squared Error (MSE)
For 2 points, it is the same as the secant line. (MSE = 0)

Proof: from secant line, we can get regression line formula.

a * x1 + b = y1      ... (1)
a * x2 + b = y2      ... (2)

(1) * x1 + (2) * x2:

a * (x1²+x2²) + b * (x1+x2) = (y1*x1 + y2*x2)      ... (3)

(1) + (2):

a * (x1+x2) + 2 * b = (y1+y2)      ... (4)

(3) and (4) are exactly n=2 linear regression formula.

[Thomas Klemm]

(07-09-2024 10:37 AM)PedroLeiva Wrote:  could you please give me an example?

This is the example given by MNH:

HP-15C

f CLEAR Σ
74 ENTER
.39 Σ+
75 ENTER
.65 Σ+

.46 f ŷ,r

74.26923077



HP-42S

CLΣ
74 ENTER
.39 Σ+
75 ENTER
.65 Σ+

.46 FCSTY

74.2692307692

[PedroLeiva]

Thanks to both, Albert and Thomas. I have just found the formula and example in "HP25 Application Programs". If X1= 7.3, Y1= 1.9879; x2= 7.4, Y2= 2.0015; for Xn= 7.37, then Yn= 1.9974. The program only takes 16 steps (pages 85-86)

[Thomas Klemm]

(07-09-2024 12:22 PM)PedroLeiva Wrote:  I have just found the formula and example in "HP25 Application Programs".

We've been there: Lagrangian Interpolation Post: #20

[PedroLeiva]

Yes, you are right. It was in March 2019, I was forgotten, sorry

[Johnh]

That simple use of linear regression with two points is really useful! Actually it may be more useful in day-to-day engineering than the more common use of LR with multiple points. Thanks! And it's so simple that there's no point in making a dedicated program.

Today's example for me in structural design: Design standards give us wind pressure coefficients at various building heights. To get the values at intermediate heights, we interpolate from the nearest values above and below.