fx-991CW: Sums with a Step Not Equal to One

fx-991CW: Sums with a Step Not Equal to One

Step ≠ 1? An Adjustment is Necessary

The summation function on the Casio fx-991 CW calculator, and a lot of other advanced scientific calculators, calculates the sum of f(x) from x = a to x = b, with an increment of 1.

But what if have a sum of following:

log(2) + log(4) + log(6) + log(8): sum(f(x) from x = 2 to x = 8, step = 2)

Clearly, the increment is not 1, but 2 instead. Believe it or not, we can still use the summation function. We will need to adjust both the function and the limits to fit the requirements of the summation function.

1. Adjust f(x) to f(step * x)

2. Adjust the limits to x = a ÷ step and x = b ÷ step

For this particular problem:

1. f(x) becomes log(2 * x)

2. The limits become x = 2 ÷ 2 = 1 to x = 8 ÷ 2 = 4 (this allows the step to be 1)

Which transforms:

sum(log(x) from x = 2 to x = 8, step = 2)

to:

Σ(log(2 * x) from x = 1 to x = 4)

Sum: 2.584331224

A Couple More Examples

Example 1:

sum(x^2 from x = 0.5 to x = 2, step = 0.5), f(x) = x^2

The key here is the step (also known as the increment). The adjustments needed are:

1. f(x) becomes (0.5 * x)^2 = (x / 2)^2

2. The limits become x = 0.5/0.5 = 1 to x = 2/0.5 = 4

Sum: 15/2 = 7.5

Example 2:

sum(e^(x) from x = 1.5 to x = 2.1, step = 0.1), f(x) = e^(x)

The adjustments needed are:

1. f(x) becomes e^(0.1 * x) = e^(x / 10)

2. The limits become x = 1.5/0.1 = 15 to x = 2.1/0.1 = 21

Sum: 43.1994368

In Summary…

For the sum:

sum(f(x), x = a to x = b, step = Δx):

1. Adjust f(x) to f(Δx * x)

2. Adjust the limits: x = a/Δx to x = b/Δx

3. Summation function looks like this: Σ( f(Δx * x), x = a/Δx to x = b/Δx)

Hope you find this useful! Take care,

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

The author does not use AI engines and never will.

RPN with HP 15C & DM32: Expanding Linear Regression

RPN with HP 15C & DM32: Expanding Linear Regression

“Linearize” the Equation

The linear regression mode of the HP 15C and Swiss Micros DM32 can be expanded to fit different curves for bi-variate data.

Curves can be set up for linear regression if we can get the equation into the form:

y = b + m * x ⇒ f(y) = b + m * g(x)

where b is the y-intercept and x is the slope. The correlation is r and r^2 can be used to determine the relationship between x (independent) and y (dependent) variables. If r^2 is close to 1, the better the curve fits to the data.

Note: f(y) and g(x) are functions that contain only one term (sin y, cos y, e^y, 1/y, y^2 …. sin x, cos x, e^x, 1/x, x^2, ….)

The DM32 labels the slope as m, y-intercept as b. Each of the parameters can be recalled separately.

The HP 12C labels the slope as a, y-intercept as b. Both a and b are calculated by the key sequence [ f ] [ Σ+ ] (L.R.): slope is on the y-stack and the y-intercept is on the x-stack. To get the correlation, enter any (valid) number and press [ f ] [ . ] (y-hat, r) [ x<>y ].

Let’s illustrate this for with a couple of examples.

y = 1 / (b + m * e^(-x))

1/y = b + m * e^(-x)

We have the equation in the required form with the following adjustments: x’ = e^(-x), y’ = 1/x.

y = b * m^x

ln y = ln (b * m^x)

ln y = ln b + ln (m^x)

ln y = ln b + x * ln m

We have the equation in the required form with the following adjustments: x’ = x, y’ = ln y.

Now note that we have ln b and ln m. This will require an adjustment when the linear regression is calculated. To get the “true” slope and y-intercept in this case, we must calculate e^b and e^m.

To employ different curve fittings, I use (at least) two programs:

Entering data (DM32, HP15C):

LBL #

g(x)

x<>y

f(y)

x<>y

Σ+

R/S

GTO #

Entering data:

CLEAR Σ (CLΣ)

y1 ENTER x1 XEQ/GSB #

yn ENTER xn R/S

You can enter as many data points as you like.

Calculating the Parameters:

DM32:

LBL @

b

(adjustments if needed)

R/S

m

(adjustments if needed)

R/S

r

x^2

RTN

HP 15C:

LBL @

L.R.

(adjustment if needed)

R/S

x<>y

(adjustment if needed)

R/S

1

y-hat, r

x<>y

x^2

RTN

Example 1: y = 1 / (b + m * e^(-x))

Linearized: 1 / y = b + m * e^(-x)

Adjustments: x’ = e^(-x), y’ = 1 / y, no adjustment to b or m

Enter Data:

HP 15C Code	HP 15C Key	DM32
42, 21, 11	LBL A	LBL D
16	CHS	+/-
12	e^x	e^x
34	x<>y	x<>y
15	1/x	1/x
34	x<>y	x<>y
49	Σ+	Σ+
31	R/S	R/S
22, 11	GTO A	GTO D

Calculating the Parameters:

HP 15C Code	HP 15C Key	DM32
42, 21, 1	LBL 1	LBL R
42, 49	L.R.	b
31	R/S	R/S
34	x<>y	m
31	R/S	R/S
1	1	r
42, 48	y-hat, r	x^2
34	x<>y	RTN
43, 11	x^2
43, 32	RTN

Example:

x	y
0.0	0.125
0.1	0.13
0.2	0.134
0.3	0.138
0.4	0.143
0.5	0.147

Results (FIX 5):

	HP 15C	DM32
Intercept (b)	4.98312	4.98312
Slope (m/a)	3.01575	3.01575
r^2	0.99841	0.99841

Example 2: y = ln(b + m * e^(-x))

Linearized: e^y = b + m * e^(-x)

Adjustments: x’ = e^(-x), y’ = e^(y), no adjustment to b or m

Enter Data:

HP 15C Code	HP 15C Key	DM32
42, 21, 12	LBL B	LBL E
16	CHS	+/-
12	e^x	e^x
34	x<>y	x<>y
12	e^x	e^x
34	x<>y	x<>y
49	Σ+	Σ+
31	R/S	R/S
22, 12	GTO B	GTO E

Calculating the Parameters:

HP 15C Code	HP 15C Key	DM32
42, 21, 1	LBL 1	LBL R
42, 49	L.R.	b
31	R/S	R/S
34	x<>y	m
31	R/S	R/S
1	1	r
42, 48	y-hat, r	x^2
34	x<>y	RTN
43, 11	x^2
43, 32	RTN

Example:

x	y
0.98	1.946
0.99	1.942
1.00	1.938
1.01	1.934
1.02	1.929
1.03	1.925

Results (FIX 5):

	HP 15C	DM32
Intercept (b)	3.99987	3.99985
Slope (m/a)	8.00051	8.00056
r^2	0.99844	0.99844

*differences may be due to rounding error in the internal algorithms

Example 3: y = √(b + m * x^2)

Linearized: y^2 = b + m * x^2

Adjustments: x’ = x^2, y’ = y^2, no adjustment to b or m

Enter Data:

HP 15C Code	HP 15C Key	DM32
42, 21, 13	LBL C	LBL F
43, 11	x^2	x^2
34	x<>y	x<>y
43, 11	x^2	x^2
34	x<>y	x<>y
49	Σ+	Σ+
31	R/S	R/S
22, 13	GTO C	GTO F

Calculating the Parameters:

HP 15C Code	HP 15C Key	DM32
42, 21, 1	LBL 1	LBL R
42, 49	L.R.	b
31	R/S	R/S
34	x<>y	m
31	R/S	R/S
1	1	r
42, 48	y-hat, r	x^2
34	x<>y	RTN
43, 11	x^2
43, 32	RTN

Example:

x	y
1.05	2.066
1.25	2.337
1.45	2.620
1.65	2.912
1.85	3.209
2.05	3.511

Results (FIX 5):

	HP 15C	DM32
Intercept (b)	1.40009	1.40009
Slope (m/a)	2.59996	2.59996
r^2	1.00000	1.00000

Example 4: y = b * m^x

Linearized: ln y = ln b + x * ln m

Adjustments: x’ = x, y’ = ln y, result adjustments: e^b, e^m

Enter Data:

HP 15C Code	HP 15C Key	DM32
42, 21, 14	LBL D	LBL G
34	x<>y	x<>y
43, 12	LN	LN
34	x<>y	x<>y
49	Σ+	Σ+
31	R/S	R/S
22, 14	GTO D	GTO G

Calculating the Parameters:

HP 15C Code	HP 15C Key	DM32
42, 21, 2	LBL 2	LBL S
42, 49	L.R.	b
12	e^x	e^x
31	R/S	R/S
34	x<>y	m
12	e^x	e^x
31	R/S	R/S
1	1	r
42, 48	y-bar, r	x^2
34	x<>y	RTN
43, 11	x^2
43, 32	RTN

Example:

x	y
0.84	2.358
0.87	2.363
0.90	2.369
0.93	2.374
0.96	2.379
0.99	2.385

Results (FIX 5):

	HP 15C	DM32
Intercept (b)	2.21302	2.21302
Slope (m/a)	1.07843	1.07843
r^2	0.99916	0.99919

*differences may be due to rounding error in the internal algorithms

Expanding Linear Regression Table

Regression	X	Y	B = ITC	M = SLP
Logarithmic: y = b + m * ln x	ln x	y	b	m
Exponential: y = b * e^(m*x)	x	ln y	e^b	m
Inverse: y = b + m/x	1/x	y	b	m
Power: y = b * x^m	ln x	ln y	e^b	m
General Exponential: y = b * m^x	x	ln y	e^b	e^m
Simple Logistic: y = 1/(b + m * e^(-x))	e^(-x)	1/y	b	m
Square Root Linear: y = √(b + m * x)	x	y^2	b	m
Cosine: y = b + m*cos(ω(x – ϕ)) With ϕ = the point (x) nearest to zero where the trough or peak begins ω = (2*π)/period (radians) or ω = 360°/period (degrees)	cos(ω(x – ϕ))	y	b	m
Logarithmic-Linear-Exponential: y = ln(b + m * e^(-x))	e^(-x)	e^y	b	m
Square Root Quadratic: y = √(b + m * x^2)	x^2	y^2	b	m

Eddie

All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.