RPN: Absolute Value Equations with the HP 11C and DM41X

RPN: Absolute Value Equations with the HP 11C and DM41X

Introduction: Solving |z * w + y| = x

Today’s blog focuses on solving the absolute value equation:

|z * w + y| = x

for the variable w, and the values of z, y, and x are given and are on the Classic RPN stack.

For example, in the problem |5 * w + 7| = 2, the stack would be set up as:

t: (anything, it doesn’t matter)

z: 5

y: 7

x: 2

(This is why I set the variable as w instead of the usual x.)

One approach is to use memory registers and other uses only stack operations. Today’s algorithm focuses on the latter.

Caution: In the above equation, x will always be non-negative. The equation will never be valid if x is negative.

The Algebra

Solve for w:

|z * w + y| = x

This leads us to solve the two equations:

(I)

z * w + y = -x

z * w = -x – y

w = -x/z – y/z

Let w- = -x/z – y/z = (-x/z) + (-y/z)

(II)

z * w + y = x

z * w = x – y

w = x/z - y/z

Let w+ = x/z – y/z = (x/z) + (-y/z)

Then:

w- = -x/z – y/z

w- = (-x/z) + (-y/z)

w- = (-x/z) + (-y/z) + (-x/z) + (x/z)

w- = 2 * (-x/z) + (x/z) + (-y/z)

w- = 2 * (-x/z) + w+

RPN Code: HP 11C (adoptable for other RPN calculators)

LBL A	001	42, 21, 11	Program start
R↓	002	33
R↓	003	33
X<>Y	004	34
R↓	005	33
1/x	006	15
×	007	20
LST x	008	43, 36
X<>Y	009	34
R↓	010	33
×	011	20
LST x	012	43, 36
R↑	013	43, 33
X<>Y	014	34
R↓	015	33
X<>Y	016	34
CHS	017	16
X<>Y	018	34
+	019	40
ENTER	020	36
ENTER	021	36
LST x	022	43, 36
-	023	30
LST x	024	43, 36
-	025	30
RTN	026	43, 32	Program end

RPN Code: DM41X (HP 41C series, no module is required)

LBL “ASBEQ”

RDN

RDN

X<>Y

RDN

1/x

×

LAST X

X<>Y

RDN

×

LAST X

R↑

X<>Y

RDN

X<>Y

CHS

X<>Y

+

ENTER

ENTER

LAST X

-

LAST X

-

RTN

Notes:

* RDN is shown as R↓

* To enter R↑, press XEQ, ALPHA, R, SHIFT, ENTER, ALPHA

Subroutines Used

We used several techniques to manipulate the stack. They are presented below:

Rotate stack from X, Y, Z, T to Z, X, Y, T

R↓

R↓

X<>Y

R↓

Source:

Ball, John A. Algorithms for RPN Calculators John Wiley & Sons: New York. 1978. ISBN 0-471-03070-8. pg. 78

Multiply Y and Z by 1/X. Stack: X, Y, Z, T → X, Y/X, Z/X, T

1/x

×

LAST X

X<>Y

R↓

×

LAST X

R↑

X<>Y

Change X, Y to Y+ X, Y – X

+

ENTER

ENTER

LAST X

-

LAST X

-

The resulting stack is:

z

z

y + x

y – x

Doing LAST X, -, twice is effectively subtracting whatever is in L register twice.

Examples

|4 * w – 6| = 5

Stack:

z: 4

y: -6

x: 5

Results:

y: 2.75

x: 0.25

|2 * w + 5| = 3

Stack:

z: 2

y: 5

x: 3

Results:

y: -1

x: -4

Eddie

All original content copyright, © 2011-2026. 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.

RPN: HP 11C: Transferring Between Bases (Common/Natural)

RPN: HP 11C: Transferring Between Bases (Common/Natural)

All algorithms were tested with the HP 11C.

Between the Exponential Function and Common-Antilog

e^α = 10^ß

Given α, what is ß?

ß = log(e^α)

RPN:

<input α>

e^x

LOG

Examples (Fix 6):

e^1.05 = 10^ß

ß ≈ 0.456009

e^(-2.2) = 10^ß

ß ≈ -0.955448

Given ß, what is α?

α = ln(10^ß)

RPN:

<input ß>

10^x

LN

Examples (Fix 6):

e^α = 10^5.4

α ≈ 12.433960

e^α = 10^0.366

α ≈ 0.842746

Between the Natural Logarithm and Common Logarithm

log α = ln ß

Given α, what is ß?

ẞ = exp(log α)

RPN:

<input α>

LOG

e^x

Examples (Fix 6):

log 17 = ln ß

ß ≈ 3.422766

log 317 = ln ß

ß ≈ 12.195405

Given ß, what is α?

α = 10^(ln ß)

RPN:

<input ß>

LN

10^x

Examples (Fix 6):

log α = ln 425

α ≈ 1,127,428.915

log α = ln 9.81

α ≈ 192.044677

Eddie

All original content copyright, © 2011-2026. 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.

fx-991 CW: Finding the Eigenvalues of a 2 x 2 Matrix

fx-991 CW: Finding the Eigenvalues of a 2 x 2 Matrix

Finding the Eigenvalues of a 2 x 2 Matrix

To find the eigenvalues of a matrix M, the characteristic polynomial must be solved in terms of λ:

det( M – λ * I) = 0

where:

M = a square matrix of dimension n x n

I = an identity matrix of size n x n.

The identity matrix is a square matrix in which all elements have a value 0 except the diagonal elements, which have the value 1.

A 2 x 2 identity matrix: [ [ 1, 0 ] [ 0, 1 ] ]

A 3 x 3 identity matrix: [ [ 1, 0, 0 ] [ 0, 1, 0 ] [ 0, 0, 1 ] ]

For a 2 x 2 matrix:

M = [ [ a, b ] [ c , d ] ]

The characteristic polynomial used to find the eigenvalues are:

det( M – λ * I) = 0

det( [ [ a, b ] [ c , d ] ] - λ * [ [ 1, 0 ] [ 0, 1 ] ] ) = 0

det( [ [ a, b ] [ c , d ] ] - [ [ λ, 0 ] [ 0, λ ] ] ) = 0

det( [ [ a - λ, b ] [ c , d - λ ] ] ) = 0

(a – λ) * (d – λ) – b * c = 0

λ^2 – (a + d) * λ + (a * d – b * c) = 0

Note:

trace(M) = a + d

det(M) = a * d – b * c

The characteristic equation to be solved is:

λ^2 – (a + d) * λ + (a * d – b * c) = 0

λ^2 – trace(M) * λ + det(M) = 0

Casio fx-991CW Algorithm

The algorithm will involve two apps: Matrix and Equation. Here is a way of finding the eigenvalues of a 2 x 2 matrices using only the fx-991 CW calculator without the need for writing anything down.

Note: The variables I use in the procedure is just for illustrative purposes. You can use any variables you want to designate the corner elements, the trace, and the determinant. The point is to be organized.

Variables used in this procedure:

A = upper-left element

B = lower-right element

C = trace = A + B

D = determinant

Settings: It is assumed that the MathI/MathO Input/Output mode and a+bi is selected for Complex result.

The screen shots are generated using the ClassPad Math (classpad.net) emulator for the fx-991CW and illustrates finding the eigenvalues of the matrix:

MatA = [ [ 4, 2 ] [ 5, 4 ] ]

Matrix App

Step 1: Press the [ HOME ] key, select the Matrix app.

Step 2: Use the TOOLS key to define a Matrix of dimension 2 x 2.

Step 3: Enter the matrix’s elements. Register each element is registered by using [ OK ] or [ EXE ]. Be careful not to press [ EXE ] or [ OK ] without entering a value first, as we need to keep the matrix editing screen up.

Step 4: Go to element (1,1) (upper-left hand corner), press the [ VARIABLE] key, go to variable A, press [ OK ], and select Store. Then go to element (2,2) (lower-right hand corner), press the [ VARIABLE] key, go to variable B, press [ OK ], and select Store.**

Step 5: Without entering or editing an element, press either [ EXE ] or [ OK ] to leave the matrix editor. When the message “Press [TOOLS] to define Matrix.” appears, we are in the calculation mode of the Matrix app.

Note: If you leave the matrix edit mode before storing the corner elements, you can go back into matrix edit mode by pressing [ TOOLS ], selecting your matrix, and then selecting Edit. You can check to see if corner values are stored by pressing [ VARIABLE ].

Step 6: Press [ SHIFT ] [ 4 ] (A) + [ SHIFT ] [ 7 ] ( B ) [ EXE ]. This calculates the matrix’s trace. Then press [ VARIABLE ], choose C, press [ OK ], choose Store.

Step 7: Press [ CATALOG ], select the Matrix sub-menu, then the Matrix Calc sub-menu, and select Determinant. Then use the catalog to grab the matrix and press [ EXE ] to calculate the matrix. Use the variable list to store the value in D (similar procedure in Step 6).

When transferring between apps, the values stored in the memory registers A-F, x, y, and z are retained, even when the fx-991CW is turned off.

Equations App

Step 8: Press [ HOME ] and select the Equation app and press [ OK ]. Select Polynomial, ax²+bx+c (2^nd order polynomial, quadratic equation).

Step 9: Enter the following coefficients: 1 x² – C x + D (note the minus sign on C).

Step 10: Press [ OK ]. The first eigenvalue is displayed. Press the down arrow ([↓]) to get the other eigenvalue. 

 An optional step is to use the [ VARIABLE ] key to store the results (like in E or F, for example). 

 Another optional step is to press [ FORMAT ], select Decimal to see the decimal approximation.

The results are:

C = trace = 8

D = determinant = 6

Eigenvalues:

λ1 = 4 + √10

λ2 = 4 - √10

Other Examples

Find the eigenvalues of Mat B = [ [ -8, 1 ] [ 16, 7 ] ]

Results:

C = trace = -1

D = determinant = -72

Eigenvalues:

λ1 = 8

λ2 = -9

Find the eigenvalues of Mat C = [ [ -5, -7 ] [ 3, - 2] ]

Results:

C = trace = -7

D = determinant = 31

Eigenvalues:

λ1 = (-7 + 5 * √3 * i) / 2

λ2 = (-7 - 5 * √3 * i) / 2

I hope you find this useful and beneficial. Until next time,

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.