--- DANSYSX: Complex cubic equations ---

2016

Both LibreOffice Calc and DANSYS are able to do basic complex arithmetic, but I want DANSYSX to be able to do advanced complex mathematics. There are a number of algorithms on Jean Pierre Moreau's excellent website (http://jean-pierre.moreau.pagesperso-orange.fr/index.html) that I want to adapt to this purpose.

The first is an algorithm to determine the roots of a cubic equation with complex coefficients. It uses the cubic formula (like the quadratic formula, but for cubic functions):

z(1...3) = cube root (-q/2 + square root(q^2/4 + p^3/27)) + cube root(-q/2 - square root(q^2/4 + p^3/27))

where: p = (c/a) - (b^2/3a^2)
           q = (2b^3/27a^3) + (d/a) - (bc)/3a^2)

I want the complex mathematics programs to be in their own folder so I use Tools>Macros>Organize Macros>LibreOffice Basic>Organizer to create the new folder, Complex under DANSYSX.ods>Standard. To begin programming, I open the LibreOffice Basic Macros dialog and select DANSYSX.ods>Standard>Complex and click the Edit button.

I am more comfortable programming in an environment where indexes run from 1 to whatever (instead of beginning at 0), so I add the Option Base 1 to the top of the editor Window, and change the name of the blank macro from Main to CompCube. I will also want the Macro to be a function so I change the header from Sub to Function and the result looks like this:

REM  *****  BASIC  *****

Option Base 1
Function CompCube

End Function

Now, everything is ready for me to start programming. I have to decide what I want my program to look like - the inputs and outputs. And I want to add a descriptive comment to the top of the code.

Looking at the Moreau algorithm, I see that I will need to input four complex numbers and I want to stick to the LibreOffice convention of designating complex numbers as strings of the form a±bi, so I want to pass four strings to the function. I will also want to output three complex numbers because a cubic equation will have up to three unique roots. So I'll start with the following header:

REM  *****  BASIC  *****

Option Base 1
Function CompCube(Z1 AS STRING, Z2 AS STRING, Z3 AS STRING, Z4 AS STRING)
'The following code determines the roots of a cubic equation
'defined by four complex coefficients:
'a z^3 + b z^2 + c z + d = 0
'The algorithm is an adaptation of one (CRoot3) from the Jean Pierre Moreau
'website: http://jean-pierre.moreau.pagesperso-orange.fr/Basic/croot3_bas.txt
'Several subroutines are used to perform elementary complex
'calculations and are shared with other DANSYSX programs.
'The global variables CMPN 1, 2, ..., 6 are also used to
'transfer complex variables between subroutines.

End Function

Most of the code will be in upper caps. This is not required by LibreOffice Basic, but I have found that I catch errors a lot quicker if everything is capitalized.

The Function statement names the program CompCube. The "Comp" part is a convention I established in DANSYS. Functions that deal with complex numbers begin with "Comp". The variables passed to the function (also called "parameters" or "arguments") are defined in the parentheses following the function name.

One of the flaws in LibreOffice Basic (all programming languages have weak spots) is that you can't transfer matrix values to and from functions and subroutines. (When I say "can't", I mean that I looked in all the resources I could find and tried to figure it out and couldn't - there may be a way but it remains a mystery to me.) I will have to transfer two value matrices (representing complex numbers), so I will store each value in a 1x2 array called CMPN1, CMPN2, to CMPN6. These arrays are defined as global variables (variables that can be used by all routines in DANSYSX and that won't lose their values between calls) which as designated in the Structures folder as, for instance:

Global CMPN1(1,2)

This generates a global array that has 1 row and 2 columns.

Global variables have to be defined outside of any function or subroutine. I collect them in the Structures folder.

I will also have to output three strings representing complex numbers so, to call this function from a spreadsheet, the user will select a 3x1 range of cells and type "=CompCube()' Inside the parentheses will either be four strings or the addresses of four cells containing strings.

I have written programs before that take complex numbers and have saved the code to do it in a KeyNote file of other handy code snippets that I use over and over. I will do that in the initialization section of the program.

Now, I go through the original code and figure out all the variables that I will need and declare them. At first guess, I come up with:

DIM A(2), B(2), C(2), D(2), U(2), V(2), W(2)
DIM DET(2), P(2), Q(2), SDET(2), U1(2), U2(2)
DIM V1(2), V2(2), ZT1(2), ZT2(2), Z(9,2)

The DIM statement is simple. It is just DIM followed by a list of variables separated by commas.

All the variables but the last are arrays with two places. Those are complex numbers.

Using my tried and true complex number decoders in the initialization section to load A, B, C, and D from Z1, Z2, Z3, and Z4, I wrote:

DIM A(2), B(2), C(2), D(2), U(2), V(2), W(2)
DIM DET(2), P(2), Q(2), SDET(2), U1(2), U2(2)
DIM V1(2), V2(2) ZT1(2), ZT2(2), Z(9,2)
DIM I AS INTEGER, ST AS STRING
DIM OutMat(3,1) AS STRING

OutMat is a convention I use to output arrays. I load all the data I want to display into an array and I set the function equal to that array. In this case, I'll be displaying three complex number strings - 3 rows and 1 column.

Notice that the last two variables are explicitly specified to be integers. LibreOffice allows you to specify variables as INTEGER (between values of -32768 and 32767), LONG (integers between -2147483648 and 2147483647), SINGLE (floating-point values between 3.402823E38 and 1.401298R-45), DOUBLE (floating-point values between 1.79769313486232E308 and 4.94065645841247E-324), CURRENCY (64 bit numbers displayed as fixed decimal numbers with 15 non-whole and 4 decimal places ranging from -922337203685477.5808 to 922337203685477.5807), STRING (character strings holding up to 65,535 characters), BOOLEAN (stores TRUE/1 or FALSE/0), and DATE (date and time values stores in an internal format.) Constants can also be declared using the keyword CONST (instead of DIM).

Variable types can also be specified by tagging a declaration symbol to the end of a variable name as follows:

Single !
Double #
Currency @
String $
For instance, DIM Money@ would declare the variable Money as the type Currency.

ST=""
FOR I=1 TO LEN(Z1)
IF MID(Z1,I,1)="+" OR MID(Z1,I,1)="-" THEN
A(1)=VAL(ST)
ST=""
END IF
ST=ST & MID(Z1,I,1)
NEXT I
A(2)=VAL(ST)

ST=""
FOR I=1 TO LEN(Z2)
IF MID(Z2,I,1)="+" OR MID(Z2,I,1)="-" THEN
B(1)=VAL(ST)
ST=""
END IF
ST=ST & MID(Z2,I,1)
NEXT I
B(2)=VAL(ST)

ST=""
FOR I=1 TO LEN(Z3)
IF MID(Z3,I,1)="+" OR MID(Z3,I,1)="-" THEN
C(1)=VAL(ST)
ST=""
END IF
ST=ST & MID(Z3,I,1)
NEXT I
C(2)=VAL(ST)

ST=""
FOR I=1 TO LEN(Z4)
IF MID(Z4,I,1)="+" OR MID(Z4,I,1)="-" THEN
D(1)=VAL(ST)
ST=""
END IF
ST=ST & MID(Z4,I,1)
NEXT I
D(2)=VAL(ST)

MSGBOX A(1)
MSGBOX A(2)
MSGBOX B(1)
MSGBOX B(2)
MSGBOX C(1)
MSGBOX C(2)
MSGBOX D(1)
MSGBOX D(2)

Each block of code beginning with ST works like this: First a blank string is generated in the string variable ST. Then, the following loop looks at each character of an input string Z. If the character is a + or a -, the first cell of a two place array is loaded with the numerical value of whatever is in ST and ST is emptied. If the character is anything else, it is tagged onto the right end of the string in ST and the next character comes up for review.  This goes on until each character of the input string (LEN(Z) is the number of characters in string Z) has been considered, then the numerical value of whatever is left in ST is loaded into the second cell of the two place variable.

The MID(Z,I,1) extracts 1 character from the ith position of string Z.

MSGBOX is a command that causes the following to be displayed in a message box. In this case, I use it to make sure that the input values are being transferred to the complex value arrays. They work so I'm happy. The MSGBOX commands will be deleted.

The next step requires that I add a subroutine that performs complex division and one that performs complex multiplication. I design these so that they work on the complex numbers stored in CMPN1 and CMPN2 and return an answer in CMPN3. You notice that the CMPN global variables act like a stack of registers for complex numbers. Here are CDIV and CMUL:

SUB CDIV()
'Subroutine to divide two complex numbers.
DIM CC(2), D AS DOUBLE, Z(2)

'Determine the value of the denominator
D=CMPN2(1,1)*CMPN2(1,1)+CMPN2(1,2)*CMPN2(1,2)
IF D<1e-15 THEN
MSGBOX "Complex divisiion by 0"
EXIT SUB
ELSE
CC(1)=CMPN2(1,1)
CC(2)=-CMPN2(1,2)
CMPN4(1,1)=CMPN1(1,1)
CMPN4(1,2)=CMPN1(1,2)
CMPN5(1,1)=CMPN2(1,1)
CMPN5(1,2)=CMPN2(1,2)
CMPN2(1,1)=CC(1)
CMPN2(1,2)=CC(2)
CMUL
Z(1)=CMPN3(1,1)
Z(2)=CMPN3(1,2)
CMPN2(1,1)=CMPN5(1,1)
CMPN2(1,2)=CMPN5(1,2)
CMPN3(1,1)=CMPN3(1,1)/D
CMPN3(1,2)=CMPN3(1,2)/D
END IF
END SUB

SUB CMUL()
'DETERMINE THE PRODUCT OF CMPN1 AND CMPN2
'SAVE THE RESULT IN CMPN3
CMPN3(1,1)=CMPN1(1,1)*CMPN2(1,1)-CMPN1(1,2)*CMPN2(1,2)
CMPN3(1,2)=CMPN1(1,1)*CMPN2(1,2)+CMPN1(1,2)*CMPN2(1,1)
END SUB

You will recognize most of the arithmetic operators. * is LibreOffice's multiplication operator (instead of x) and an exponent is denoted by the symbol ^. For instance, x squared would look like: x^2.

The difference between a function and a sub (or subroutine) is that the function returns a value and a sub does something but does not output a value. For instance, CMUL calculates the product of two complex numbers and loads them into a global array, but does not return the product to the program that called the sub. That program has to get the value from the global array. Here's the part of CompCube that uses CDIV and retrieves the result:

CMPN1(1,1)=C(1)
CMPN1(1,2)=C(2)
CMPN2(1,1)=A(1)
CMPN2(1,2)=A(2)
CDIV
U(1)=CMPN3(1,1)
U(2)=CMPN3(1,2)
MSGBOX U(1)
MSGBOX U(2)

This divides the complex number in C by the one in A. In order for CDIV to work C has to be loaded into CMPN1 and A has to be loaded into CMPN2. To call CDIV, I simply type CDIV. Then to get the result back as U, I load CMPN3 into U. Again, the MSGBOX commands display the result and, after I'm satisfied that CDIV works, I'll delete them. That's called "debugging". I prefer to debug as I go.

Now, to complete the calculation of P:

CMPN1(1,1)=B(1)
CMPN1(1,2)=B(2)
CMPN2(1,1)=B(1)
CMPN2(1,2)=B(2)
CMUL
V(1)=CMPN3(1,1)
V(2)=CMPN3(1,2)

CMPN1(1,1)=A(1)
CMPN1(1,2)=A(2)
CMPN2(1,1)=A(1)
CMPN2(1,2)=A(2)
CMUL
W(1)=CMPN3(1,1)
W(2)=CMPN3(1,2)

W(1)=3*W(1)
W(2)=3*W(2)

CMPN1(1,1)=V(1)
CMPN1(1,2)=V(2)
CMPN2(1,1)=W(1)
CMPN2(1,2)=W(2)
CDIV
ZT1(1)=CMPN3(1,1)
ZT1(2)=CMPN3(1,2)
P(1)=U(1)-ZT1(1)
P(2)=U(2)-ZT1(2)

You should be able to follow that. It's the same process I used to calculate c/a. It checks out, so I can continue.

The next piece of the puzzle is Q. It's a straight forward but long calculation (you see why they just teach the quadratic formula in high school and leave the cubic and quartic monsters for folks that are too curious for their own good.) That section of code looks like:

'Calculate q=2b^3/27a^3 + d/a - bc/3a^2
CMPN1(1,1)=B(1)
CMPN1(1,2)=B(2)
CMPN2(1,1)=V(1)
CMPN2(1,2)=V(2)
CMUL
W(1)=CMPN3(1,1)
W(2)=CMPN3(1,2)

W(1)=2*W(1)
W(2)=2*W(2)

CMPN1(1,1)=A(1)
CMPN1(1,2)=A(2)
CMPN2(1,1)=A(1)
CMPN2(1,2)=A(2)
CMUL
U(1)=CMPN3(1,1)
U(2)=CMPN3(1,2)

CMPN1(1,1)=U(1)
CMPN1(1,2)=U(2)
CMPN2(1,1)=A(1)
CMPN2(1,2)=A(2)
CMUL
V(1)=CMPN3(1,1)
V(2)=CMPN3(1,2)

V(1)=27*V(1)
V(2)=27*V(2)

CMPN1(1,1)=W(1)
CMPN1(1,2)=W(2)
CMPN2(1,1)=V(1)
CMPN2(1,2)=V(2)
CDIV
Q(1)=CMPN3(1,1)
Q(2)=CMPN3(1,2)

CMPN1(1,1)=D(1)
CMPN1(1,2)=D(2)
CMPN2(1,1)=A(1)
CMPN2(1,2)=A(2)
CDIV
U(1)=CMPN3(1,1)
U(2)=CMPN3(1,2)

Q(1)=Q(1)+U(1)
Q(2)=Q(2)+U(2)

CMPN1(1,1)=B(1)
CMPN1(1,2)=B(2)
CMPN2(1,1)=C(1)
CMPN2(1,2)=C(2)
CMUL
U(1)=CMPN3(1,1)
U(2)=CMPN3(1,2)

CMPN1(1,1)=A(1)
CMPN1(1,2)=A(2)
CMPN2(1,1)=A(1)
CMPN2(1,2)=A(2)
CMUL
V(1)=CMPN3(1,1)
V(2)=CMPN3(1,2)

V(1)=3*V(1)
V(2)=3*V(2)

CMPN1(1,1)=U(1)
CMPN1(1,2)=U(2)
CMPN2(1,1)=V(1)
CMPN2(1,2)=V(2)
CDIV
W(1)=CMPN3(1,1)
W(2)=CMPN3(1,2)

Q(1)=Q(1)-W(1)
Q(2)=Q(2)-W(2)

It's very repetitive and allows me to copy, paste,and modify most of it from earlier code. Then there is det:


'Calculate DET = q^2/4 + p^3/27

CMPN1(1,1)=Q(1)
CMPN1(1,2)=Q(2)
CMPN2(1,1)=Q(1)
CMPN2(1,2)=Q(2)
CMUL
U(1)=CMPN3(1,1)
U(2)=CMPN3(1,2)
U(1)=U(1)/4
U(2)=U(2)/4

CMPN1(1,1)=P(1)
CMPN1(1,2)=P(2)
CMPN2(1,1)=P(1)
CMPN2(1,2)=P(2)
CMUL
V(1)=CMPN3(1,1)
V(2)=CMPN3(1,2)

CMPN1(1,1)=P(1)
CMPN1(1,2)=P(2)
CMPN2(1,1)=V(1)
CMPN2(1,2)=V(2)
CMUL
W(1)=CMPN3(1,1)
W(2)=CMPN3(1,2)
W(1)=W(1)/27
W(2)=W(2)/27

DET(1)=U(1)+W(1)
DET(2)=U(2)+W(2)

Now, I need the square root of DET, and for that, I need the following subroutine that calculates the square root of the complex number stored in CMPN1.

SUB CSQRT()
'Determine the square root of CMPN1
'Save the result in CMPN3.
DIM R AS DOUBLE
R=SQR(CMPN1(1,1)*CMPN1(1,1)+CMPN1(1,2)*CMPN1(1,2))
CMPN3(1,1)=SQR((R+CMPN1(1,1))/2)
CMPN3(1,2)=SQR((R-CMPN1(1,1))/2)
IF CMPN1(1,2)<0 THEN CMPN3(1,2)=-CMPN3(1,2)
END SUB

The result is stored in CMPN3. SQR is the LibreOffice command to take the square root of the value in parentheses following it.

Now, the section of CompCube that calculates the square root of DET is:

'Calculate the square root of DET

CMPN1(1,1)=DET(1)
CMPN2(1,2)=DET(2)
CSQRT
SDET(1)=CMPN3(1,1)
SDET(2)=CMPN3(1,2)

V(1)=-Q(1)/2+SDET(1)
V(2)=-Q(2)/2+SDET(2)

If you look back up at the cubic formula, you will see where those last two statements came from.

So, I need the three cube roots of V and to get that, I need a subroutine to calculate the cube root of CMPN1 and store the results in CMPN2, CMPN3, and CMPN4, and that subroutine will need a function to determine the phase of CMPN1.

Here's the function CPHASE:

FUNCTION CPHASE(NUM,DENOM)
'Returns the phase of a complex number between -pi and pi
DIM PPI AS DOUBLE, CP AS DOUBLE
PPI=4*ATN(1)
IF ABS(DENOM)<1e-15 THEN
IF NUM<0 THEN
CPHASE=-PP1/2
ELSE
CPHASE=PPI/2
END IF
ELSE
CP=ATN(NUM/DENOM)
IF DENOM<0 THEN
CPHASE=CPHASE+PPI
ELSE
CPHASE=CP
END IF
END IF
END FUNCTION

Notice that the header says "FUNCTION" instead of "SUB". This code takes two arguments: NUM and DENOM, and returns a value. The value is determined by the "CPHASE=" statements. When you set a function's name equal to a value, you are outputing the value. To call the function, you set some variable equal to the function name (You'll see that below). ATN is a LibreOffice Basic function that calculates the arctangent of some number. ABS is the function that returns the absolute value of a number. This function is basically a decisiion tree that determines what value CPHASE will take.

If DENOM is less than 1E-15 (or 1 x 10-15), then if NUM is negative CPHASE is -pi/2, else CPHASE is pi/2. If DENOM is more than 1E-15, then if DENOM is negative, CPHASE is the arctangent of NUM/DENOM + pi, otherwise, it is just the arctangent of NUM/DENOM.

Here's the subroutine CCUBRT, which calculates the cube roots of a complex number (there will be three cube roots).

SUB CCUBRT()
'Calculates the three cube roots of CMPN1.
'The results are stored in CMPN2, CMPN3, and CMPN4
DIM PPI AS DOUBLE, R AS DOUBLE, T AS DOUBLE, TT AS DOUBLE

PPI=4*ATN(1)
R=SQR(CMPN1(1,1)*CMPN1(1,1)+CMPN1(1,2)*CMPN1(1,2))
R=R^(1/3)
T=CPHASE(CMPN1(1,2),CMPN1(1,1))
T=T/3
CMPN2(1,1)=R*COS(T)
CMPN2(1,2)=R*SIN(T)
TT=T-(2*PPI/3)
CMPN3(1,1)=R*COS(TT)
CMPN3(1,2)=R*SIN(TT)
TT=T+(2*PPI/3)
CMPN4(1,1)=R*COS(TT)
CMPN4(1,2)=R*SIN(TT)
END SUB

The statement that calls CPHASE is:

T=CPHASE(CMPN1(1,2),CMPN1(1,1))

It feeds the values stored in CMPN1 to the function and transfers the result to the variable T.

The part of CompCube that calculates the cubed roots of V is:

CMPN1(1,1)=V(1)
CMPN1(1,2)=V(2)
CCUBRT
U(1)=CMPN2(1,1)
U(2)=CMPN2(1,2)
U1(1)=CMPN3(1,1)
U1(2)=CMPN3(1,2)
U2(1)=CMPN4(1,1)
U2(2)=CMPN4(1,2)

U, U1, and U2 stores the resulting cubed roots.

And the part that calculates the cube roots of W is almost exactly the same except for the variables used:

W(1)=-Q(1)/2-SDET(1)
W(2)=-Q(2)/2-SDET(2)

CMPN1(1,1)=W(1)
CMPN1(1,2)=W(2)
CCUBRT
V(1)=CMPN2(1,1)
V(2)=CMPN2(1,2)
V1(1)=CMPN3(1,1)
V1(2)=CMPN3(1,2)
V2(1)=CMPN4(1,1)
V2(2)=CMPN4(1,2)

The next section collects all the possible roots of the equation. Only three will be appropriate, but the nine possible roots will all have to be tested.

'Find all possible solutions to the formula.

Z(1,1)=U(1)+V(1)
Z(1,2)=U(2)+V(2)
Z(2,1)=U(1)+V1(1)
Z(2,2)=U(2)+V1(2)
Z(3,1)=U(1)+V2(1)
Z(3,2)=U(2)+V2(2)
Z(4,1)=U1(1)+V(1)
Z(4,2)=U1(2)+V(2)
Z(5,1)=U1(1)+V1(1)
Z(5,2)=U1(2)+V1(2)
Z(6,1)=U1(1)+V2(1)
Z(6,2)=U1(2)+V2(2)
Z(7,1)=U2(1)+V(1)
Z(7,2)=U2(2)+V(2)
Z(8,1)=U2(1)+V1(1)
Z(8,2)=U2(2)+V1(2)
Z(9,1)=U2(1)+V2(1)
Z(9,2)=U2(2)+V2(2)

To test these values, I have to use them to evaluate the cubic equation. I use another subroutine for that, and that subroutine requires a function to return the absolute value of the complex number. I also realized that I needed to save the complex coefficients for this routine so I went back and did that in MatArray1, a global array I use to transfer matrices around.

'Load equation into MatArray1
REDIM MatArray1(4,2)
MatArray1(1,1)=A(1)
MatArray1(1,2)=A(2)
MatArray1(2,1)=B(1)
MatArray1(2,2)=B(2)
MatArray1(3,1)=C(1)
MatArray1(3,2)=C(2)
MatArray1(4,1)=D(1)
MatArray1(4,2)=D(1)

The sizes of dynamic arrays like the global MatArrays can be changed any time using the REDIM (redimension) command.

Here are the test routine and absolute value function:

FUNCTION CABS(ARG1, ARG2)
'This function returns the absolute value of a complex number
'ARG1+ARG2 i
CABS=SQR(ARG1*ARG1+AGR2*ARG2)
END FUNCTION

SUB CTEST()
'This subroutine tests a solution set of a cubic equation for appropriate
'values.
'The output is z1=a z^3 + b x^2 + c z + d
'In other words, it evaluates the original cubic equation.
'It gets the equation from MatArray1.
'The value of the complex variable is in CMPN4.
DIM U(2), V(2), W(2), Z1(2)

CMPN1(1,1)=MatArray1(1,1)
CMPN1(1,2)=MatArray1(1,2)
CMPN1(2,1)=CMPN4(1,1)
CMPN1(2,2)=CMPN4(1,2)
CMUL
U(1)=CMPN3(1,1)
U(2)=CMPN3(1,2)

CMPN1(1,1)=CMPN4(1,1)
CMPN1(1,2)=CMPN4(1,2)
CMPN1(2,1)=U(1)
CMPN1(2,2)=U(2)
CMUL
V(1)=CMPN3(1,1)
V(2)=CMPN3(1,2)

CMPN1(1,1)=CMPN4(1,1)
CMPN1(1,2)=CMPN4(1,2)
CMPN1(2,1)=V(1)
CMPN1(2,2)=V(2)
CMUL
W(1)=CMPN3(1,1)
W(2)=CMPN3(1,2)

Z1(1)=W(1)
Z1(2)=W(2)

CMPN1(1,1)=MatArray1(2,1)
CMPN1(1,2)=MatArray1(2,2)
CMPN1(2,1)=CMPN4(1,1)
CMPN1(2,2)=CMPN4(1,2)
CMUL
U(1)=CMPN3(1,1)
U(2)=CMPN3(1,2)

CMPN1(1,1)=U(1)
CMPN1(1,2)=U(2)
CMPN1(2,1)=CMPN4(1,1)
CMPN1(2,2)=CMPN4(1,2)
CMUL
V(1)=CMPN3(1,1)
V(2)=CMPN3(1,2)

Z1(1)=Z1(1)+V(1)
Z1(2)=Z1(2)+V(2)

CMPN1(1,1)=MatArray1(3,1)
CMPN1(1,2)=MatArray1(3,2)
CMPN1(2,1)=CMPN4(1,1)
CMPN1(2,2)=CMPN4(1,2)
CMUL
U(1)=CMPN3(1,1)
U(2)=CMPN3(1,2)

Z1(1)=Z1(1)+U(1)+MatArray1(4,1)
Z1(2)=Z1(2)+U(2)+MatArray1(4,2)

CMPN3(1)=Z1(1)
CMPN3(2)=Z1(2)

END SUB

After all that I had some small bugs (small bugs = disaster) and had to do some fine tuning. Debugging got rid of the bugs and my test subroutine was too sensitive for the tiny round off errors I produced so I had to change the test criterion:

IF X<0.001 THEN
to the more liberal:

IF X<10 THEN

Now, it works, and it work within input precision.

I'll be programming a lot more and it will be available at the Timeline. The basics of LibreOffice Basic are here and they should give you enough to figure out what I'm doing in future programs. I have a large, complex project coming up that will give me a chance to share some of my problem solving techniques with you.

--- Computer software ---

2016

There are a few software packages that I like and a few sites that also make me happy and entertain me. I've already talked about a few in the article, "Open Source and Freeware." Here are a few more.

I have a soft spot (somewhere in my head, I'm sure.) for the old fashion Windows Help files and you can still compose your own with ShalomHelp. But you'll have a hard time connecting your help files to anything. Just use ShalomHelp to read them. Anyway, ShalomHelp is available at several download sites such as Softonic (http://shalom-help-maker-shm.en.softonic.com). Be careful with these download sites. Some are not monitored very well and you are liable to pick up something you don't want. Also, many of them try to sneak in some extra software with the one you're downloading so be sure and read the information on the installers carefully.

Except for "riders" (those extra programs that get snuck in), I've never had any problems with Softonic, Tucows, or CNET. That doesn't mean they never have corrupted software - it just means I've never had a problem with them.

Python is my current favorite non-LibreOffice Basic programming language. It's both fun and powerful, which is why it's also popular. It gives you constant feedback while you compose. If you type in 5+7, it will immediately return 12. The caveat is that older versions of Python are not necessarily compatible with current versions so, if you're going to use a textbook or user guide to learn Python, make sure the text goes with the version of Python that you have. Here's their site:

https://www.python.org

Since I'm a statistician, Free Statistics would reasonably be one of my favorite software venues. This site offers a lot of surprisingly powerful free, open source, and limited edition statistics software at:

http://freestatistics.altervista.org

--- Useful things ---

2016

An advantage of walking is that you only need what you can carry, or, put another way, you only carry what you can carry. My years as a backpacker comes in handy. It's surprising how much you learn when you have to worry about size and weight.

I find that I don't have to worry too much. In the first place, I'm at that age where I can wear just about anything I want without drawing attention ("When I'm old, I shall wear purple...."). My main consideration is that I'm in Colorado which is a meteorologist's nightmare. I walked to the shopping center (less than a mile) a couple of days ago. When I left, it was warm and fair. I was chased home by a thunderstorm. That night, some folks around here was coerced into wearing sleeves. From September to June, if I go on a long hike, I have to plan to remove layers as the day goes on, and I'm never sure if I will need to. Luckily, I get hot in the upper 50s when I'm active.

It's been below freezing here and I've always worn overalls and coveralls hiking. A flannel shirt, sweater, and leather coat is the most I've ever needed for warmth. You, of course, need to plan for your weather and your body heat.

I wear regular clothes and inexpensive shoes. Socks make a difference. Shelling out for the microfibre wool and such is worthwhile if you don't like blisters and such, but I go the way of the bandaid and moleskin when I feel a hotspot developing. I've had a lifelong relationship with pain and it doesn't bother me much. Pain avoidance, I find, is a individualistic thing.

I have caps for short hikes and big, floppy expedition hats for long hikes. I've had cataracts removed from both eyes and Colorado's sun is not the same sun I knew in Alabama. It's closer and less filtered. I try to keep the sun out of my eyes and I find most sunglasses to be ineffective.

Secondly, hiking requires something to carry everything in. I'm a Rob Liefeld hiker. I like pockets and lots of them, which is one thing that draws me to overalls. I hate digging through a backpack to find a piece of equipment.

I have two backpacks, one for short, nontechnical hikes and just walking to the store - the other for long, technical hikes and for hauling back 60 pounds of groceries from the store.

I'll be distinguishing between technical hikes and nontechnical hikes. Technical hikes are project oriented, requires equipment, and usually involves the collection of quantitative data. Nontechnical hikes are more casual, requires little equipment, and usually involves observation.

My big backpack can handle a laptop and I have several laptop size inserts with pockets and straps, that can be used to outfit myself with a portable laboratory. American Science and Surplus (bless their little bitty comedic hearts) often carry those kinds of things at prices that a retiree can handle.

I also keep a couple of waist packs (or "fanny packs", if you must) for my photographic material. One of those I almost never carry on hikes because it carries equipment for the SLRD camera, which I only use for portrait photography. I would not want to have to carry that huge thing on a hike (although I wouldn't mind driving up to a ridge one afternoon with it.)

I carry two cameras. I use my regular digital camera for most of my photography. That saves batteries for my phone. That,  I use for much more than photographs. My phone camera is used for closeups, telephotos, and microphotos. It's much better and has much better stabilization than my regular camera.

The phone also carries a library of apps, guidebooks and maps that I use for technical hikes.

I like to be a model of what people on a limited income can do to enjoy their world so I try to avoid expensive equipment and activities. My most expensive piece of equipment is my laptop which, admittedly, is a little pricey, although not nearly so much as when I bought it. But I assume that, if you're reading this, you have access to a computer anyway. My computer is my home lab. I have many pieces of equipment that plugs into the USB port, things that, a few years ago, would have cost a laboratory enough that they would have to save up awhile to buy it. For instance, you can now buy a spectroscope for a computer or smartphone for less than ten dollars.

According to where you live, rain gear is important. In Colorado, definitely. In the southeast, it's a sometimes thing. And in Arizona, do you ever need it? A good, light rain jacket is inexpensive and easy to pack. I've never had any need at all for rain pants. As long as my shoes are relatively waterproof, I'm happy. Actually, I don't mind rain or being wet as long as I'm not also cold. Years working outside in the rain has made me rather blase about most weather conditions.

Werewolves don't get sick easily and heal quickly, but I carry a small first aid kit anyway, mostly for blisters. Plantar blisters plague me and I can slap a bandaid of swatch of moleskin on one and I'm good to go. I carry only what I think I might use. If I get a cut, I'll wipe it off with a moist towelette and smear on some Neosporin. I don't even cover it. If It's bleeding, I let it bleed. There's no better antiseptic/antibiotic in the world than blood. On long hikes, I carry suntan lotion (the spray on greaseless kind) because I'm light skinned and tend to burn easily, and I carry bug spray because biting insects find me tasty.

If I'm gone long, I carry a roll of toilet paper. There are many rest stops on Bear Creek - other places, maybe not so much.

I've taken to wearing a biker's mirror on my glasses, not so much for the bikers; in my area, they're very polite and warn you when they're coming up behind you, but for stalkers. We have a few furry ones who are not very much trouble but, I would want a photo. Briefly, I'm out there to observe my world and I want as much coverage as I can get.

Other than that, I choose the equipment I will need for the project I have at hand. I choose inexpensive and compact tools and I pack so that I can get to what I want when I want it.

Uh, I almost forgot one of the most important pieces of hiking equipment. I always carry an old, worn t-shirt tucked into my back pocket. I sweat a lot and that works much better than a handkerchief to keep the sweat out of my eyes. I can also twist it between the sleeves, flip it over my head so the tail falls over the back of my neck, and tie the twisted part around my forehead for a quick do-rag.

--- Lectures for lifelong learning ---

2016

I've mentioned that I'm a lifelong learner. That's my hobby but saying that is like saying I only eat food. My idea of lifelong learning is to try to pick up as much and as varied information as I can in this life. Adventuring, for me is just another phase of lifelong learning.

My bookmarks for online sources of lectures make up a long list. Still, I have some favorites - more coverage, better quality - and I'll share a few with you. Four are sources of free downloads (called "opencourseware") and one is commercial but well worth every cent.

Many colleges and universities make their content available to anyone. Only their students can get credit for taking the courses but, for lifelong learners, credit isn't the important part - it's the experience and the information. MIT is preeminent. They provide opencourseware in just about every field from sailing to differential equations and medical technology. The courses vary from "Here's the books we use. Read them." to full video lectures with textbook, tests, and software. But the video lectures they do have are excellent. Here's a link:

http://ocw.mit.edu/index.htm

Another excellent source of lectures is Gresham College. They have several series of special lectures, not particularly opencourseware. The topics are varied but the speakers are active in their fields, informative, and engaging. Gresham College can be found here:

http://www.gresham.ac.uk/attend

Academic Earth brings together many of the best open courses from many schools. They try to keep the quality high and, for the most part, succeed. Here's a link to that site:

http://academicearth.org

Another source that brings together different sources is the Wikipedia article on opencourseware:

https://en.wikipedia.org/wiki/OpenCourseWare

It provides a worldwide sample of educational opportunities.

The Teaching Company is a commercial source of courses. The lecture series are rather expensive but every item goes on sale at least once during the year and, then, the prices are very affordable. The company auditions their lecturers and use customer feedback to choose future topics and lecturers. One thing I like about these products is that the lecturers are active in their fields and I almost always hear things that I've never heard before. I've never bought a Teaching Company product and regretted the purchase. Here is their home site:

http://www.thegreatcourses.com