This program is freeware, but should be registered by sending me an email at SSchmidt@nospam.dk. This is merely so that I can track the use of my software – no licensing fee is necessary. You are free to distribute this program to anyone, as long as this document is included and they register by email as well.

I cannot take responsibility for any damage or data loss caused by this program – it is written in 100% SysRPL and there could be bugs in it. This also means that it will not run on the HP48 series.

If you have any suggestions, additions or find any bugs in the code, you’re welcome to contact me per the above-mentioned email. Please include details about ROM revision and flag settings if you’re reporting a bug.

Credits

The library is coded and compiled directly on the HP49G, which has proven to be a great tool for this too. A thanks goes to ACO for making it possible – including these programming tools on the calculator itself has made the job a lot easier.

Mr. Bernard Parisse (CAS author on the HP49G) has been a big help explaining his software to me. I have had especially great use of three pieces of software on the HP49G, which are Emacs, Keyman & Nosy.

Requirements & Installation

You need to copy the library (library #1125) to the calculator (with HPComm for example) and store it in a port (any port should do).

To store it in port 2 for example, you will need to recall the library to the stack, type 2 and STO.

Hold down ON and press F3, release both keys and the calc should warm boot (warning: you’ll loose all stack contents, but calc memory will remain intact). When this is done, your port 2 should contain the entry for the library (press left shift APPS to enter the Filer):

Running BYTES on the library on the stack should yield #C37h and 3729 bytes.

This document is showing stack syntax for RPN mode, but algebraic mode works too.

Polynomial

This library is a small set of commands to handle polynomials as vectors of coefficients – in this document called vector polynomials, and denoted [ coefficients ]. On several occasions on comp.sys.hp48, there has been requested a way to extract the coefficients from a polynomial. This is now possible with this library, as it was on the HP48 with NeoPolys. I’m not trying to take any fame or glory away from NeoPolys, nor have a looked at the code therein – this is a totally independent development.

The commands in this library are listed below.

A®P

Algebraic to polynomial command:

The basic command used to convert an algebraic expression or equation into its two vector polynomials – the numerator and the denominator.

Level 2	Level 1	à	Level 2	Level 1
symbolic	global name	à	[ coefficients_Num ]	[ coefficients_Denom ]
number_x	global name	à	[ number_x ]	[1]

This command is indifferent to the algebraic form of the Level 2 argument, as different forms don’t affect performance much. The two conversions below are the same polynomial, the first written in a compact form, a form one would most likely see it in, the second is the expanded form, revealing that it is actually a 10^th degree polynomial, divided by a 7^th degree polynomial.

The first conversion is calculated in 1.91 seconds, and the second one is calculated in 1.51 seconds (yielding the same result). This would make it seem obvious that it is best to expand expressions before using A®P, but the picture is different when considering that the HP49G will spend 2.89 seconds expanding the first form to the second one. In short, it doesn’t matter what form you feed into A®P.

If the denominator is [1] you can delete it, since the numerator then represents the complete polynomial.

Equations are also valid input.

You can combine al sorts of coefficients, reals, complex numbers, integers and symbolics.

Irrational input is handled in a special way. The goal of A®P is that P®A always shall return the original expression, even though the coefficient vector contains the variable itself (e.g. X). Generally, an irrational part will always end up being the constant part of the vector.

When the vector polynomial contains the polynomial variable, as in the above two examples, it will generally not be valid to use the commands in this library that are variable dependent, namely PDER and PINT.

P®A

Polynomial to algebraic command:

The basic command used to convert a vector polynomial into its algebraic representation.

Level 2	Level 1	à	Level 1
[ coefficients ]	global name	à	symbolic_Polynomial

Basically, all it does is to set the global name to an iterating power, while multiplying it with the corresponding coefficient.

P2®A

Polynomial fraction to algebraic command:

Converts a vector polynomial numerator and denominator into its algebraic representation.

Level 3	Level 2	Level 1	à	Level 1
[ coefficients_Num ]	[ coefficients_Denom ]	global name	à	symbolic_Polynomial

This is the reciprocal to A®P.

P3®A

Polynomial partial fraction to algebraic command:

Converts a vector polynomial and a vector polynomial numerator/denominator into its algebraic representation.

Level 3	Level 2	Level 1	Level 3	à	Level 1
[ coeff ]	[ coeff_Num	[ coeff_Denom ]	global name	à	symbolic_Polynomial

This command can convert the output of PDIV into its algebraic representation.

RAT?

Rationality test command:

Tests whether an expression or equation is rational in a given variable. Possible outputs are true or false, denoted by 1. or 0. respectively.

Level 2	Level 1	à	Level 1
symbolic	global name	à	0. or 1.
number	global name	à	1.

Even though an expression is rational, doesn’t mean that it is a polynomial. It does mean however, that it is a polynomial fraction consisting of a numerator polynomial and a denominator polynomial in that variable. An expression can be rational in some variables and irrational in others.

PDEG

Polynomial degree command:

Returns the degree/order of a polynomial. It will accept an algebraic as well as a vector polynomial.

Level 2	Level 1	à	Level 2	Level 1
symbolic	global name	à	n_Deg,Num	n_Deg,Denom
	[ coefficients ]	à	n_Deg,Num	0

Since PDEG must return both the degree of the numerator as well as of the denominator, if it was given an algebraic, it also returns two values when given a vector polynomial. This is to make programmable use of it simple. If the input was a vector, then the Level 1 output is zero and can be discarded.

PMONIC

To monic polynomial command:

Converts a vector polynomial into its monic representation.

Level 1	à	Level 1
[ coefficients ]	à	[ coefficients_Monic ]

A monic polynomial is a polynomial whose leading coefficient is 1, so ultimately, this command simply divides all coefficients by the leading coefficient.

PSIMP

Simplify polynomial command:

Simplifies a vector polynomial. Basically, it will call EVAL on numeric coefficients (coefficients consisting of only reals, complex numbers or integers), and COLLECT on all other symbolic constructs. This is necessary because no commands in this library simplify their output. This means that vector polynomials can contain large constructs that can be simplified, as well as leading zeroes.

The reason that COLLECT is called instead of FACTOR, is that COLLECT only factors over the integers, which yields a “nicer” result:

à but à

à and à

The reason that COLLECT doesn’t get called upon numeric constructs is obvious:

à while à

Since COLLECT factors numeric constants in symbolic constructs, the result can get mangled. Notice the denominator of the second to last coefficient in the example below:

I considered calling SIMPLIFY before COLLECT to take care of transcendental functions, but I thought it a waste of time, since it’s rarely necessary.

PADD

Add polynomials command:

Adds two vector polynomials.

Level 2	Level 1	à	Level 1
[ coefficients ]	[ coefficients ]	à	[ coefficients ]

The vectors may differ in length.

PSUB

Subtract polynomials command:

Subtracts two vector polynomials.

Level 2	Level 1	à	Level 1
[ coefficients ]	[ coefficients ]	à	[ coefficients ]

The vectors may differ in length.

PMUL

Multiply polynomials command:

Multiplies two vector polynomials.

Level 2	Level 1	à	Level 1
[ coefficients ]	[ coefficients ]	à	[ coefficients ]

The vectors may differ in length.

PDIV

Polynomial division command:

Divides two vector polynomials by long division. This will produce a quotient (Level 2/Level 1), and a remainder.

Level 2	Level 1	à	Level 3	Level 2	Level 1
[ coeff_Num ]	[ coeff_Denom ]	à	[ coeff_Quotient ]	[ coeff_Rem,Num ]	[ coeff_Rem,Denom ]

If one polynomial is divided by another polynomial of higher degree, then the quotient will be zero, and the remainder will be the original fraction. An example will illustrate how it works:

à º

…and when Level 2 is [0], there is no remainder, and the two lower levels can be discarded:

à º

PPOW

Polynomial to a power command:

Takes a vector polynomial to a given power.

Level 2	Level 1	à	Level 1
[ coefficients ]	n_Power	à	[ coefficients ]

The execution time can be very long for high powers. takes ~35 seconds to calculate (but the calc takes ~50 seconds to expand the same expression, so maybe it’s not that long anyway J).

PDER

Polynomial derivative command:

Takes the derivative of a vector polynomial.

Level 1	à	Level 1
[ coefficients ]	à	¶[ coefficients ]

Remember that the derivative is not valid if the polynomial variable is present in the vector.

PINT

Polynomial integration command:

Integrates a vector polynomial.

Level 1	à	Level 1
[ coefficients ]	à	ò[ coefficients ]

Remember that the integral is not valid if the polynomial variable is present in the vector. As in any integration, you can add an arbitrary constant while maintaining integrity.

PMATEVAL

Polynomial matrix evaluation command:

Evaluates a matrix in a polynomial.

Level 3	Level 2	Level 1	à	Level 1
symbolic	global name	[ matrix ]	à	[ matrix ]

To evaluate a matrix in a polynomial, is best illustrated with an example:

Consider the matrix , in the polynomial .

This means

PSYSSOLVE

Solve polynomial system command:

Solves a polynomial system, like the built-in LINSOLVE does.

Level 2	Level 1	à	Level 1
[ symbolic_System ]	[ global name_Vars ]	à	[ solutions ]

The example system here is still considered simple, even though PSYSSOLVE does well. A 10x10 system, for example, is considered complex, but far too big to use as example. Just take my word for it, that this command will solve such systems up to 10 times faster than the built-in functions.

solved for , and .

SOLVE: 13.45 s.

LINSOLVE: 16.72 s.

PSYSSOLVE: 12.82 s.

LAGUERRE

N’th LaGuerre polynomial command:

Returns the n’th order LaGuerre polynomial, similar to the built-in commands TCHEBYCHEFF, HERMITE and LEGENDRE.

Level 1	à	Level 1
n_Order	à	[ coefficients_LaGuerre ]

Negative orders are not allowed. The 4^th order LaGuerre polynomial :

Vectors on the HP49G

A vector on the HP49G can look like this; [1 3 –5 ‘X+2’ 9], can contain reals (e.g. 2.2514784455), complex numbers (e.g. (1.2,4.7)), integers (e.g. 5) and symbolics (e.g. ‘X’ or ‘(A-2)/7’), or a mix of them. There are different ways on the calc to create and edit these vectors.

Creating vectors

There are four (simple) ways to create a vector on the calc.

Create in the edit-line – this is done with the []-brackets, found with left shift ´ (multiply). When you press this key, you’ll have the []-brackets appear in the edit-line, and a blinking cursor placed between them (see Figure 1). All objects you enter have to be separated by space (SPC), except when you’re entering symbolics, which are enclosed by ¢-delimiters, found with right shift EQW (see Figure 2). Press ENTER when finished (see Figure 3). You cannot create an empty vector.

Figure 1 Figure 2 Figure 3

Transform a list – this is done by creating a list in much the same way as above, with the {}-brackets, found with left shift + (add). All other steps are the same. The built-in command AXL toggles between a list and a vector (see Figure 4, 5 & 6) – you’ll not be able to transform an empty list into a vector though.

Figure 4 Figure 5 Figure 6

Assemble from stack – this is done with the ®ARRY command. The arguments to ®ARRY (when creating a vector) are the vector length in a list in stack Level 1, and the vector elements above that (see Figure 7 & 8).

Figure 7 Figure 8

Use the Matrix Writer (MTRW) – you access this by pressing left shift EQW (see Figure 9). Here you just enter the single elements by typing them in as in the command line discussed earlier, pressing ENTER after each element. Be sure to check that the menu label is bulleted – if it looks like this: , then you have to press F2 to make the MTRW output a vector instead of a one-dimensional array. While in the MTRW, you can access the stack by pressing HIST, and the Equation Writer by pressing the EQW key.

When you’ve input all elements, press ENTER to have the vector put on the stack (see Figure 10 & 11).

Figure 9 Figure 10 Figure 11

Editing vectors

You can edit vectors in much the same way as creating them. You need to have the vector on stack Level 1 for all of these methods to work.

Edit in the command line – pressing left shift down-arrow does this. The same rules apply when you add or edit items, as when you created a vector from scratch.

Transform to list – this is again done with AXL. The list form has many advantages over the vector form, when it comes to adding and deleting items. You can use + to add an element either to the beginning or the end of the list (depending on the lists position in either Level 1 or 2 in the stack). You can also append two lists with +. Then there is HEAD, TAIL, REVLIST and SORT (probably not that useful).

Edit in the Matrix Writer (MTRW) – this is the place you end up when you press down-arrow. Now you can add elements to the end of the vector, or you can edit the elements themselves, either with the EDIT menu label (F1), which will launch a command line for editing, or with left shift EDIT, that’ll launch the Equation Writer if the item to be edited is a symbolic.

The disadvantage of the MTRW is that it can be difficult to make big changes in the vector. It is easier with the list method to add items in the middle of the vector, or to add elements to the beginning aso.

Vector arithmetic

When dealing with vectors there are many operations that are valid on them. I’ll concentrate here on the ones that have relation to polynomials.

Negation (+/-) – you can negate all elements of a vector with the +/-button.
Multiplication (´) – you can multiply all elements of a vector with reals, complex numbers, integers and symbolics, with the ´button.
Division (¸) – you can divide all elements of a vector with reals, complex numbers, integers and symbolics, with the ¸button.
If you want another operation done on all elements of a vector, you’ll have to convert it to a list with AXL first. Then, due to the HP49Gs normal list processing, will all commands like Ö, SIN, EXP aso. be executed on all elements of the list.

Relationship w/ SymbToolz

The commands A®P, P®A and RAT? did reside in the SymbToolz library up until v1.1. From SymbToolz v2.0 (not released yet) onwards, this library will be necessary for it to run. This library is the natural place for such commands, and therefore they have been removed from SymbToolz.

Revision History

v1.0: Initial public release.

v2.0: Added commands P2®A, P3®A, PMATEVAL and PSYSSOLVE.

v2.1: Fixed a bug that affected some of the commands in this library – numeric vector (type 3) input could cause a TTRM.