PinaCAS is a computer algebra system for the TI-84 Plus CE calculators that shows its work. It is a fork of PineappleCAS by Nathan Farlow, which provides the simplifier, identities, derivatives, and the calculator GUI. PinaCAS adds integration and differential equations, in addition to aforementioned showing of work. Both use the imath library for arbitrary precision math.
- Indefinite integration: linearity, a table of elementary integrals, polynomial division, u-substitution, and integration by parts
- Step-by-step work for derivatives and integrals, shown in a scrollable viewer with fractions, exponents, roots, and integral signs drawn as in print
- Differential equations entered with primes, such as
Y''+2Y'=3X, classified by order and linearity - Checking that a function solves a differential equation and its initial conditions
- Solving first order equations (separable, linear, exact, Bernoulli, homogeneous, and y' = f(ax + by)), linear equations with constant coefficients by undetermined coefficients or variation of parameters, second order equations by reduction of order, and linear equations with polynomial coefficients by power series about an ordinary point, with initial value problems
- Integral and Solve DE functions in the GUI
- An editor for the input and output variables inside the app, with a live preview of the expression
- Much faster simplification, especially when showing work
- Runs as a Flash app named PinaCAS, so it is not limited to the 64 KB size of a program, and uses spare RAM for a larger heap
- The TI-Basic interface is removed
- Builds with the LLVM-based CE C toolchain, with imath and app_tools as git submodules
Expressions are read from and written to the Y= variables, strings, and Ans. To edit one without leaving PinaCAS, select its box, such as Input, and press [y=]. The editor works like the OS: 2nd and alpha select the second function or letter of a key, and 2nd alpha locks alpha. [math] opens a menu of tokens without a key of their own, such as =, ', and abs(, and 2nd [math] and 2nd [apps] type = and ' directly. Under the tokens is a preview of the expression, or why it cannot be parsed. [enter] saves, [mode] cancels, and [clear] empties the line. Ans cannot be edited.
Type the equation into a Y= variable or string, in the editor or the OS. Write derivatives of the unknown function with primes from the ANGLE menu (2nd, APPS), and the equals sign from the TEST menu (2nd, MATH). For example, y'' + 16y = 0 is Y''+16Y=0. Choose the independent variable with "Respect to", which is X by default. Any letter can be the unknown function, so x'' + w²x = F cos(gt) is X''+W²X=Fcos(GT) with respect to T.
Add initial conditions after the equation, separated by commas: y'' + 16y = 0, y(0) = 2, y'(0) = -2 is Y''+16Y=0,Y(0)=2,Y'(0)=-2. Enter a differential form M dx + N dy = 0 divided by dx, as M+NY'=0.
Solve DE classifies the equation and solves it when it can, writing the solution to the output variable. First order equations are solved by separating variables, with an integrating factor when linear, as exact equations (with an integrating factor of x or y alone when needed), or by the Bernoulli, homogeneous, or y' = f(ax + by) substitutions. An initial condition determines the constant, and the solution is solved for the function when each inverse step is unambiguous, so Y'=X²e^(6Y),Y(8)=0 gives -ln(1025-2X³)/6. Otherwise the implicit solution is written as an equation, like Y²=X²+C.
Linear equations with constant coefficients are solved from the roots of the characteristic equation, such as Y''+16Y=0 giving Acos(4X)+Bsin(4X). The arbitrary constants are A, B, C, and so on, skipping letters in the equation, and initial conditions determine them. A coefficient may be a letter when the equation has the form X''+W²X=…, which is taken as an oscillator with W positive. When the right side is not zero, a particular solution comes from undetermined coefficients if each term is a polynomial times e^(ax) times cos(bx) or sin(bx), and otherwise, for second order equations, from variation of parameters. To use reduction of order on a homogeneous second order linear equation, add a known solution after the equation: 4X²Y''+Y=0,Y=√(X)ln(X). Equations it cannot solve yet are classified, and linear ones are written in standard form.
Linear equations with polynomial coefficients and right side that no other method solves, like Y''+XY'+Y=0, are solved with a power series about an ordinary point. Check "Power series" to use it for any such equation. The work shows the series and its derivatives as sums, shifts their indices, separates the first terms, and finds the recurrence relation, then the coefficients until each solution has the chosen number of nonzero terms (1 to 9, 4 by default) or ends. The series is about the point of the initial conditions, or about another point given after the equation like Y''+(X-1)Y=0,X=1, and about 0 otherwise.
To check a solution, put it in another variable, either as Y=2cos(4X)-1/2sin(4X) or just 2cos(4X)-1/2sin(4X). On the Solve DE page, check "Verify solution" and choose that variable under "Solution in". PinaCAS differentiates the solution, substitutes it into both sides of the equation, and checks each initial condition.
These are from PineappleCAS, and show the simplifier.
PinaCAS is a Flash app. The calculator only accepts apps signed by TI over the link cable, so PinaCAS comes with an installer program that writes the app from AppVars, like Cesium's installer.
- Download the latest release and send
PINACAS.8xpand thePinaCAS.*.8xvAppVars to your calculator, or sendPinaCAS.b84, which bundles them. - Send the C libraries,
clibs.8xg, from the release or from https://github.com/CE-Programming/libraries/releases. - Run
PINACAS. It installs the app and offers to delete the installer files. - Open PinaCAS from the APPS menu. Navigate the GUI with the arrow keys, press enter on GUI elements to change their value, and press clear to exit.
On OS 5.5 or newer, TI removed the ability to run assembly programs, so run the installer from a shell like Cesium after jailbreaking with arTIfiCE. On OS 5.3 and 5.4, run prgmPINACAS from the home screen, and on older versions run Asm(prgmPINACAS).
To update, first delete the installed app in Mem Management (2nd, +, 2, then Apps), since the installer does not replace it.
Download and install the latest CE C toolchain from https://github.com/CE-Programming/toolchain
On RHEL-like distributions (e.g. Fedora), scripts/install-toolchain.sh installs or updates it in /opt/CEdev (override with CEDEV_PREFIX), along with the matching clibs.8xg for the calculator.
Compile for calculator:
git clone --recursive https://github.com/james-pre/PinaCAS
cd PinaCAS
make
This builds the app, bin/PinaCAS.8ek, splits it into the bin/PinaCAS.*.8xv AppVars, builds the installer bin/PINACAS.8xp from app_tools, and bundles them into bin/PinaCAS.b84. make debug builds the same files without optimization into bin/debug, with dbg_printf output shown in CEmu's console.
The calculator program compiles consistently on Ubuntu, but Windows has a problem with it. Executing make on Windows many times seems to work for some reason. (I blame the compiler!)
Run in CEmu:
scripts/emu.sh --libs
Builds PinaCAS, starts CEmu with the ROM at tmp/TI84+CE.rom (override with CEMU_ROM), sends the installer and AppVars, and runs the installer through Cesium. --debug uses the debug build, and --libs also sends clibs.8xg, which is only needed once. Set CEMU_CESIUM_APP to Cesium's number in the APPS menu if it is not 4. Delete the installed app before reinstalling, as on a calculator.
Compile for PC:
git clone --recursive https://github.com/james-pre/PinaCAS
cd PinaCAS
make pc
This builds bin/pinacas and does not need the CE toolchain. make check runs the tests in tests.txt, and make format formats the source with clang-format.
The version shown on the calculator comes from the latest vX.Y.Z git tag. Builds after a tag add the commit count and hash as build metadata, like 2.0.0+5.g1a2b3c4.
- James Prevett: PinaCAS
- Nathan Farlow: PineappleCAS, which PinaCAS is built on
- Michael J. Fromberger: the imath library
- Adriweb and Mateo: help and contributions to PineappleCAS
PineappleCAS is licensed under the MIT, all copyright belongs to Nathan for his amazing work on the original project. This fork is licensed under the GPL.