the resources and conventions of the operating system. As new hardware resources become necessary and available, the operating system must grow to manage those resources effectively and as transparently as possible to the applica tions built on the system. The operating system (and system programming language) in the HP 48G/GX is the RPL operating system, first used in the HP 18C and HP 28C and subsequently in a number of other machines including the HP 28S, HP 48S/SX, and now, with extensions, in the HP 48G/GX. HP 48G/GX Fundamentals The key concept underlying the operation of the calculator is the idea of objects on the stack. A stack is a data structure that is similar to a stack of cafeteria trays. The clean trays are added to the top of the stack, and as trays are needed, they are removed from the top of the stack. This type of last in, first out ordering characterizes the HP 48G/GX stack. All operations take their arguments (if any) from the stack and return their results (if any) to the stack. There is only one data stack in the HP 48G/GX. This resource is shared by the user and the system RPL programmer, who must take great care to make sure that any objects that be long to the user are preserved through the operation of sys tem RPL programs. For example, the user may have a few numbers sitting on the stack, then decide to plot the graph of a function. The system RPL program that runs when the DRAW key is pressed does many operations that require the use of the stack, such as recalling the plotting parameters, checking that they are valid, calculating the range over which to plot, evaluating the user's function, and converting the function values to pixel coordinates. After the graph is com plete (or if the drawing of the graph is interrupted by the user), when the user sees the stack again, the same numbers that were there to begin with should not have been disturbed. Instead of trays, users may collect various types of numeric, symbolic, and graphic objects on the HP 48G/GX stack. The types of objects available in the HP 48G/GX include real and complex numbers, real and complex arrays, binary integers, names, characters, strings, tagged objets, algebraic objects, unit objects, and graphic objects. There are also backup ob jects, library objects, directories, programs, and lists. (HP 48 object types are discussed in more detail in reference 2.) In a keyperfunction calculator, there is a single key that the user needs to press to get the machine to perform any opera tion, such as cosine. The HP 48G/GX has many more opera tions than the 49 keys on the keyboard, so there needs to be a way to access all the functionality without assigning one operation to each key on the keyboard. This is accomplished through the use of menus and softkeys. The top row of keys on the keyboard do not have anything printed on them be cause they correspond to menu labels that appear along the bottom of the screen. These keys are called softkeys, and their meaning changes whenever the corresponding labels on the screen are changed by the software. HP 48S/SX Memory Controller Configurations We will now discuss the memory controller configurations used in the HP 48S/SX and how these are used in imple menting the various types of expanded address modes de veloped for these products. The next section outlines the

System RAM 32K

Port 1 128K Higher Priority

Port 2 128K

Extra 2K Covered ROM 32K Larger Addresses

System ROM 224K

Unused Controller Address Space
Fig. 2. Standard memory control ler configuration for the HP 48S/SX calculator. Memory sizes are in bytes.
differences in configuration between the HP 48S/SX and the HP 48G/GX and discusses how these differences are used to extend and refine the expanded address technology to pro vide access to a total of 4.75M bytes of code and data as transparently as possible. The CPU bus architecture first developed for the HP 71 and used in all HP calculators since that time has several useful features. One of the nicest is its address configuration capa bilities. All chips attached to the bus are required to be able to change, on command of the bus, the range of addresses that evoke a response from the chips. Such a system elimi nates, once and for all, the inconvenience and headache of configuring jumper switches on cards designed to plug into the machine. For a consumer product like a calculator this is not only a nicety, it is a necessity. In the early days of the architecture (HP 71 to HP 28C), the CPU bus lines were actually routed around the circuit board and any RAM, ROM, or memory mapped I/O that was at tached to the bus had to be custommade with the bus inter face attached. This had the advantage of allowing an arbitrary number of parts to be added to the system with assurance that the system would be capable of handling all of them in one way or another. It had the grave disadvantage of putting a price premium on such essential items as ROM and RAM. In the secondgeneration CPU chip, a fixed number of mem ory controllers were included onboard the CPU. The CPU bus was then, for all practical purposes, completely hidden within the CPU itself. The combination of external standard
RAM or ROM together with one of the internal memory controllers was then equivalent (so far as the CPU bus is concerned) to a standard bus device. In the standard device implementations, the size of the device (that is, the address space occupied by the device) is de signed into the device. In the secondgeneration chip, the size of the controllers was mask programmed at the time of man ufacture since we knew exactly what size each controlled device would be. With the advent of plugins for the HP 48S/SX, the configu ration capabilities of the memory controllers had to be ex panded to include varying the apparent size of the memory controller to conform with the device being plugged in. This is one of the many advanced features in the thirdgeneration, HP 48S/SX implementation of the architecture. This resizing feature, in addition to allowing plugins of various sizes, also presented the opportunity to explore expanded address modes, which we have come to call the covered" technol ogy, for reasons that will be apparent shortly. The thirdgeneration CPU chip has six memory controllers. In the HP 48SX, these are allocated to memory mapped I/O, system RAM, port 1, port 2, and system ROM, and there is one extra controller. Their configuration in the usual state is shown in Fig. 2. The memory controllers are shown with their sizes and locations in the address space (00000h to FFFFFh). They are also pictured as having a vertical location in priority space." In the CPU bus definition the devices are chained, with the result that devices closest to the CPU on

System RAM (Shrunken)

Port 1 128K Higher Priority Covered ROM 32K Unused
Covered Code to be Executed In-Place System ROM

Larger Addresses

Fig. 3. Executeinplace configu ration for HP 48S/SX covered code.

Mailbox in System RAM

Port 1 (128K) Higher Priority Unused Covered ROM 32K Larger Addresses
Covered Code and Data to Copy to Mailbox System ROM
Fig. 4. Copytomailbox configu ration for HP 48S/SX covered data.
the chain have the first opportunity to respond to bus re quests. In consequence, if two devices are configured with overlapping address ranges, the one closer to the CPU on the chain effectively hides the more distant one. In Figs. 2 to 12, higher priority can be interpreted as closer to the CPU" or hides those below." As shown in Fig. 2, the memory controller for system RAM hides the section of ROM shown as covered. This is the reason for the name covered" technology. Fig. 3 shows more detail of the covered ROM and the first way in which it is used. In one section of the covered ROM there is assembly language code (mostly math routines) that requires no RAM resources outside the CPU for execution. This code is executed inplace in the covered ROM by shrinking and/or moving the memory controller for system RAM so that the relevant section of code is temporarily un covered. When the routine finishes execution, system RAM is returned to its normal configuration. A second set of routines, all of which only need access to a fixed set of locations within system RAM, can execute with system RAM in any one of 16 locations, as long as they themselves are not currently covered by system RAM. Fig. 4 shows a second way in which the covered ROM is used. In this case, code and data (mostly data) are copied from covered ROM to a mailbox at a fixed location in system RAM. After the copy is completed, system RAM is returned
to its normal configuration and the code and data are avail able to the rest of the system. Coders using this data must remain aware that it is volatile and can be destroyed by an other fetch of data from covered ROM. In this sense, this method is not transparent. Another way in which covered ROM is used is shown in Fig. 5. It is as transparent as the executeinplace method but entails fewer restrictions on the code and data that can be included. In the HP 48SX code, this system is usually tied to the execution of ROMPTRs. Recall that ROMPTRs are RPL objects that substitute for hard addresses of objects whose precise location is not known in advance (and in fact might not even be present.) They are midway between hard ad dresses that only change at compile/link time and identifiers whose corresponding objects may move between subsequent calls at run time. If, during the conversion of a ROMPTR to an address, it is determined that the corresponding object lives in covered ROM, the object is copied from covered ROM, through the mailbox, to the TEMPOB (temporary object) area. The address of its new location in the TEMPOB area is then returned. Fig. 6 shows a comparison of a named ROM word (keyword or command) as it would exist in covered ROM and as copied to the TEMPOB area. Although we'll refer back to Fig. 6 later, for now notice that in addition to the object itself, an addi tional piece is added to the image in the TEMPOB area. This piece is a ROMPTR preceding the object itself. This allows

TEMPOB Area

System RAM Port 1 (128K) Higher Priority
Covered ROM Words to Copy to the TEMPOB Area System ROM Covered ROM 32K Larger Addresses Unused
Fig. 5. CopytoTEMPOB configura tion for HP 48S/SX covered ROM words.
Mark and Link Property List Flags ROMPTR Body ROM Word Body Property List Item Property List Item Property List Item Property List Item Property List Item ROMPTR Preceding Object

DOROMP

These features required increasing the usable address space from 0.5M bytes to 4.75M bytes, an 850% increase over pre vious machines. While the HP 48G/GX has CPU functionally equivalent to the thirdgeneration CPU discussed above and thus has six memory controllers, these controllers are configured and used differently. Fig. 7 shows the standard HP 48GX config uration. The controller previously allocated to port 2 is now used as a bank switch control, and the extra controller is now allocated to port 2. Furthermore, there are now as many as 34 layers over the last 128K bytes of address space. Eliminated in this configuration is the HP 48S/SX covered ROM. This means that all of the functionality included in the HP 48S/SX can be accessed more quickly. Two things that are visibly enhanced are plotting (since the math routines are not covered) and screen update (since the font bitmaps are not covered.) Since there are a great many more covered places to access, however, there are many more temporary" con figurations to keep track of while working with the covered data. To simplify the system, we use only a single covered tech nique, namely, covered ROM word access, with appropriate modifications. Without this simplification, the number of access method and configuration combinations would be unmanageable. Moreover, this is the only feasible method of covered access to code written for the HP 48S/SX or not expressly written for the the new configuration. Fig. 8 shows the configuration while copying an object from a bank of port 2 to the TEMPOB area. Port 1 is unconfigured. In the unconfigured state, the controller responds to only a handful of bus commands and acts as if it weren't there for data access. Fig. 9 shows the configuration while copying an object from the second half of the upper system ROM. In this case, both ports are unconfigured. Fig. 10 shows the configuration while copying an object from the first half of the upper system ROM. Since a controller move or resize operation takes many more CPU resources than configure or unconfigure, it is often necessary to copy objects from this section, through a mailbox, and then into the TEMPOB area.

User Interface

With ease of learning and ease of use the primary goals for the new HP 48G/GX calculator, the user interface and many builtin applications have been largely redesigned. Input forms provide the common starting point for the new and rewritten applications in the HP 48G/GX. Looking much like dialog boxes in an Apple Macintosh or Microsoft

Title Label

Help Line Menu
Fig. 13. A typical input form.
Windows PC, input forms provide a fillintheblanks guide to the input needed for a task, plus applicationspecific menu keys for acting on that input. For selecting an application in a particular topic and for picking an input from among several choices we developed choose boxes, a type of popup menu that suggests alterna tives and narrows the input focus. We designed message boxes to make feedback to the user more manageable within our increasingly crowded display space. Message boxes appear on top of whatever the user is working on and provide more flexibility for formatted mes sages and icons than the twoline, fixedlocation error mes sages they replace. They also preserve the context that can otherwise be lost when something surprising happens within an application. Input Forms An input form provides both a means to enter data pertinent to an application and operations that permit the user to direct actions. Visually, an input form consists of (see Fig. 13): A title suggesting the form's purpose One or more fields, typically with explanatory labels, which are used to gather and display user input A help line that details the input expected in the selected field Menu keys that provide more options for working within or exiting the input form. Each input form field can be one of four types. Most input forms, such as the Set Alarm and I/O Transfer input forms, con tain several or all types of fields. Text fields are used to enter arbitrary HP 48G/GX objects like real numbers and matrices; the object types allowed are specific to each text field. In Fig. 14, a text field is used to enter an alarm message in the Set Alarm input form. When a single choice among several is required, list fields are used to eliminate invalid input and to help focus user actions. To select an entry in a list field, a choose box is displayed. In Fig. 15, a list field is used to specify the trans fer format in the I/O Transfer input form. Sometimes only a simple yesno, doordon't type of choice is needed. For this we use check fields. Fig. 16 shows how the overwrite (OVWR) field is used to specify whether or not an existing variable should be overwritten. Finally, when arbitrary input is possible but logical choices are also available, combined text/list fields are employed. In

Fig. 14. Using a text field in an input form.
the Transfer input form, the Name field is a combined field that permits new names to be entered or the names of exist ing HP 48G/GX variables or PC files to be selected (see Fig. 17). As the figures illustrate, each of the three base field types has associated with it a dedicated menu key that triggers the unique feature of that field type. This feature is an important part of how we maintained a calculator keyperfunction style interface within the constraints of a small display and with no pointing device. In other graphical user interfaces, visual elements such as list arrows are activated by mouse clicks to elicit different behaviors from fields. In the HP
Fig. 15. Using a list field in an input form.
Fig. 18. A typical choose box.
a specific application from among several. Fig. 18 shows the choose box that is displayed when the STAT key is pressed to perform statistical calculations. When circumstances require, choose boxes can include any or all of several advanced features. The Memory Browser application, for example, is actually a maximumsize choose box embellished with a title, multichoice capability, and a custom menu (see Fig. 19).
Fig. 16. Using a check field in an input form.
48G/GX, the user's finger acts as the pointing device, trigger ing the desired behavior by pressing the appropriate action button for each field. Consistent location of the three types of action buttons helps the user navigate an input form confidently. Some input form menu keys perform applicationspecific operationsfor example, DRAW in Plotting. In the second row of the input form menu are more advanced input form opera tions for resetting a field or the entire form, displaying the object types allowed in a field, and temporarily accessing the user stack to calculate or modify a field value. Choose Boxes Choose boxes are used to make a choice in an input form list field. They are also used in most subject areas to choose
Message Boxes Message boxes are used primarily for reporting errors that require attention before proceeding. For example, if the user attempts to enter a vector in the EXPR field of the Integrate input form, a message box appears to inform the user of the problem (see Fig. 20). Some applications also use message boxes to give additional information. For example, in the Solve Equation input form, the user can press INFO any time after a solution has been found to review the solution and determine how it was calculated (see Fig. 21). Input Form Implementation For the HP 48S/SX, we developed an RPL tool called the parameterized outer loop3 to speed development of new interfaces such as the MatrixWriter by automating routine key and error handling and display management. The input forms in the new HP 48G/GX embrace this conceptin fact, the input forms engine is a parameterized outer loop appli cationand take it one step farther to automate routine mat ters of application input entry and selection of options. The input forms engine brings a uniform interface to all new HP 48G/GX applications. While narrowly focusing the task of application development by managing command input tasks, the input forms engine also leaves much room for the customization that helps opti mize the HP 48G/GX for ease of use. Since an important measure of our progress towards our goals for the calculator

Fig. 17. Using a combined text/list field in an input form.
Fig. 19. The Memory Browser: a complex choose box.
Fig. 20. A typical message box.
was to be feedback from typical users throughout the devel opment cycle, we designed the input forms engine from the ground up to be highly customizable. This was accom plished in a programmerfriendly manner by including over fifty hooks into the input forms engine's responses to exter nal and internal events. External events are triggered by us ers and include lowlevel events such as key presses and highlevel events such as completion of a field entry. Inter nal events are usually activated by external events, such as formatting a completed field entry for proper display. A single external event can trigger a half dozen or more inter nal events, all of which are customizable. Input form applications can customize any or all formlevel events such as title display or field events such as displaying a help line. Each field has a field procedure associated with it, and the entire form has a form procedure associated with it. Whenever an event occurs, the appropriate field or form procedure is called with an identifying event number and perhaps additional information. If the procedure does not customize the event, it returns FALSE to the input forms en gine. If it does customize the event, the procedure performs the custom behavior and returns TRUE. In this manner, every event first queries the proper form or field procedure to determine if custom behavior is needed, then handles the event normally only if it isn't customized. If a form or field has no custom behavior, it specifies a default procedure that quickly responds FALSE to all event queries. The reason for a form procedure and multiple field proce dures is to spread the burden of customization throughout the form. Since each field procedure only checks for the events that pertain to it, and since the form procedure only checks for formlevel events, no single event processing is slowed by a highly customized form that would otherwise have to compare the event number against a lengthy list of event and field combinations. For the HP 48G/GX project we needed another layer of regularity not enforced by the input forms engine. Because we sought and reacted to usability feedback almost until the
code was released to production, the user interface details for each subject area were subject to constant change. It was imperative, therefore, that we maintain a strict and formal division between unchanging and wellunderstood tasks such as getting and saving problem domain information and calculating resultsand the user interface details that were changing regularly. We developed a set of conventions that were embodied in what we called translation files. We used naming rules and constrained responsibilities to greatly miti gate the effects of user interface changes on the underlying problemsolving functionality. For example, one RPL word in the plotting translation file has the simple task of reading the current horizontal plot range from calculator memory. Since the word has no presumptions about how and when it will be called, references to it could be (and often were) changed around as the fields populating the plotting input form were worked out. Choose Box Implementation The choose box engine is very much like the input forms engine. For customization, the programmer can supply a choose procedure that responds to 26 messages. A feature of choose boxes that simplifies their use is the optionheavily used by the builtin applicationsof items that encapsulate both display and evaluation data. For exam ple, when an angle measuredegrees, radians, or gradsis to be chosen in certain input forms, the choose box engine displays plain descriptions but returns an RPL program that sets the selected angle measure. This circumvents the need for branching according to the returned object and simplifies the extension of choices. Results: Benefits and Costs Initial feedback from the educational advisory committee and user reviews suggests that the use of input forms and other graphical user interface elements has greatly improved the ease of use of the HP 48G/GX over the HP 48S/SX. However, the path we took to this accomplishment was more challenging than we planned. Event customization, originally conceived as a means to ex tend the functionality of input forms in unforeseen ways, turned out to be a key component of our ability to prototype new user interface ideas rapidly. As their name may imply, the original intent of input forms was very modest compared to the role they now play. We designed input forms to be the standard means by which applications gather data for a task. One or more input forms would be displayed as neces sary within the context of another, undefined, application context. This original concept is applied successfully throughout the calculator. For example, in the Memory Browser, when NEW is pressed to create a new variable, an input form is used to get the information required (see Fig. 22). In this context, the user can do only three things in the input form: enter data, cancel the form, or accept (OK) the form. This simple but effective behavior was the model used for the original input form design. As the project developed, however, it became apparent that an input form could serve not only as a information gatherer but also as an action director. Input forms thus graduated from simple dialog boxes to fullfledged application environments.

3D Plotting

The functionality described in this section is a suite of 3D graphing and viewing utilities for the HP 48G/GX. We had several requirements to consider in creating these routines. Our aims were that they be psychologically effective and require only a small amount of code. In exploring visualization techniques on a variety of ma chines we found that increasing realism" (raytraced, Phongshaded, hiddenline, etc.) in the graphical presenta tion of functions of two variables did not necessarily corre late with increasing ease of comprehension. The HP 48G/GX routines represent the results of some of these experiments (including timetocompletion as an important factor). All of the 3D plotting routines are intended as seamless ex tensions of the other builtin plotting utilities. In particular, they share the same standard user interface and are selected as alternative plot types. The 3D plotting routines are SLOPEFIELD, WIREFRAME, YSLICE, PCONTOUR, GRIDMAP, and PARSURFACE.
SLOPEFIELD The SLOPEFIELD plot type draws a lattice of line segments
whose slopes represent the function value at their center point. Using SLOPEFIELD to plot f(x,y) allows your eye to pick out integral curves of the differential equation dy/dx = f(x,y). It is quite useful in understanding the arbitrary constant in antiderivatives.
The number of lattice points per row is determined by Nx and the number of lattice points per column is determined
Z Zhigh Y Yfar Ynear Zlow Xleft View Screen Xright X View Screen (Xe, Ye, Ze) Ynear Xleft Xright X 1 Unit View Volume Y Yfar Top View

(Xe, Ye, Ze)

Fig. 23. VPAR parameters in relation to the view volume.

(2,3.6 )

Top View
Y Plotted Point (0,2.8 ) Superimposed Integral Curve (Xe, Ye, Ze) View Screen X View Screen (Xe, Ye, Ze) 1 Unit X
Fig. 24. SLOPEFIELD plot of dx/dt = sin(xt).
by Ny. The input region sampled is given by XlefttXtXright and YneartYtYfar.
Fig. 25. Perspective projection of a point in the view volume onto the view screen.
The input form in this case allows the user to: Choose or enter the defining expression for the function to be plotted Choose the names of the two variables (identical to INDEP and DEPEND) Choose Xleft and Xright (default to their current value, or XRNG if no current value) Choose Ynear and Yfar (default to their current value, or YRNG if no current value) Choose Nx and Ny (default to their current value or 13 and 8 if no current value) Verify and/or choose RADIANS, DEGREES, or GRADS mode.
The input form in this case allows the user to: Choose or enter the defining expression for the function to be plotted Choose the names of the two variables (identical to INDEP and DEPEND) Choose Xleft and Xright (default to their current value, or XRNG if no current value) Choose Ynear and Yfar (default to their current value, or YRNG if no current value) Choose Zlow and Zhigh (default to their current value, or default YRNG if no current value) Choose Xe, Ye, and Ze (default to their current value, or 0, -1, 0 if no current value) In trace mode for SLOPEFIELD, the arrow keys jump the cursor Choose Nx and Ny (default to their current value or 13 and 8 if no current value) from sample point to sample point indicating both the coor Verify and/or choose RADIANS, DEGREES, or GRADS mode. dinates of the sample point and the value of the slope at that point. In trace mode for WIREFRAME, the arrow keys jump the cursor Example Problem: Determine graphically whether all solutions of the differential equation dx/dt = sin(xt) with initial condi tions 3.0tx(0)t3.1 satisfy 2.8tx(t)t3.6 for all t in [0,2]. from sample point to sample point and the display indicates all three coordinates of the sample point.

Solution: Choose SLOPEFIELD plot type and enter SIN(X*T) as the current equation. Choose T as the independent variable and X as the dependent variable. Choose 0 as Xleft, 2 as Xright, 2.8 as Ynear, and 3.6 as Yfar. Verify RADIANS mode, and draw the result. As seen in Fig. 24, almost all of the integral curves in this region leave the window either through the top or the bottom. Therefore, not all the integral curves satisfy 2.8tx(t)t3.6 for t in [0,2].
WIREFRAME The WIREFRAME plot type draws an obliqueview, perspec
Example Problem: Determine graphically whether the surface defined by z = x4 - 4x2y2 + y4 is, at the origin, concave up, concave down, or neither. Solution: Choose WIREFRAME plot type and enter X^44*X^2*Y^ 2+Y^4 as the current equation. Choose X and Y as the inde pendent and dependent variables. Choose -1 for Xleft, 1 for Xright, -1 for Ynear, 1 for Yfar, -1 for Zlow, and 1 for Zhigh so that the view volume surrounds the origin. Choose 4 for Xe, -10 for Ye, and 3 for Ze to give a distant, oblique view of the graph. As seen in Fig. 27, the graph displays a monkey saddle" which is neither convex nor concave at the origin.
tive, 3D plot of a wireframe model of the surface deter mined by z = f(x,y). The function determined by the current equation is sampled in a grid with Nx samples in each row and Ny samples in each column. Each sample is perspective projected onto the view screen along the line connecting the sample and the eye point (see Fig. 25). Neighboring samples are connected by straight lines. The sampled region is determined by the base of the view vol ume (Xleft, Xright, Ynear, Yfar). The region of the view screen represented in the PICT GROB (graphics object3) and hence on the display is determined by the projection of the view volume on the view screen (see Fig. 26).

New Interactive Features

The picture environment, which is invoked automatically when graphs are drawn or by pressing the PICTURE key, al lows the user to interact with a graph. The user can move

View Volume View Screen

Fig. 26. Relationship of view volume and eye point to XRNG and
Animation The ANIMATE command is a program that was easy to write and that the user could have written in userRPL program ming language, but we added it for the sake of convenience. Also, it is used as part of Yslice 3D plotting. It sets up a loop that repeatedly puts graphics objects into the PICT dis play area.

Fig. 1. A custom input form created by INFORM. Field Specifications Reset and Current Values 5: Personal Information 4: { Name: { } { } Bldg: Phone: { } Notes: { } {} } 3: { } 2: { } 1: { }

INFORM

Title Field Expander Column Count (3) and Tab Width (5)
columns in the grid is specified as one of INFORMs arguments, and each fields width is determined by the width of its label and by the user-supplied tab width, which places invisible tab stops within each column to help align fields vertically. A field can span multiple columns with a special field-expander specification. Help text and object type restrictions can be included for any field, but arent required. Fig. 1 shows an example of a custom input form created by INFORM. Notice that, despite the relative simplicity of the input arguments, an input form with aligned fields of varying widths is presented. This technique for building input forms proved so valuable that it was used to create the Solve Equation input form, which changes according to the number and names of variables in the equation to be solved.
past with success, but it can be timeconsuming, expensive, and risky. Another approach sometimes available is to consult standard computational libraries used by the professional scientific community. Several such publicdomain libraries are available that represent the current state of the art. In some develop ment environments these libraries can be used directly. In others, they can at least provide highquality methods and implementations that when judiciously used facilitate meet ing tight development schedules at low cost. We found the LAPACK library4 of FORTRAN 77 numerical linear algebra subroutines particularly helpful in this regard. As usual, code was reused whenever possible to achieve timely and reliable implementations. In addition to the source code for the HP 48S/SX and its Equation Library card, we had implementations dating from the HP 71 Math Pac that were revised for the HP 48S/SX but didn't find ROM space in that product. While reusing code, we took advantage of the HP 48G/GX CPU clock speedup and larger RAM environment over the HP 48S/SX to reconsider some of our previous implementa tion tradeoffs in an effort to achieve greater accuracy. In
some cases we decided to employ more computational effort and to store intermediate values in higher precision to achieve more accurate results. New Mathematical Features The HP 48G/GX includes many new mathematical features over those provided by the HP 48S/SX. These are array manipulations, additional linear algebra operations, a poly nomial root finder and related operations, two differential equations solvers and associated solution plotters, discrete Fourier transforms, and financial loan computations. The array manipulation commands are primarily pedagogical tools. These include a random array generator and com mands to add or delete rows or columns of matrices or ele ments of vectors, decompose matrices into or create matrices from row or column vectors, extract diagonal elements from a matrix or create a matrix from its diagonal elements, per form elementary row and column operations, and compute the rowreduced echelon form of a matrix. We significantly improved and expanded the linear algebra functionality of the HP 48G/GX over the HP 48S/SX. The determinant, linear system solver, and matrix inverter were

revised to be more accurate through additional computation and by storing all intermediate values in extended precision. We added a command to compute a condition number of a square matrix, which can be used to measure the sensitivity of numerical linear algebra computations to rounding errors, a command to compute a solution to an underdetermined or overdetermined linear system by the method of least squares, commands to compute eigenvalues and eigenvectors of a square matrix, commands to compute the singular value decomposition of a general matrix, and commands to com pute related matrix factorizations and functions. These linear algebra commands accept both real and complex arguments and perform all intermediate computation and storage in extended precision. The HP 48G/GX has commands to compute all roots of a real or complex polynomial, to construct a monic polynomial from its roots, and to evaluate a polynomial at a point. The polynomial root finder is a modification of the HP 71 Math Pac's PROOT command, extended to handle complex coeffi cients. It uses the Laguerre method with deflation for fast convergence and constrained step size and an alternate initial search strategy for reliability. The HP 48G/GX has commands to compute the discrete Fourier transform or the inverse discrete Fourier transform of real or complex data. These commands were leveraged from the HP 71 Math Pac's FFT and IFFT commands, requiring the data lengths to be a nonzero power of 2, and were modified slightly to match the customary definitions of these trans formations. Finally, we included timevalueofmoney commands. These commands have appeared in our financial calculators and were available on the HP 48SX Equation Library card. Since engineering feasibility studies must include at least rudimen tary timevalueofmoney computations it seemed useful to include these commands in the HP 48G/GX. Differential Equation Plotting The HP 48G/GX contains two differential equation solvers and solution plotters. These solvers and solution plotters can be accessed via their input forms or invoked programmati cally via commands. We provide a programmatic interface to the differential equation solvers and their subtasks so the user can use them with the calculator's general solver feature to determine when a computed differential equation solution satisfies some condition, or to implement custom differential equation solvers from their subtasks. In implementing the differential equation solution plots, one challenge was to identify and implement good solution meth ods. Another challenge was to merge this new plot type with the new 3D plot types described earlier and with the existing HP 48SX plot environment in a backwardcompatible manner. The HP 48G/GX specifically solves the initial value problem, consisting of finding the solution y(t) to the firstorder equa tion y(t) = f(t,y) with the initial condition y(t0) = y0. Here y(t) denotes the first derivative of a scalarvalued or vector valued solution y with respect to a scalarvalued parameter t. Higherorder differential equations can be expressed as a

* Stiff problems typically have solution components with large differences in time scale. More information is needed by a solver to compute a solution efficiently.
Acknowledgments We would like to acknowledge the rest of the software de velopment team: Jim Donnelly, Gabe Eisenstein, Max Jones, and Bob Worsley. Bill Wickes also contributed software. Clain Anderson and Ron Brooks from the marketing depart ment were involved in the daytoday design process and kept us informed about user needs. Dennis York oversaw both the R&D and marketing aspects of the project, which helped generate synergy between R&D and marketing. We would like to thank Dan Coffin, our manual writer, and John Loux from the technical support group for going out of their way to participate in the design work and for providing many valuable ideas.

References

1. W.C. Wickes, An Evolutionary RPN Calculator for Technical Pro fessionals," HewlettPackard Journal, Vol. 38, no. 8, August 1987, pp. 1117. 2. W.C. Wickes and C.M. Patton, The HP 48SX Scientific Expandable Calculator: Innovation and Evolution," HewlettPackard Journal, Vol. 42, no. 3, June 1991 pp. 612. 3. T.W. Beers, et al, HP 48SX Interfaces and Applications," Hewlett Packard Journal, Vol. 42, no. 3, June 1991 pp. 1321. 4. E. Anderson, et al, LAPACK Users' Guide, SIAM, Philadelphia, 1992.
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