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FCA 2.8 Resident Floating Point Scientific Calculator by Bob Eyer [73230,2620] Apr 28, 1992 ┌────────────────────────────────────────────────────────────────┐ │ │ │ Like the idea of a BBS program that works at a basic │ │ level with no configuration at all--a miniature BBS │ │ which can be made as sophisticated as you like, but │ │ which can be run as a simple utility if you forget how │ │ to use it? Try HOSP. Lots of people like it. And you │ │ can have a copy for only $25 (the basic shareware │ │ charge plus $5). That contribution pays for HOSP as │ │ well as some four dozen other utilities. See SHARE.TXT │ │ for details. For preview, see HOST30.ZIP in IBMBBS. │ │ │ └────────────────────────────────────────────────────────────────┘ Syntax ------ FCA [/U] Hotkey: Rightshift-/ Modes ----- places COLOR y RAD/DEG TWO/NAP/TEN STUF/D DMS/NODMS Review ------ ST Registers: E PI A B C F K L M N Q R S X Y Z Data entry areas ---------------- ADD MEAN REGRESSION Binary ops ---------- + - * / ^ CO PE Unary operations ---------------- SQ/SQRT EXP/LOG SIN/ASN COS/ACS TAN/ATN SINH COSH TANH ! HEX/DEC X= Y= Probability ----------- BIN NORM POIS STUD See the note about Errors below. Changes in FCA 2.8 ------------------ - Addition of Hex, decimal conversions. - Bug in data entry areas which affected the sign of entered numbers now removed. - Exponential notation now uses 'D' rather than 'e'. - FCA 2.8 is essentially FCAP 1.3; latter pgm no longer offered as retail enhancement of FCA. INTRODUCTORY NOTE: FCA is a floating point companion alternative to CA (see program listing in Shareware Notice above), which is restricted to four function fixed point arithmetic. FCA supports at least 30 functions. CA uses about 12K of memory; FCA uses about 40K. In the 40K class, competing floating-point calculators offer far fewer functions. FCA is not a general formula evaluator. If you need to do many different kinds of calculations on the same text in one shot, it may be more appropriate to use Lotus 1-2-3 or similar, print results as a file, edit, and then insert into your text. FCA, however, does provide a simple means of re-using input data for subsequent calculations (see section below about Automatic Store/Recall and also the discussion about register operations). Also, FCA provides a method of adding many numbers before injecting the sum to text (See Special Accumulator section below). FCA's floating point emulator will take advantage of the coprocessor, if one is installed. This advantage is noticeable primarily with regard to the probability functions. The slowest and least accurate function is the Student probability integral, which may require up to 8 seconds and beyond to find results arising from certain rare combinations of input variables on a 386 machine WITH installed 80387. This function works up to 60 times more slowly when run on a machine lacking a coprocessor. While the transcendentals generally have an accuracy of the order of 12 decimal places, Gosset's Student t is generally accurate to only 5. See below about the STUD function. FCA's real number range, apart from sign, is roughly between 1D-294 and 1.7D+308. This is much larger than is possible in the companion program CA, and is also larger than the usual range for many scientific calculators. FCA does not provide a menu or picture of a calculator - so as to avoid obscuring portions of the underlying editing application, and to avoid requiring the user to employ only 25 x 80 screens (some TSR calculators require this, to make use of Line 25 for calculator monitoring). FCA responds equally well in 50 x 80 or 44 x 132 video environments; generally, it is indifferent to the manner in which the video raster is defined. FCA is designed to respond just to the hotkey, which provides merely a 'calculation window', in which the user enters his calculations or mode changes. Hitting ENTER after supplying a calculation or mode change to the window, simply executes the job and returns to the application. Where a result is generated, it is pasted directly to the underlying application where the cursor was last located. In examples seen below, EACH calculation or mode change is preceded by invoking the hotkey. This, however, does not apply to special areas dealing with more complex problems, such as MEAN and REGRESSION. FCA is also designed to return the cursor to the initial position in the main calculation window, to permit making mode adjustments without line skipping. Once a task is completed, you may exit the window merely by hitting ENTER one additional time or by using the spacebar to delete the window. The old window always disappears in an editing environment after the result is injected to the text, but mode change information will, in general, remain. Modes/review ------------ FCA provides 6 groups of mode selections - number of decimal places to which to round results, colour of calculation window, whether to use Radian or Degree measure, what base to use for the EXP and LOG operations, whether to echo the result to display or paste to text, and whether to use DMS format in trig calculations. The current mode situation may be viewed simply by entering ST in the calculation window (ST is short for 'status'). Mode changes are entered directly in the calculation window and the new mode specs are resummarised in the calculation window, just as though ST had been issued. x Entry merely of a number, such as 5, will cause FCA to operate so as to round all results to, say, five decimal places. Entry of 0 means that no formatting will occur. The default value is 0. COLOR y: Entry of 'COLOR 30' will cause the calculation window to have a bright yellow foreground and a blue background. For colour details see below. The default colour scheme is black on white (112). Color 0 automatically converts to 112, so as to avoid black on black. RAD/DEG: Entry of RAD in the window causes FCA to assume all angles entered as arguments in trig functions are in radian measure. Entry of DEG puts FCA into Degree mode. The default is Degree mode. TWO/NAP/TEN: FCA supports three bases for use with the LOG or EXP functions. TWO means base 2; NAP is short for the Napierian base or base e (e=2.718...); and TEN means base 10. The default is base e. STUF/D: STUF means stuff the result to the underlying application, and D means display only at TSR video level. Register operations ------------------- FCA supports pasting 11 registers direct to text. For example, after invoking the hotkey, we simply enter e [= 2.7182818284590 (the use of the left bracket here merely signifies that the result appears at the underlying application, not at TSR level) In addition, the user may simply enter A, B, or C to examine the last value stored in FCA's "store/recall" facilities (see below), as well as the general storage register X, and the statistics registers. Binary operations ----------------- FCA supports seven two-variable ('binary') operations: + - * / ^ CO PE, or add, subtract, multiply, divide, raise to a power, combinations and permutations, respectively. For example, 1991-1917 [= 74 7D-6+3D-4 [= 0.000307 When entering exponentially formatted numbers one must specify a sign (+ or -) immediately after the 'D' symbol. FCA automatically distinguishes between the sign of an exponent and an addition or subtraction operation between two numbers. Additional examples: 35/34 [= 1.0294117647058 35*34 [= 1190 35^34 [= 3.150021D+52 (6 places) 10 CO 4 [= 210 (210 combinations of 10 things taken 4 at a time) Unary operations ---------------- FCA also supports 10 unary transcendental functions, grouped by inverses, as well as three hyperbolic functions, and the factorial function. Here, a blank must separate the name of the operation and the number which the operation takes as its argument. Examples, ! 6 [= 720 (factorial 6) HEX 12000 [= 2EE0 DEC 2EE0 [= 12000 SIN 45 [= 0.7071067811865 COS 1 [= 0.9998476951563 LOG 2 [= 0.6931471805599 EXP 1 [= 2.7182818284590 SQRT 2 [= 1.4142135623731 If you wish to use Base 10 logarithms, just enter TEN in the window, and then LOG 2 [= 0.3010299956639 Note: the EXP function is in fact an antilog function, since it is subject to the same range of base changes as is the LOG function. This may be inconvenient to some users who assume that the exponential function must always have base e, but implementing a separate antilog function seemed, in the circumstances, merely to be useless duplication of what is, essentially, a quite flexible function. The internal setup for the hyperbolic functions is the same as for the trig functions. If you select DEGree mode, FCA will convert your degree measure into the radian equivalent before calculating the function. If you select RADian mode, FCA will do no such conversion, but will inject your argument directly to the function. If DMS is selected, FCA will convert decimal degrees to degrees, minutes and seconds and set places to 6. Automatic store/recall ---------------------- If you are doing several calculations which involve use of the same term, you may reduce typing further by using variables. Calculation window variables are A, B, and C. The first time you execute a calculation with numbers, the first number is always stored into A, the second number into B, and the result into C. These values can be re-used, simply by employing these variables in subsequent calculations [except, of course, for the fact that, as each new calculation is done, the value of C will be updated with the new result]. Example (after entering 3 in the window): Suppose we wish to perform the following calculations - 34.21102 x 435, and 34.21102 / 355.5 Here, each calculation uses the same initial term. We proceed as follows, each time by hitting Rightshift-slash, and entering the calculation shown: 34.21102 * 435 [= 14881.794 a/355.5 [= 9.623D-02 The second calculation above could be repeated merely by entering a/b The user may also proceed to obtain results for the other three operations, using the same numbers, as follows: a*b [= 12162.018 a+b [= 389.711 a-b [= -321.289 The output C-variable can also be used in calculations. For example, we may first calculate with no scientific notation 2/3 [= 0.667 Here, A = 2, B = 3, and C is the result in brackets. Now, if we multiply the result C by 3, we should get back the numerator A: C * 3 [= 2.000 This example illustrates the fact that FCA, like handheld calculators with the store/recall function, stores results in a separate register before rounding. It is this separate register that is used for input, when the user employs C in a calculation. [otherwise, the user might get back 2.001]. This type of 'result protection' on use of a previous result is not, however, found in most TSR calculators. All these remarks apply also to FCA's trig functions. For example (using 5 decimal places), SIN 89 [= 0.99985 We may now get back the value of the argument, simply by performing the inverse (ARCSINE or ASN) on the result: ASN C [= 89.00000 The same principle also works with squares and squareroots, as well as logs and exponentials. For example, LOG 2 [= 0.69315 but the argument '2' may be had by performing the inverse on the result: EXP C [= 2.00000 You may also store a previous input or result into either of the extra storage registers X or Y. Just use, for example, X= C or Y = B to store the value of C into X or B into Y. Note that FCA is indifferent on whether the entry formula is 'X=C' or 'X = C'. X may be recalled simply by entering X by itself, or by using it as an argument in any function. MEAN (Total, Mean and Standard deviation) ----------------------------------------- Entry of the keyword MEAN in the calculation window will transfer control to a special area of FCA which displays the following type of prompt: X> Numbers entered at this prompt are totalled and in passing FCA also computes the mean and standard deviation for all numbers entered, displaying these results in the stated order to the left of the prompt. If the next number in the series is the same as the previous one, you may simply enter X at the prompt; FCA interprets X in the MEAN window as the value last entered. The same principle applies where the user wishes merely add the previous total to itself: use the result register C. The data displayed to the left of the prompt are displayed as rounded to a minimum of 3 places. When finished entering data, simply hit the Esc key to paste the Total to the underlying application. The other data are available by entering the following variable names: C - Total M - Mean S - Standard deviation N - Number of data entered. F - Degrees of freedom (for MEAN calculations this will be N - 1). The main result, the total, may be accessed by using register C. The values listed above may be used, for example, in the Normal distribution to obtain automatic calculation of a z-score (see below). For the interest of statistics people, the definition of standard deviation used in FCA assumes the "n - 1" basis. That is, it is based on multiplying the the root mean square of the data by the square root of the ratio of N to N - 1. The purpose of the n-1 basis is to obtain a more realistic value for the standard deviation in small sample applications than is possible with a mere variance calculation. The value of this routine is primarily to obtain quick results on small amounts of data found directly on the screen in the underlying application. If you have a large job to do, involving many points, your best bet is to use a large program such as LOTUS 1-2-3, not a small program like FCA. REGRESSION (regression analysis) -------------------------------- Entry of the keyword REGRESSION in the calculation window will transfer control to the REGRESSION area of FCA. Here, the prompt will change between X> and Y> and FCA will issue a beep to the speaker each time a Y value is entered. Three running results are displayed to the left of the prompt: The Galton-Edgeworth correlation coefficient, the y-intercept of the regression line, and the slope of that line. The previous value of X may be entered at either prompt by using X, and the previous value of Y may be entered at either prompt by using Y. If you enter C (the result register) the current correlation coefficient will be entered at the prompt in question. Avoid, unless you really want to do that. But otherwise, operation of the REGRESSION area is similar to that in MEAN. On use of Esc to exit, FCA will paste the correlation coefficient to the underlying application. To paste the y-intercept, use K; and to paste the slope, use L. The regression line has the standard form, Y = K + LX As usual, N is the number of pairs of data entered in the REGRESSION window. The correlation coefficient may be obtained by entering R. F, the number of degrees of freedom, is also calculated by FCA, and in the REGRESSION window, will always be set to N - 2. BIN --- This function returns the cumulative binomial probability of the occurrence of X or fewer events in a sample of size N, where the probability of a single event is Q, as well as the point probability in register Y. The syntax is BIN X N Q corresponding to the summation of all values of the binomial frequency distribution for occurrences ranging between 0 and X. Taking an example from Spiegel's Statistics (Schaum Outline Series, p 127), find the probability that at most 2 bolts (i.e. either 0, 1, or 2) will be defective in a sample of 4 taken from a production process in which 20% of all bolts produced are defective. Here, we use BIN 2 4 0.2 [= 0.9728 the same answer given by Spiegel. The probability that exactly 2 bolts will be defective in this problem is obtained merely by entering the Y register Y [= 0.1536 POIS ---- This is the Poisson cumulative probability of the occurrence of X or fewer events in a sample of size N, where the probability of a single event is Q - using same syntax as BIN: POIS X N Q The 'integration' logic used by FCA here is the same as for the binomial distribution. That is, the result found is the summation of the individual Poisson probabilities from 0 to X. The point probability is found in register Y. The primary reason why this distribution is needed to supplement the Binomial, is that the latter has a restricted range, owing to the fact that it uses the factorial function to operate on the sample size. This function explodes beyond the number range intelligible to FCA for arguments larger than 170. The Poisson distribution also uses the factorial function, but only on the value of X. It is therefore capable of dealing with very large sample sizes. The disadvantage of the Poisson is that it is only an approximation to the Binomial. Using another example out of Spiegel's text, we may find the probability that more than 2 individuals out of a sample of 2000 will suffer a bad reaction from a certain injection, if the probability that any individual will suffer is 0.001. This problem is clearly beyond the capabilities of the binomial distribution. The problem is solved first by finding the probability that at least 2 will suffer. This is: POIS 2 2000 0.001 [= 0.6767 The probability that more than 2 will suffer is obviously 1 minus this result: 1 - C [= 0.3233 which is the same as Spiegel's result. The probability that exactly 2 persons will suffer in this problem is found in register Y: Y [= 0.2707 NORM ---- The syntax for the normal probability is NORM [X M S F] | [Z] The form here indicates two modes of data entry; one which involves inputting the sample mean X, the population mean M, the sample standard deviation S, and the degrees of freedom F associated with that sample - or simply the z-score. The value of F should be equal to N - 1; that is, it should be one minus the sample size. FCA will automatically calculate the appropriate z-score if the first data entry method is used. The result is found in register Z. The terminology used here is compatible with the terminology used in the MEAN section of FCA. Since FCA only uses the absolute value of the z-score to calculate the probability of the null hypothesis, it does not matter whether MEAN is used to compute population or sample means. Thus, after entering the sample in MEAN, the user may exit and test a particular value of X, say 5, against those results by entering NORM 5 M S F FCA will pick up the values of M, S, F and proceed to calculate the z-score for the X-value 5. The result returned in C (and pasted to text, if STUF mode is active) is the probability of the null hypothesis - that is, the 'one-tailed' cumulative Normal probability associated with values of X equal to or greater than 5. If only one parameter is entered, NORM assumes that this value is the raw z-score. For example, the one-tailed Normal probability for a z-score of 1.96 is: NORM 1.96 [= 2.500D-02 which checks exactly with any standard table of Normal probabilities. 1.96 will be recognised by statisticians as the usual z-score associated with defining a two-tailed 95% confidence interval about a mean [2.500e-02 = 1 - 0.95/2]. The Normal ordinate for the input value of 1.96 is available in Y: Y STUD ---- The Normal distribution is not very useful except on large samples, for which reason the Student distribution was developed earlier this century by Gosset. The syntax is - STUD [X M S F] | [Z F] Same remarks for these variables and their purpose as given above for NORM. Note that, if the calculation of the t-score is avoided and direct entry is desired, the user must enter degrees of freedom F along with the score. For example, if the t-score is 12.706 based on a sample size only two data, so that there is only 1 degree of freedom, then the probability of the null hypothesis is STUD 12.706 1 [= 2.500D-02 which agrees exactly with statistical tables on the Student distribution. FCA does not yet support recovery of the Student ordinate values. The point about the importance of large samples for the Normal distribution may be seen by noting how many degrees of freedom are necessary to bring the Student probability into line with the Normal, using a constant score of 1.96. This can be seen in the following brief chart: Z = 1.96 Degrees of freedom Student probability ------------------ ------------------- 1 0.150 10 3.922D-02 30 2.967D-02 100 2.639D-02 500 2.528D-02 5000 2.503D-02 Note how the Student probability converges to the Normal as very large amounts of data are used to calculate the sample mean. Note also the probability with a sample size of 31 (F = 30): the value there is close enough to the Normal value of 2.500e-02 to let the Normal present a reasonably accurate picture of a two-tailed 95% confidence interval on that sample size. This is the reason why most statisticians don't consider a Normal test to be good unless the sample is based on at least 30 pieces of data. In text editing applications, where the user is evaluating a small number of results, say, sample sizes of the order of 3 to 12 pieces of data, and wants a quick analysis, the Normal distribution will not be helpful at all: The Student distribution will be required. And it is quite handy to have such a distribution immediately accessible at the touch of a hotkey. I have referred here to the "t-score"; but the syntax of the STUD function uses the variable Z. This is due to the fact that FCA uses the same definition for the standard deviation throughout all its statistics routines, resulting in the fact that the z-score and the t-score use the same formulas. This being the case, it was not necessary to distinguish a "T" variable for separate use with the Student distribution. Wait time --------- The Student integral is a relatively complex and slowly converging infinite series, and so, certain combinations of values will cause FCA to display the message Wait ... for an appreciable length of time. This means that the integral is being calculated. As stated above in the introductory portion of this file, FCA will take full advantage of a numeric coprocessor, if one is installed. Experiment shows that Student probability calculations run between 30 and 60 times faster when when there is an installed coprocessor. Normally, this wait time problem is not significant. But for very large t-scores and small values of F, execution time for the Student probability may prove to be substantial, even for fast machines. For example, STUD 64 1 [= 4.973D-03 requires 8 seconds on a 20 Mhz 386/387 machine. On a 486 machine, the expectation is that this result would require somewhat less than 2 seconds, since all such machines have numeric coprocessor capability built-in. On a 6 MHz AT without coprocessor, however, this calculation will require more than 22 MINUTES. So, if you see 'Wait ...' just wait: your machine is NOT hung. And avoid Z values substantially larger than 10, unless you have a coprocessor, or are prepared to wait for the result. Execution time rises about as the square of Z and drops in rough linear proportion to the value of F. Nearly always, however, the user will not be considering such large t-scores and will have samples which are larger than just two pieces of data (1 degree of freedom). For example, a t-score of 2.78 based on a calculated mean of a sample of size 5 - STUD 2.78 4 [= 2.491D-02 executes virtually instantaneously on a 386/387, and requires about 3 seconds on a 6 MHz AT without coprocessor. Prompt Colour ------------- As stated above, to change the colour of the calculation window, simply enter COLOR x in the window, where x is a COLOR number. The default is 112, which describes black foreground on a white background. 0 is impossible (black on black), and so FCA converts that number to 112 automatically. Foreground and Background colours may be determined by using the following table: Back Fore Bright Fore ---- ---- ----------- Black 0 0 8 Blue 16 1 9 Green 32 2 10 Cyan 48 3 11 Red 64 4 12 Magenta 80 5 13 Brown 96 6 14 White 112 7 15 The correct COLOR number is found merely by adding the Foreground number to the Background number desired. For example, Bright Green on Blue background is 10 + 16 = 26. Avoid setting COLOR above 127. Values above that limit will produce blinking displays. In my estimation the COLORs best for the eye are 10, 11, 14, 15, 26, 27, 30, 31, 74, 75, 78, 79 and 112. But you may have other ideas. Example, COLOR 75 sets the window to Bright Cyan on a Red background. Errors ------ FCA supports error reports as follows - xyz: illegal - This message occurs where your entered instruction, 'xyz', is not recognised by FCA. For example, 'TAN89' is illegal; should be 'TAN 89'. Read examples in documentation above to be sure you understand how to enter expressions. Illegal X setting - This means that you have tried to save the value of unsupported register into X. The supported registers are listed on the help screen. Zero divide error - Attempt to divide by zero. This can be an explicit mistake, like '5 / 0', or it may be an attempt to calculate the value of a unary function which divides two other functions to obtain its result. For example, the TAN, or Tangent of an angle, is really the ratio of the Sine and the Cosine of that angle; but the Cosine of 90 degrees is zero, so trying to find the Tangent of 90 will generate this error. Negative base error - You tried raising a negative number to a power (cannot be done on real numbers). Negative argument error - You tried taking the LOG of a negative number (cannot be done on real numbers). FCA also adds numerous other specific error message readouts, providing diagnostics on values of special variables, especially in reference to the probility and statistics functions. These are all self-explanatory. ---------------------- End of documentation