_Title "Taylor"

'program deriv by Oliver Aberth



'the original program computed the derivatives of the expression

'I changed it to compute the taylor series instead

'it was included as an example in the Basic interpreter called

'precision Basic which ran on CP/M and IMB compatible PC's



'there are shortcomings to the program, for example it fails to compute

'the series for sin(x)/x at x=0 due to division by 0.



'if the number of terms is 0 then it simply evaluates the expression

'involving the variable x using the value of the expansion point.



10 Dim A$(26), C#(200), E%(200), K%(200)

20 PI# = 3.141592653589793#

30 M% = 60

40 J% = 0

50 A$(2) = "x": A$(3) = "(": A$(4) = "Acos(": A$(5) = "Asin("

60 A$(6) = "Atan(": A$(7) = "Cos(": A$(8) = "Cosh(": A$(9) = "Exp("

70 A$(10) = "Log(": A$(11) = "Sin(": A$(12) = "Sinh(": A$(13) = "Sqr("

80 A$(14) = "Tan(": A$(15) = "Tanh(": A$(16) = "-": A$(17) = "+"

90 A$(21) = "+": A$(22) = "-": A$(23) = "*": A$(24) = "/": A$(25) = "^": A$(26) = ")"

100 Cls

110 Print

120 Print Tab(10); "Program to expand f(x) into a Taylor series"

130 Print

140 Input "Enter by number the highest term to be calculated ", N%

150 If Int(N%) <> N% Or N% < 0 Then 140

160 Print

170 Print "specify f(x) by entering successive elements of f(x) by code below"

180 Print "(entering a zero will delete the last f(x) element)"

190 Print

200 Print "C denotes any constant"

210 J% = 1: K% = 1: E9% = 0: E3% = 17

220 Print

230 GoTo 260

240 Print

250 Cls

260 Print "f(x)=";

270 If J% = 1 Then 360

280 For I% = 1 To J% - 1

    290 E% = E%(I%)

    300 If E% <> 1 Then 340

    310 K1% = K%(I%)

    320 Print C#(K1%);

    330 GoTo 350

    340 Print A$(E%);

350 Next I%

360 Print: Print

370 If E3% = 6 Then 450

380 Print "C X ( acos asin atan cos cosh exp  log sin sinh sqr  tan tanh";

390 If E3% = 17 Then Print "  -  +": GoTo 410

400 Print

410 Print "1 2 3  4    5    6  7    8  9    10  11  12  13  14  15";

420 If E3% = 17 Then Print "  16  17": GoTo 510

430 Print

440 GoTo 510

450 Print "+  -  *  /  ^  ";

460 If E9% > 0 Then Print ")": GoTo 480

470 Print "end of f(x)"

480 Print "1  2  3  4  5";

490 If E9% > 0 Then Print "  6": GoTo 510

500 Print "          6"

510 Input "Enter code integer ", E%

520 If Int(E%) <> E% Or E% < 0 Or E% > E3% Then 240

530 If E% > 0 Then 690

540 If J% = 1 Then 240

550 J% = J% - 1

560 E% = E%(J%)

570 If E% < 21 Then 610

580 E3% = 6

590 If E% = 26 Then E9% = E9% + 1

600 GoTo 240

610 If E% < 16 Then 640

620 E3% = 17

630 GoTo 240

640 E3% = 17

650 If E%(J% - 1) = 16 Or E%(J% - 1) = 17 Then E3% = 15

660 If E% = 1 Then K% = K% - 1

670 If E% >= 3 Then E9% = E9% - 1

680 GoTo 240

690 If E3% = 6 Then 860

700 If E3% = 15 Then 740

710 If E% <= 15 Then 740

720 E3% = 15

730 GoTo 920

740 If E% <> 1 Then 800

750 Input "Enter constant ", A$

760 C#(K%) = Val(A$ + "#")

770 K%(J%) = K%: K% = K% + 1

780 E3% = 6

790 GoTo 920

800 If E% <> 2 Then 830

810 E3% = 6

820 GoTo 920

830 E9% = E9% + 1

840 E3% = 17

850 GoTo 920

860 E% = E% + 20

870 If E% <> 26 Then 910

880 If E9% = 0 Then 950

890 E9% = E9% - 1

900 GoTo 920

910 E3% = 17

920 E%(J%) = E%

930 J% = J% + 1

940 GoTo 240

950 E%(J%) = 27

960 N1% = N% + 1

970 Dim O%(M%), V%(M%), S#(M%, N1%)

980 S#(0, 1) = 1#

990 If N% < 2 Then 1030

1000 For L% = 2 To N%

    1010 S#(0, L%) = 0#

1020 Next L%

1030 For L% = 1 To M%

    1040 S#(L%, N1%) = 0#

1050 Next L%

1060 Print

1070 For L% = 1 To M%

    1080 If S#(L%, N1%) <> 0# Then Stop

1090 Next L%

1100 Print "    press return to end or enter expantion point"

1101 Input "Enter x value at which series is to be expanded ", A$

1110 If A$ = "" Then End

1120 Z# = Val(A$ + "#")

1130 C#(0) = Z#

1140 Cls

1150 S#(0, 0) = C#(0): O% = 0: V% = 0: S# = 0#: J% = 0: O%(0) = 0: E3% = 17

1160 J% = J% + 1: E% = E%(J%)

1170 If E3% = 6 Then 1540

1180 If E3% = 15 Then 1250

1190 If E% < 16 Then 1250

1200 E3% = 15

1210 If E% = 17 Then 1160

1220 O% = O% + 1

1230 O%(O%) = 30

1240 GoTo 1160

1250 If E% = 1 Then 1380

1260 If E% = 2 Then 1340

1270 If E% = 3 Then 1300

1280 O% = O% + 1

1290 O%(O%) = E% + 30

1300 O% = O% + 1

1310 O%(O%) = 10

1320 E3% = 17

1330 GoTo 1160

1340 V%(V%) = 0

1350 V% = V% + 1

1360 E3% = 6

1370 GoTo 1160

1380 For I% = 1 To M%

    1390 If S#(I%, N1%) = 0# Then 1430

1400 Next I%

1410 Print "Stack filled"

1420 Stop

1430 V%(V%) = I%

1440 V% = V% + 1

1450 K% = K%(J%)

1460 S#(I%, 0) = C#(K%)

1470 If N% < 1 Then 1510

1480 For K% = 1 To N%

    1490 S#(I%, K%) = 0#

1500 Next K%

1510 S#(I%, N1%) = 1#

1520 E3% = 6

1530 GoTo 1160

1540 If E% > 24 Then 1580

1550 O2% = 20

1560 If E% > 22 Then O2% = 22

1570 GoTo 1600

1580 O2% = 30

1590 If E% > 25 Then O2% = 20

1600 If O%(O%) <= O2% Then 1630

1610 GoSub 1880

1620 GoTo 1600

1630 If E% = 27 Then 1720

1640 If E% = 26 Then 1690

1650 O% = O% + 1

1660 O%(O%) = E%

1670 E3% = 17

1680 GoTo 1160

1690 If O%(O%) <> 10 Then Stop

1700 O% = O% - 1

1710 GoTo 1160

1720 If O%(O%) <> 0 Or V% <> 1 Then Stop

1730 I% = V%(0)

1740 S#(I%, N1%) = 0#

1750 Print

1751 Print "if there are more than 20 terms to be printed then the program"

1752 Print "will pause after printing 20 terms and wait for a keypress"

1753 Print "to print another 20 terms and so on until all terms are printed"

1754 Input "press return to continue", A$

1756 Print

1760 Print "function    = ";

1770 If S#(I%, 0) >= 0# Then Print " ";

1780 Print S#(I%, 0)

1790 If N% < 1 Then 1870

1800 For K% = 1 To N%

    1810 Print "A"; K%;

    1820 Print Tab(14); "= ";

    1830 If S#(I%, K%) >= 0 Then Print " ";

    1840 Print S#(I%, K%)

    1850 If (K% Mod 20) = 0 Then Input "", A$

1860 Next K%

1865 Input "", A$

1870 GoTo 1060

1880 O1% = O%(O%): O% = O% - 1

1890 For K% = 1 To M%

    1900 If S#(K%, N1%) = 0# Then 1930

1910 Next K%

1920 Stop

1930 S#(K%, N1%) = 1#

1940 Z# = V% - 1#

1950 K2% = V%(Z#)

1960 S#(K2%, N1%) = 0#

1970 If O1% >= 30 Then 2890

1980 V% = Z#

1990 Z# = Z# - 1#

2000 K1% = V%(Z#)

2010 V%(Z#) = K%

2020 S#(K1%, N1%) = 0#

2030 If O1% = 21 Then 2080

2040 If O1% = 22 Then 2120

2050 If O1% = 23 Then 2160

2060 If O1% = 24 Then 2240

2070 GoTo 2340

2080 For L% = 0 To N%

    2090 S#(K%, L%) = S#(K1%, L%) + S#(K2%, L%)

2100 Next L%

2110 Return

2120 For L% = 0 To N%

    2130 S#(K%, L%) = S#(K1%, L%) - S#(K2%, L%)

2140 Next L%

2150 Return

2160 For L% = 0 To N%

    2170 Z# = 0#

    2180 For M1% = 0 To L%

        2190 Z# = Z# + S#(K1%, M1%) * S#(K2%, L% - M1%)

    2200 Next M1%

    2210 S#(K%, L%) = Z#

2220 Next L%

2230 Return

2240 Z1# = S#(K2%, 0)

2250 For L% = 0 To N%

    2260 Z# = S#(K1%, L%)

    2270 If L% = 0 Then 2310

    2280 For M1% = 1 To L%

        2290 Z# = Z# - S#(K2%, M1%) * S#(K%, L% - M1%)

    2300 Next M1%

    2310 S#(K%, L%) = Z# / Z1#

2320 Next L%

2330 Return

2340 If N% < 1 Then 2390

2350 For L% = 1 To N%

    2360 Z# = S#(K2%, L%)

    2370 If Z# <> 0# Then 2800

2380 Next L%

2390 Z1# = S#(K1%, 0)

2400 Z# = S#(K2%, 0)

2410 Z2# = Z# + 1#

2420 If Z1# = 0# Then If Int(Z#) = Z# And Z# > 0# Then 2550

2430 S#(K%, 0) = Z1# ^ Z#

2440 If N% < 1 Then 2540

2450 For L% = 1 To N%

    2460 Z# = 0#: Z3# = 0#

    2470 For M1% = 1 To L%

        2480 Z4# = S#(K%, L% - M1%) * S#(K1%, M1%)

        2490 Z3# = Z3# + Z4#

        2500 Z# = Z# + M1% * Z4#

    2510 Next M1%

    2520 S#(K%, L%) = (Z# * Z2# / L% - Z3#) / Z1#

2530 Next L%

2540 Return

2550 For K3% = 1 To M%

    2560 If S#(K3%, N1%) = 0 Then If K3% <> K2% Then 2590

2570 Next K3%

2580 Stop

2590 S#(K%, N1%) = 0#

2600 Z2# = Z#

2610 For L% = 0 To N%

    2620 S#(K%, L%) = 0#

    2630 S#(K3%, L%) = S#(K1%, L%)

2640 Next L%

2650 S#(K%, 0) = 1#

2660 K4% = K%

2670 Z# = Int(Z2# / 2)

2680 Z1# = Z2# - Z# - Z#

2690 Z2# = Z#

2700 If Z1# = 0 Then 2770

2710 K1% = K3%: Z# = K2%: K2% = K4%: K% = Z#: K4% = Z#

2720 GoSub 2160

2730 If Z2# > 0 Then 2770

2740 S#(K%, N1%) = 1#

2750 V%(V% - 1) = K%

2760 Return

2770 K1% = K3%: Z# = K2%: K2% = K3%: K% = Z#: K3% = Z#

2780 GoSub 2160

2790 GoTo 2670

2800 V%(V%) = K2%

2810 S#(K2%, N1%) = 1#

2820 V% = V% + 1: K2% = K1%

2830 GoSub 3070

2840 O1% = 23

2850 GoSub 1890

2860 O1% = 39

2870 GoSub 1890

2880 Return

2890 V%(Z#) = K%

2900 If O1% <> 30 Then 2950

2910 For L% = 0 To N%

    2920 S#(K%, L%) = -S#(K2%, L%)

2930 Next L%

2940 Return

2950 If O1% <> 39 Then 3060

2960 S#(K%, 0) = Exp(S#(K2%, 0))

2970 If N% < 1 Then 3050

2980 For L% = 1 To N%

    2990 Z# = 0#

    3000 For M1% = 1 To L%

        3010 Z# = Z# + M1% * S#(K%, L% - M1%) * S#(K2%, M1%)

    3020 Next M1%

    3030 S#(K%, L%) = Z# / L%

3040 Next L%

3050 Return

3060 If O1% <> 40 Then 3190

3070 Z2# = S#(K2%, 0)

3080 S#(K%, 0) = Log(Z2#)

3090 If N% < 1 Then 3180

3100 For L% = 1 To N%

    3110 Z# = 0#

    3120 If L% = 1 Then 3160

    3130 For M1% = 1 To L% - 1

        3140 Z# = Z# + M1% * S#(K2%, L% - M1%) * S#(K%, M1%)

    3150 Next M1%

    3160 S#(K%, L%) = (S#(K2%, L%) - Z# / L%) / Z2#

3170 Next L%

3180 Return

3190 If O1% = 37 Or O1% = 38 Or O1% = 41 Or O1% = 42 Or O1% = 44 Or O1% = 45 Then 3210

3200 GoTo 3550

3210 For K3% = 1 To M%

    3220 If S#(K3%, N1%) = 0 Then If K3% <> K2% Then 3250

3230 Next K3%

3240 Stop

3250 If Not (O1% = 37 Or O1% = 38) Then 3270

3260 Z# = K%: K% = K3%: K3% = Z#

3270 Z# = S#(K2%, 0)

3280 If O1% = 38 Or O1% = 42 Or O1% = 45 Then 3330

3290 S#(K%, 0) = Sin(Z#)

3300 S#(K3%, 0) = Cos(Z#)

3310 Z1# = -1#

3320 GoTo 3380

3330 S#(K%, 0) = Exp(Z#)

3340 S#(K%, 0) = .5# * (S#(K%, 0) - 1# / S#(K%, 0))

3350 S#(K3%, 0) = Exp(Z#)

3360 S#(K3%, 0) = .5# * (S#(K3%, 0) + 1# / S#(K3%, 0))

3370 Z1# = 1#

3380 If N% < 1 Then 3490

3390 For L% = 1 To N%

    3400 Z# = 0#: Z2# = 0#

    3410 For M1% = 1 To L%

        3420 Z3# = M1% * S#(K2%, M1%)

        3430 Z# = Z# + S#(K3%, L% - M1%) * Z3#

        3440 Z2# = Z2# + S#(K%, L% - M1%) * Z3#

    3450 Next M1%

    3460 S#(K%, L%) = Z# / L%

    3470 S#(K3%, L%) = Z2# * Z1# / L%

3480 Next L%

3490 If Not (O1% = 44 Or O1% = 45) Then Return

3500 S#(K3%, N1%) = 1#

3510 V%(V%) = K3%

3520 V% = V% + 1

3530 O1% = 24

3540 GoTo 1890

3550 If O1% <> 43 Then 3610

3560 Z1# = S#(K2%, 0)

3570 Z2# = 1.5#

3580 S#(K%, 0) = Sqr(Z1#)

3590 K1% = K2%

3600 GoTo 2440

3610 For K3% = 1 To M%

    3620 If S#(K3%, N1%) = 0 Then If K3% <> K2% Then 3650

3630 Next K3%

3640 Stop

3650 Z1# = -1#

3660 If O1% = 36 Then Z1# = 1#

3670 S#(K3%, 0) = 1# + Z1# * S#(K2%, 0) * S#(K2%, 0)

3680 If N% < 1 Then 3760

3690 For L% = 1 To N%

    3700 Z# = 0#

    3710 For M1% = 0 To L%

        3720 Z# = Z# + S#(K2%, M1%) * S#(K2%, L% - M1%)

    3730 Next M1%

    3740 S#(K3%, L%) = Z# * Z1#

3750 Next L%

3760 If O1% <> 36 Then 3890

3770 S#(K%, 0) = Atn(S#(K2%, 0))

3780 Z1# = S#(K3%, 0)

3790 If N% < 1 Then 3880

3800 For L% = 1 To N%

    3810 Z# = 0#

    3820 If L% = 1 Then 3860

    3830 For M1% = 1 To L% - 1

        3840 Z# = Z# + M1% * S#(K3%, L% - M1%) * S#(K%, M1%)

    3850 Next M1%

    3860 S#(K%, L%) = (S#(K2%, L%) - Z# / L%) / Z1#

3870 Next L%

3880 Return

3890 S#(K3%, N1%) = 1#

3900 S#(K%, N1%) = 0#

3910 V%(V% - 1) = K3%

3920 Z1# = S#(K3%, 0)

3930 S#(K%, 0) = Sqr(Z1#)

3940 Z2# = 1.5#

3950 If N% < 1 Then 4060

3960 For L% = 1 To N%

    3970 Z# = 0#

    3980 Z3# = 0#

    3990 For M1% = 1 To L%

        4000 Z4# = S#(K%, L% - M1%) * S#(K3%, M1%)

        4010 Z3# = Z3# + Z4#

        4020 Z# = Z# + Z4# * M1%

    4030 Next M1%

    4040 S#(K%, L%) = (Z# * Z2# / L% - Z3#) / Z1#

4050 Next L%

4060 Z1# = S#(K%, 0)

4070 X# = S#(K2%, 0)

4080 If Abs(X#) = 1 Then Y# = Sgn(X#) * .5 * PI# Else Y# = Atn(X# / Sqr(1# - X# * X#))

4090 S#(K3%, 0) = Y#

4100 If N% < 1 Then 4190

4110 For L% = 1 To N%

    4120 Z# = 0#

    4130 If L% = 1 Then 4170

    4140 For M1% = 1 To L% - 1

        4150 Z# = Z# + M1% * S#(K%, L% - M1%) * S#(K3%, M1%)

    4160 Next M1%

    4170 S#(K3%, L%) = (S#(K2%, L%) - Z# / L%) / Z1#

4180 Next L%

4190 If O1% = 35 Then Return

4200 S#(K3%, 0) = S#(K3%, 0) - .5# * PI#

4210 O1% = 30

4220 GoTo 1890