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+--- Thread: Newton had a fun way to approximate general roots... (/showthread.php?tid=884)
RE: Newton had a fun way to approximate general roots... - Kernelpanic - 09-15-2022
Ok, I didn't understand almost anything, but what the heck.
In case anyone is interested: A function that can be used to calculate the n-te root from a basis. Certainly still room for improvement when it comes to larger numbers, in terms of clean formatting.
Code: (Select All)
'N-te Wurzel berechnen - 15. Sept. 2022
'sqrtn(x, n) = x ^ (1 / n)
Option _Explicit
Declare Function Nte_Wurzel(n As Double, x As Double) As Double
Dim As Double n, x
Print
Input "Wurzelexponent: ", n
Input "Radikand : ", x
Print
Print Using "Die ##.# Wurzel aus ###.# ist: ##.###"; n, x, Nte_Wurzel(n, x)
End
Function Nte_Wurzel (n As Double, x As Double)
Dim wurzelx As Double
wurzelx = x ^ (1 / n)
Nte_Wurzel = wurzelx
End Function![[Image: n-te-Wurzel2022-09-15.jpg]](https://i.ibb.co/f1yKB9F/n-te-Wurzel2022-09-15.jpg)
RE: Newton had a fun way to approximate general roots... - Pete - 09-15-2022
I think I saw that formula when I was search around online. I couldn't use it, because I have not yet created a decimal power algorithm in my string math routines. For instance input 2 as the root and 16 as the number: The equation 1 / 2 = .5. Raising x ^.5 is not in my routines yet.
I may need to find a method to calculate decimal powers soon, but that method would have to be based on */+-. No logs.
Thanks for posting this, as it is great for small numbers. Very efficient!
Pete
RE: Newton had a fun way to approximate general roots... - Kernelpanic - 09-15-2022
(09-15-2022, 09:14 PM)Pete Wrote: I think I saw that formula when I was search around online. I couldn't use it, because I have not yet created a decimal power algorithm in my string math routines. For instance input 2 as the root and 16 as the number: The equation 1 / 2 = .5. Raising x ^.5 is not in my routines yet.
I may need to find a method to calculate decimal powers soon, but that method would have to be based on */+-. No logs.
Thanks for posting this, as it is great for small numbers. Very efficient!
Pete
The formula comes from here: Quick Basic: Mathematische Funktionen
I entered the 16th root of base 2. The result is correct - of course, if you check it, you get the usual inaccuracy: 1.99... instead of 2.
![[Image: n-te-Wurzel-2-2022-09-15.jpg]](https://i.ibb.co/gT0kpt5/n-te-Wurzel-2-2022-09-15.jpg)
RE: Newton had a fun way to approximate general roots... - Kernelpanic - 09-15-2022
Wait, the link has an error - but I don't know why. I have no problems.
Ok, a screenshot:
![[Image: n-te-Screenshot-Seite2022-09-15.jpg]](https://i.ibb.co/w6KwpD6/n-te-Screenshot-Seite2022-09-15.jpg)
RE: Newton had a fun way to approximate general roots... - Pete - 09-16-2022
Well back to working with Euler approximations. So based on his work, I put together what I think is a pretty cool way to get decimal roots like 7^2.3, 8 ^.7, etc. calculated without using logs.
So if this works, it can be adopted to use with string math routines I have already made.
DECIMAL POWERS ROUTINE
Code: (Select All)
WIDTH 120, 42
_SCREENMOVE 0, 0
DO
INPUT "Number: "; num
LINE INPUT "Power: "; power$
IF INSTR(power$, ".") THEN
t = VAL(MID$(power$, INSTR(power$, ".")))
power$ = MID$(power$, 1, INSTR(power$, ".") - 1)
a = 1
b = num
x = 0
y = 1
highc = b
lowc = a
highz = 1
DO
t1 = a * b
c = SQR(t1)
t1 = x + y
z = t1 / 2
IF z = t THEN EXIT DO
IF z > t THEN
highz = z: highc = c
ELSEIF z < t THEN
lowz = z: lowc = c
END IF
PRINT a, b, highc, lowc, z, highz, lowz
a = highc: b = lowc
x = highz: y = lowz
LOOP
IF LEN(power$) THEN
a = num ^ VAL(power$) * a
END IF
PRINT "Results = "; a
ELSE
PRINT "Results = "; num ^ VAL(power$)
END IF
PRINT
LOOPPete
RE: Newton had a fun way to approximate general roots... - Pete - 09-16-2022
Okay, here is a combo of decimal powers and general roots using the formula ...
Code: (Select All)
x ^ (1 / n)...provided by @Kernelpanic in a previous post. Too tired tonight to test out much tonight, but I know it will be littered with rounding errors like cube root of 125 = 4.999something.
Code: (Select All)
WIDTH 170, 42
_SCREENMOVE 0, 0
DIM SHARED limit&&, betatest%
limit&& = 32
betatest% = 0
DO
PRINT "Press 1 for general roots or 2 for decimal powers: ";
DO
_LIMIT 30
k$ = INKEY$
IF LEN(k$) THEN
IF k$ = CHR$(27) THEN SYSTEM
IF k$ = "1" OR k$ = "2" THEN EXIT DO
END IF
LOOP
PRINT k$: PRINT
SELECT CASE k$
CASE "1"
LINE INPUT "Number: "; num$
LINE INPUT "Root: "; root$
temp$ = sm_div$("1", root$)
PRINT sm_pow$(num$, temp$)
PRINT
CASE "2"
LINE INPUT "Number: "; num$
LINE INPUT "Power: "; power$
PRINT sm_pow$(num$, power$)
PRINT
END SELECT
LOOP
SUB sm_greater_lesser (stringmatha$, stringmathb$, gl%)
compa$ = stringmatha$: compb$ = stringmathb$ ' So original variables do not get changed.
DO
WHILE -1 ' Falx loop.
IF gl% = 2 THEN EXIT WHILE ' For bypassing sign and decimal adjustments when only positive non-decimal numbers are being evaluated.
' Remove trailing zeros after a decimal point.
IF INSTR(compa$, ".") THEN
DO UNTIL RIGHT$(compa$, 1) <> "0" AND RIGHT$(compa$, 1) <> "." AND RIGHT$(compa$, 1) <> "-"
compa$ = MID$(compa$, 1, LEN(compa$) - 1)
LOOP
END IF
IF INSTR(compb$, ".") THEN
DO UNTIL RIGHT$(compb$, 1) <> "0" AND RIGHT$(compb$, 1) <> "." AND RIGHT$(compb$, 1) <> "-"
compb$ = MID$(compb$, 1, LEN(compb$) - 1)
LOOP
END IF
IF MID$(compa$, 1, 2) = "-0" OR compa$ = "" OR compa$ = "-" THEN compa$ = "0"
IF MID$(compb$, 1, 2) = "-0" OR compb$ = "" OR compb$ = "-" THEN compb$ = "0"
' A - and +
j% = 0: k% = 0
IF LEFT$(compa$, 1) = "-" THEN j% = -1
IF LEFT$(compb$, 1) = "-" THEN k% = -1
IF k% = 0 AND j% THEN gl% = -1: EXIT DO
IF j% = 0 AND k% THEN gl% = 1: EXIT DO
j&& = INSTR(compa$, ".")
k&& = INSTR(compb$, ".")
' A starting decimal and non-decimal.
IF j&& = 0 AND k&& = 1 THEN
IF compa$ = "0" THEN gl% = -1 ELSE gl% = 1
EXIT DO
END IF
IF k&& = 0 AND j&& = 1 THEN
IF compb$ = "0" THEN gl% = 1 ELSE gl% = -1
EXIT DO
END IF
' remove decimals and align.
j2&& = 0: k2&& = 0
IF j&& <> 0 OR k&& <> 0 THEN
IF j&& THEN compa$ = MID$(compa$, 1, INSTR(compa$, ".") - 1) + MID$(compa$, INSTR(compa$, ".") + 1): j2&& = LEN(compa$) - j&& + 1
IF k&& THEN compb$ = MID$(compb$, 1, INSTR(compb$, ".") - 1) + MID$(compb$, INSTR(compb$, ".") + 1): k2&& = LEN(compb$) - k&& + 1
compa$ = compa$ + STRING$(k2&& - j2&&, "0")
compb$ = compb$ + STRING$(j2&& - k2&&, "0")
END IF
EXIT WHILE
WEND
' Remove leading zeros if any.
DO UNTIL LEFT$(compa$, 1) <> "0"
compa$ = MID$(compa$, 2)
LOOP
IF compa$ = "" THEN compa$ = "0"
DO UNTIL LEFT$(compb$, 1) <> "0"
compb$ = MID$(compb$, 2)
LOOP
IF compb$ = "" THEN compb$ = "0"
' Both positive or both negative whole numbers.
SELECT CASE LEN(compa$)
CASE IS < LEN(compb$)
gl% = -1
CASE IS = LEN(compb$)
IF compa$ = compb$ THEN
gl% = 0
ELSEIF compa$ > compb$ THEN gl% = 1
ELSEIF compa$ < compb$ THEN gl% = -1
END IF
CASE IS > LEN(compb$)
gl% = 1
END SELECT
EXIT DO
LOOP
END SUB
SUB sm_add_subtract_router (stringmatha$, operator$, stringmathb$, runningtotal$)
DIM AS _INTEGER64 a, c, s
a1$ = stringmatha$: b1$ = stringmathb$
s = 18: i&& = 0: c = 0
a$ = stringmatha$: b$ = stringmathb$: op$ = operator$
IF op$ = "-" THEN
IF LEFT$(b$, 1) = "-" THEN b$ = MID$(b$, 2) ELSE b$ = "-" + b$
END IF
IF INSTR(a$, ".") <> 0 OR INSTR(b$, ".") <> 0 THEN
decimal% = -1
IF INSTR(a$, ".") <> 0 THEN
dec_a&& = LEN(MID$(a$, INSTR(a$, ".") + 1))
a$ = MID$(a$, 1, INSTR(a$, ".") - 1) + MID$(a$, INSTR(a$, ".") + 1)
END IF
IF INSTR(b$, ".") <> 0 THEN
dec_b&& = LEN(MID$(b$, INSTR(b$, ".") + 1))
b$ = MID$(b$, 1, INSTR(b$, ".") - 1) + MID$(b$, INSTR(b$, ".") + 1)
END IF
' Line up decimal places by inserting trailing zeros.
IF dec_b&& > dec_a&& THEN
j&& = dec_b&&
a$ = a$ + STRING$(dec_b&& - dec_a&&, "0")
ELSE
j&& = dec_a&&
b$ = b$ + STRING$(dec_a&& - dec_b&&, "0")
END IF
END IF
IF LEFT$(a$, 1) = "-" OR LEFT$(b$, 1) = "-" THEN
IF LEFT$(a$, 1) = "-" AND LEFT$(b$, 1) = "-" THEN
sign$ = "": a$ = MID$(a$, 2): b$ = MID$(b$, 2)
ELSE
IF LEFT$(a$, 1) = "-" THEN a$ = MID$(a$, 2): sign_a$ = "-"
IF LEFT$(b$, 1) = "-" THEN b$ = MID$(b$, 2): sign_b$ = "-"
IF LEFT$(a1$, 1) = "-" THEN a1_x$ = MID$(a1$, 2) ELSE a1_x$ = a1$
IF LEFT$(b1$, 1) = "-" THEN b1_x$ = MID$(b1$, 2) ELSE b1_x$ = b1$
sm_greater_lesser a1_x$, b1_x$, gl%
IF gl% < 0 THEN
IF LEN(sign_b$) THEN sign$ = "-": SWAP a$, b$
ELSE
IF LEN(sign_a$) THEN sign$ = "-": SWAP sign_a$, sign_b$
END IF
END IF
END IF
z$ = ""
' Addition and subtraction of digits.
DO
i&& = i&& + s
x1$ = MID$(a$, LEN(a$) - i&& + 1, s)
x2$ = MID$(b$, LEN(b$) - i&& + 1, s)
IF LEN(x2$) > LEN(x1$) THEN SWAP x1$, x2$
a = VAL(sign_a$ + x1$) + VAL(sign_b$ + x2$) + c
IF x1$ + x2$ = "" AND c = 0 THEN EXIT DO
c = 0
IF a > VAL(STRING$(s, "9")) THEN a = a - 10 ^ s: c = 1
IF a < 0 THEN a = a + 10 ^ s: c = -1 ' a will never be less than 0.
tmp$ = LTRIM$(STR$(a))
z$ = STRING$(LEN(x1$) - LEN(tmp$), "0") + tmp$ + z$
LOOP
IF decimal% THEN
z$ = MID$(z$, 1, LEN(z$) - j&&) + "." + MID$(z$, LEN(z$) - j&& + 1)
END IF
' Remove any leading zeros.
DO
IF LEFT$(z$, 1) = "0" THEN z$ = MID$(z$, 2) ELSE EXIT DO
LOOP
IF z$ = "" OR z$ = "0" THEN z$ = "0" ELSE z$ = LEFT$(sign$, 1) + z$
runningtotal$ = z$
sign$ = "": sign_a$ = "": sign_b$ = "": i&& = 0: j&& = 0: decimal% = 0: c = 0
END SUB
FUNCTION sm_add$ (stringmatha$, stringmathb$)
operator$ = "+"
sm_add_subtract_router stringmatha$, operator$, stringmathb$, runningtotal$
sm_add$ = runningtotal$
END FUNCTION
FUNCTION sm_sub$ (stringmatha$, stringmathb$)
operator$ = "-"
sm_add_subtract_router stringmatha$, operator$, stringmathb$, runningtotal$
sm_sub$ = runningtotal$
END FUNCTION
FUNCTION sm_mult$ (stringmatha$, stringmathb$)
DIM AS _INTEGER64 a, c, aa, cc, s, ss
z$ = "": sign$ = "": mult&& = 0: h&& = 0: i&& = 0: j&& = 0: c = 0: decimal% = 0
zz$ = "": ii&& = 0: jj&& = 0
s = 8: ss = 18
a$ = stringmatha$: b$ = stringmathb$
IF INSTR(a$, "-") <> 0 OR INSTR(b$, "-") <> 0 THEN
IF INSTR(a$, "-") <> 0 AND INSTR(b$, "-") <> 0 THEN
a$ = MID$(a$, 2): b$ = MID$(b$, 2)
ELSE
IF INSTR(a$, "-") <> 0 THEN a$ = MID$(a$, 2) ELSE b$ = MID$(b$, 2)
sign$ = "-"
END IF
END IF
IF INSTR(a$, ".") <> 0 OR INSTR(b$, ".") <> 0 THEN
decimal% = -1
IF INSTR(a$, ".") <> 0 THEN
dec_a&& = LEN(MID$(a$, INSTR(a$, ".") + 1))
a$ = MID$(a$, 1, INSTR(a$, ".") - 1) + MID$(a$, INSTR(a$, ".") + 1)
END IF
IF INSTR(b$, ".") <> 0 THEN
dec_b&& = LEN(MID$(b$, INSTR(b$, ".") + 1))
b$ = MID$(b$, 1, INSTR(b$, ".") - 1) + MID$(b$, INSTR(b$, ".") + 1)
END IF
END IF
IF LEN(a$) < LEN(b$) THEN SWAP a$, b$ ' Needed so x1$ is always the largest for leading zero replacements.
' Multiplication of digits.
DO
h&& = h&& + s: i&& = 0
x2$ = MID$(b$, LEN(b$) - h&& + 1, s)
DO
i&& = i&& + s
x1$ = MID$(a$, LEN(a$) - i&& + 1, s)
a = VAL(x1$) * VAL(x2$) + c
c = 0
tmp$ = LTRIM$(STR$(a))
IF LEN(tmp$) > s THEN c = VAL(MID$(tmp$, 1, LEN(tmp$) - s)): tmp$ = MID$(tmp$, LEN(tmp$) - s + 1)
z$ = STRING$(LEN(x1$) - LEN(tmp$), "0") + tmp$ + z$
LOOP UNTIL i&& >= LEN(a$) AND c = 0
jj&& = jj&& + 1
IF jj&& > 1 THEN
ii&& = 0: cc = 0
aa$ = holdaa$
bb$ = z$ + STRING$((jj&& - 1) * s, "0")
' Addition only of digits.
DO
ii&& = ii&& + ss
xx1$ = MID$(aa$, LEN(aa$) - ii&& + 1, ss)
xx2$ = MID$(bb$, LEN(bb$) - ii&& + 1, ss)
IF LEN(xx1$) < LEN(xx2$) THEN SWAP xx1$, xx2$
aa = VAL(xx1$) + VAL(xx2$) + cc
IF xx1$ + xx2$ = "" AND cc = 0 THEN EXIT DO ' Prevents leading zeros.
cc = 0
IF aa > VAL(STRING$(ss, "9")) THEN aa = aa - 10 ^ ss: cc = 1
tmp$ = LTRIM$(STR$(aa))
zz$ = STRING$(LEN(xx1$) - LEN(tmp$), "0") + tmp$ + zz$
LOOP
DO WHILE LEFT$(zz$, 1) = "0"
IF LEFT$(zz$, 1) = "0" THEN zz$ = MID$(zz$, 2)
LOOP
IF zz$ = "" THEN zz$ = "0"
holdaa$ = zz$
ELSE
holdaa$ = z$ + STRING$(jj&& - 1, "0")
END IF
z$ = "": zz$ = ""
LOOP UNTIL h&& >= LEN(b$)
z$ = holdaa$
IF decimal% THEN
DO UNTIL LEN(z$) >= dec_a&& + dec_b&&
z$ = "0" + z$
LOOP
z$ = MID$(z$, 0, LEN(z$) - (dec_a&& + dec_b&& - 1)) + "." + MID$(z$, LEN(z$) - (dec_a&& + dec_b&&) + 1)
DO UNTIL RIGHT$(z$, 1) <> "0" AND RIGHT$(z$, 1) <> "."
z$ = MID$(z$, 1, LEN(z$) - 1)
LOOP
END IF
IF STRING$(LEN(z$), "0") = z$ OR z$ = "" OR z$ = "0" THEN z$ = "0" ELSE z$ = sign$ + z$
decimal% = 0: sign$ = ""
runningtotal$ = z$
sm_mult$ = z$
END FUNCTION
FUNCTION sm_div$ (stringmatha$, stringmathb$)
hold_stringmatha$ = stringmatha$: hold_stringmathb$ = stringmathb$
q$ = "": divisor$ = stringmathb$: dividend$ = stringmatha$
DO ' Falx loop.
'Strip off neg(s) and determine quotent sign.
IF LEFT$(divisor$, 1) = "-" THEN divisor$ = MID$(divisor$, 2): q$ = "-"
IF LEFT$(dividend$, 1) = "-" THEN dividend$ = MID$(dividend$, 2): IF q$ = "-" THEN q$ = "" ELSE q$ = "-"
' Quick results for divisor 1 or 0.
IF dividend$ = "0" THEN q$ = "0": EXIT DO
IF divisor$ = "1" THEN q$ = dividend$: EXIT DO
IF divisor$ = "0" THEN q$ = "Division by zero not possible.": EXIT DO
' Determine decimal direction. -1 to left, +1 to right.
gl% = 0: sm_greater_lesser divisor$, dividend$, gl%
IF betatest% AND gl% = 1 THEN PRINT divisor$; " > "; dividend$; " Move decimal to the left"
IF betatest% AND gl% = 0 THEN PRINT divisor$; " = "; dividend$
IF betatest% AND gl% = -1 THEN PRINT divisor$; " < "; dividend$; " Move deciml to the right."
IF gl% = 1 THEN ' Divisor is larger than dividend so decimal moves to the left.
div_decimal% = -1 ' Move decimal point to the left.
ELSEIF gl% = -1 THEN
div_decimal% = 1 ' Move decimal point to the right.
ELSE
' Divisor and dividend are the same number.
q$ = q$ + "1": EXIT DO
END IF
divisor_ratio_dividend% = gl%
' Strip off decimal point(s) and determine places in these next 2 routines.
dp&& = 0: dp2&& = 0: j2&& = 0
temp&& = INSTR(divisor$, ".")
IF temp&& THEN
divisor$ = MID$(divisor$, 1, temp&& - 1) + MID$(divisor$, temp&& + 1)
IF temp&& = 1 THEN
DO UNTIL LEFT$(divisor$, 1) <> "0" ' Strip off any leading zeros on divisor only.
divisor$ = MID$(divisor$, 2)
dp&& = dp&& + 1
LOOP
dp&& = dp&& + 1
ELSE
dp&& = -(temp&& - 2)
END IF
ELSE
dp&& = -(LEN(divisor$) - 1)
END IF
temp&& = INSTR(dividend$, ".")
IF temp&& THEN
dividend$ = MID$(dividend$, 1, temp&& - 1) + MID$(dividend$, temp&& + 1)
IF temp&& = 1 THEN
DO UNTIL LEFT$(dividend$, 1) <> "0" ' Strip off any leading zeros on divisor only.
dividend$ = MID$(dividend$, 2)
dp2&& = dp2&& + 1
LOOP
dp2&& = dp2&& + 1
ELSE
dp2&& = -(temp&& - 2)
END IF
ELSE
dp2&& = -(LEN(dividend$) - 1)
END IF
IF betatest% THEN COLOR 11: PRINT "Divisor decimal moves "; LTRIM$(STR$(dp&&)); ". Dividend decimal moves"; LTRIM$(STR$(dp2&&)); ". Quotent decimal ABS("; LTRIM$(STR$(dp&&)); " - "; LTRIM$(STR$(dp2&&)); ") =";: COLOR 14: PRINT ABS(dp&& - dp2&&);: COLOR 11: PRINT "+ any adjustment.": COLOR 7
dp&& = ABS(dp&& - dp2&&)
IF betatest% THEN PRINT "Divisor 1st# = "; MID$(divisor$, 1, 1); " Remainder 1st# = "; MID$(dividend$, 1, 1)
' Adjust decimal place for instances when divisor is larger than remainder the length of the divisor.
j% = 0
IF MID$(divisor$, 1, 1) > MID$(dividend$, 1, 1) THEN
j% = 1
IF betatest% THEN PRINT "Larger divisor, so move quotent decimal one place back to: ";: COLOR 14: PRINT LTRIM$(STR$(dp&&)): COLOR 7
ELSEIF MID$(divisor$, 1, 1) = MID$(dividend$, 1, 1) THEN
IF LEN(divisor$) = LEN(dividend$) THEN
IF divisor$ > dividend$ THEN j% = 1
ELSE
IF LEN(divisor$) > LEN(dividend$) THEN
temp$ = dividend$ + STRING$(LEN(divisor$) - LEN(dividend$), "0")
ELSE
temp$ = MID$(dividend$, 1, LEN(divisor$))
END IF
IF divisor$ > temp$ THEN j% = 1
END IF
IF j% THEN
dp&& = dp&& - div_decimal%
IF betatest% THEN PRINT "Larger divisor than dividend at LEN(divisor$), so move quotent decimal one place back to: ";: COLOR 14: PRINT LTRIM$(STR$(dp&&)): COLOR 7
ELSE
IF betatest% THEN PRINT "Smaller divisor than dividend at LEN(divisor$), so no quotent decimal place adjustment needed. Quotent decimal place = ";: COLOR 14: PRINT LTRIM$(STR$(dp&&)): COLOR 7
END IF
ELSE
j% = 0
IF betatest% THEN PRINT "Smaller divisor, so no quotent decimal place adjustment needed. Quotent decimal place = ";: COLOR 14: PRINT LTRIM$(STR$(dp&&)): COLOR 7
END IF
IF j% THEN dp&& = dp&& - div_decimal%
origdividend$ = dividend$
' Determine length of divisor and dividend to begin initial long divison step.
gl% = 2: sm_greater_lesser divisor$, MID$(dividend$, 1, LEN(divisor$)) + STRING$(LEN(divisor$) - LEN(dividend$), "0"), gl%
divisor_ratio_dividend% = gl%
IF gl% = 1 AND MID$(dividend$, 1, 1) <> "0" THEN
dividend$ = MID$(dividend$, 1, LEN(divisor$) + 1) + STRING$(LEN(divisor$) + 1 - LEN(dividend$), "0")
ELSE
dividend$ = MID$(dividend$, 1, LEN(divisor$)) + STRING$(LEN(divisor$) - LEN(dividend$), "0")
END IF
' Long divison loop. Mult and subtraction of dividend and remainder.
k&& = 0
IF betatest% THEN PRINT "Begin long divison loop..."
DO
SELECT CASE MID$(divisor$, 1, 1)
CASE IS < MID$(dividend$, 1, 1)
adj_rem_len% = 0
CASE IS = MID$(dividend$, 1, 1)
gl% = 2: sm_greater_lesser divisor$, MID$(dividend$, 1, LEN(divisor$)), gl%
IF gl% = 1 THEN adj_rem_len% = 1 ELSE adj_rem_len% = 0
CASE IS > MID$(dividend$, 1, 1)
adj_rem_len% = 1
END SELECT
IF j2&& = 0 THEN j2&& = LEN(divisor$) + adj_rem_len%
DO
IF LEN(divisor$) > LEN(dividend$) THEN
w3&& = 0: runningtotal$ = dividend$: stringmathb$ = "0"
IF betatest% THEN PRINT: COLOR 3: PRINT "Divisor is larger so "; dividend$; " \ "; divisor$; " =";: COLOR 5: PRINT w3&&: COLOR 7
EXIT DO
END IF
IF LEN(divisor$) = LEN(dividend$) THEN
gl% = 2: sm_greater_lesser divisor$, dividend$, gl%
IF gl% = 1 THEN
w3&& = 0: runningtotal$ = dividend$: stringmathb$ = "0"
IF betatest% THEN COLOR 9: PRINT "Length of divisor is the same as remainder but remainder is smaller so w3&& = ";: COLOR 5: PRINT "0": COLOR 7
EXIT DO
END IF
END IF
SELECT CASE LEN(dividend$)
CASE IS > 2
w3&& = VAL(MID$(dividend$, 1, 2 + adj_rem_len%)) \ VAL(MID$(divisor$, 1, 2))
IF betatest% THEN PRINT MID$(dividend$, 1, 2 + adj_rem_len%); " \ "; MID$(divisor$, 1, 2); " =";
CASE ELSE
w3&& = VAL(MID$(dividend$, 1, 1 + adj_rem_len%)) \ VAL(MID$(divisor$, 1, 1))
IF betatest% THEN PRINT MID$(dividend$, 1, 1 + adj_rem_len%); " \ "; MID$(divisor$, 1, 1); " =";
END SELECT
IF betatest% THEN COLOR 5: PRINT " " + LTRIM$(STR$(w3&&));: COLOR 7: PRINT ". Begin mult est. at or one above this number."
IF w3&& < 9 THEN w3&& = w3&& + 1 ELSE IF w3&& = 10 THEN w3&& = 9
DO
stringmatha$ = divisor$: stringmathb$ = LTRIM$(STR$(w3&&))
runningtotal$ = sm_mult$(divisor$, LTRIM$(STR$(w3&&)))
gl% = 2: sm_greater_lesser runningtotal$, dividend$, gl%
IF gl% <= 0 OR w3&& = 0 THEN EXIT DO
IF betatest% THEN COLOR 8: PRINT "Mult loop:"; w3&&; "* "; divisor$; " = "; runningtotal$: COLOR 7
w3&& = w3&& - 1
LOOP
stringmatha$ = dividend$: stringmathb$ = runningtotal$
sm_add_subtract_router dividend$, "-", stringmathb$, runningtotal$
EXIT DO
LOOP
IF betatest% THEN PRINT LTRIM$(STR$(w3&&)); " * "; divisor$; " = "; stringmathb$; " | "; stringmatha$; " - "; stringmathb$; " = "; runningtotal$; " Remainder and drop-down = ";
j2&& = j2&& + 1
drop$ = "0": MID$(drop$, 1, 1) = MID$(origdividend$, j2&&, 1)
IF runningtotal$ <> "0" THEN remainder$ = runningtotal$ ELSE remainder$ = ""
dividend$ = remainder$ + drop$
w3$ = LTRIM$(STR$(w3&&))
temp$ = ""
IF div_decimal% = -1 THEN
IF dp&& AND k&& = 0 THEN
q$ = q$ + "." + STRING$(dp&& - 1, "0")
IF w3&& = 0 THEN w3$ = ""
END IF
END IF
IF div_decimal% >= 0 THEN
IF dp&& = k&& THEN
temp$ = "."
END IF
END IF
q$ = q$ + w3$ + temp$
IF betatest% AND remainder$ = "" THEN betatemp$ = CHR$(34) + CHR$(34) ELSE IF betatest% THEN betatemp$ = remainder$
IF betatest% AND MID$(origdividend$, j2&&, 1) = "" THEN betatemp2$ = CHR$(34) + CHR$(34) ELSE IF betatest% THEN betatemp2$ = MID$(origdividend$, j2&&, 1)
IF betatest% THEN PRINT dividend$; " ("; betatemp$; " + "; drop$; ") at:"; j2&&; "of "; origdividend$; " Loop"; k&& + 1; "Quotent = ";: COLOR 14, 4: PRINT q$;: COLOR 7, 0: PRINT: SLEEP
' Check to terminate
IF div_decimal% = -1 THEN
' Decimal to left.
IF remainder$ = "" AND MID$(origdividend$, j2&&, 1) = "" OR LEN(q$) >= limit&& THEN EXIT DO
ELSE
' Decimal to right.
IF remainder$ = "" AND MID$(origdividend$, j2&&, 1) = "" AND k&& >= dp&& OR LEN(q$) >= limit&& THEN EXIT DO
END IF
IF INKEY$ = " " THEN EXIT DO
k&& = k&& + 1
LOOP
EXIT DO
LOOP
IF RIGHT$(q$, 1) = "." AND divisor$ <> "0" THEN runningtotal$ = MID$(q$, 1, LEN(q$) - 1) ELSE runningtotal$ = q$
sm_div$ = runningtotal$
stringmatha$ = hold_stringmatha$: stringmathb$ = hold_stringmathb$
END FUNCTION
FUNCTION sm_sqrt$ (sm_var$)
oldy$ = "": sqrt$ = "": custom_limit&& = limit&&: REM IF limit&& < 150 THEN custom_limit&& = 150 ELSE custom_limit&& = limit&&
sqrt_a$ = sm_var$
IF INSTR(sqrt_a$, ".") THEN
decx$ = MID$(sqrt_a$, 1, INSTR(sqrt_a$, ".") - 1)
sqrt_a$ = MID$(sqrt_a$, 1, INSTR(sqrt_a$, ".") - 1) + MID$(sqrt_a$, INSTR(sqrt_a$, ".") + 1)
IF LEN(sqrt_a$) = 1 THEN sqrt_a$ = sqrt_a$ + "0"
ELSE
decx$ = sqrt_a$
END IF
j&& = LEN(decx$)
' VAL() okay, one character eval.
IF VAL(RIGHT$(LTRIM$(STR$(j&&)), 1)) / 2 = VAL(RIGHT$(LTRIM$(STR$(j&&)), 1)) \ 2 THEN
i&& = 1 ' Even number length.
ELSE
i&& = 0 ' Odd number length.
END IF
DO
runningtotal$ = sm_sub$(z$, k$)
z$ = runningtotal$ + (MID$(sqrt_a$, i&&, 2))
IF LEFT$(z$, 1) = "0" THEN z$ = MID$(z$, 2) ' Remove leading zeros
oldy$ = ""
FOR j&& = 1 TO 10
IF i&& > 1 THEN
y$ = sm_mult$(sqrt$, "2")
y$ = y$ + LTRIM$(STR$(j&&))
ELSE
y$ = LTRIM$(STR$(j&&))
END IF
runningtotal$ = sm_mult$(y$, LTRIM$(STR$(j&&)))
sm_greater_lesser runningtotal$, z$, gl%
IF gl% > -1 THEN
IF gl% = 0 THEN
h% = 0: oldy$ = y$ ' Perfect square division.
ELSE
h% = 1
END IF
runningtotal$ = sm_mult$(oldy$, LTRIM$(STR$(j&& - h%)))
IF STRING$(LEN(z$), "0") = z$ AND runningtotal$ = "0" AND i&& >= LEN(decx$) THEN EXIT DO
IF dpx&& = 0 THEN ' Limited to && size unless converted to string.
IF i&& >= LEN(decx$) THEN
dpx&& = INT(LEN(decx$) / 2 + .5)
IF dpx&& = 0 THEN dpx&& = -1
END IF
END IF
IF betatest% < -1 THEN PRINT "Sqrt "; sqrt$; " * 2 = ";: COLOR 2, 0: PRINT LTRIM$(STR$(VAL(sqrt$) * 2));: COLOR 7, 0: PRINT LTRIM$(STR$(j&& - h%)); " * "; LTRIM$(STR$(j&& - h%)); " ="; VAL(oldy$) * (j&& - h%)
sqrt$ = sqrt$ + LTRIM$(STR$(j&& - h%))
runningtotal$ = sm_mult$(oldy$, LTRIM$(STR$(j&& - h%)))
k$ = runningtotal$
IF betatest% < -1 THEN PRINT "Remainder "; z$; " minus "; k$; " = ";
EXIT FOR
END IF
oldy$ = y$
NEXT
i&& = i&& + 2
IF LEN(z$) >= custom_limit&& THEN EXIT DO
sqrt_a$ = sqrt_a$ + "00"
LOOP
IF dpx&& THEN sqrt$ = MID$(sqrt$, 0, dpx&& + 1) + "." + MID$(sqrt$, dpx&& + 1)
sm_sqrt$ = sqrt$
END FUNCTION
FUNCTION sm_pow$ (num$, power$)
IF INSTR(power$, ".") THEN
t$ = MID$(power$, INSTR(power$, "."))
power$ = MID$(power$, 1, INSTR(power$, ".") - 1)
a$ = "1"
b$ = num$
x$ = "0"
y$ = "1"
highc$ = b$
lowc$ = a$
highz$ = "1"
DO
t1$ = sm_mult$(a$, b$)
c$ = sm_sqrt$(t1$)
t1$ = sm_add(x$, y$)
z$ = sm_div(t1$, "2")
sm_greater_lesser z$, t$, gl%
IF gl% >= 0 THEN
highz$ = z$: highc$ = c$
ELSEIF gl% = -1 THEN
lowz$ = z$: lowc$ = c$
END IF
REM PRINT a$, b$, highc$, lowc$, z$, highz$, lowz$
a$ = highc$: b$ = lowc$
x$ = highz$: y$ = lowz$
LOOP UNTIL gl% = 0
IF LEN(power$) THEN
'a = num ^ VAL(power$) * a
temp$ = num$
FOR i&& = 1 TO VAL(power$) - 1
temp2$ = sm_mult$(temp$, num$): temp$ = temp2$
NEXT
a$ = sm_mult$(temp2$, a$)
END IF
ELSE
'num ^ VAL(power$)
temp$ = num$
FOR i&& = 1 TO VAL(power$) - 1
a$ = sm_mult$(temp$, num$): temp$ = a$
NEXT
END IF
sm_pow$ = a$
END FUNCTIONPete
RE: Newton had a fun way to approximate general roots... - Pete - 09-21-2022
So back to the drawing board. Approximations would require too many rounding rules. Long division works perfectly for square roots, but I still can't get a handle on the extra hoops to apply for nth roots like cube root of 126.
Anyway, here is an optimized update of my previous square root long division method, set to 12 decimal places. It explains the process as it calculates the square root of 8.
Code: (Select All)
$CONSOLE:ONLY
DIM SHARED betatest%: betatest% = -2
num$ = "8"
PRINT sm_sqrt$(num$)
DO: _LIMIT 1: LOOP
FUNCTION sm_sqrt$ (sm_var$)
sqrt_a$ = sm_var$ ' Convert to prevent sm_var$ from being changed.
oldy$ = "": sqrt$ = "": k$ = "": z$ = ""
IF limit&& < 150 THEN custom_limit&& = 12 ELSE custom_limit&& = limit&&
IF INSTR(sqrt_a$, ".") THEN ' Number is a decimal.
decx$ = MID$(sqrt_a$, 1, INSTR(sqrt_a$, ".") - 1) ' Whole number portion of number.
sqrt_a$ = MID$(sqrt_a$, 1, INSTR(sqrt_a$, ".") - 1) + MID$(sqrt_a$, INSTR(sqrt_a$, ".") + 1) ' Remove decimal point, which converts deciaml to whole number number.
IF LEN(sqrt_a$) = 1 THEN sqrt_a$ = sqrt_a$ + "0"
ELSE
decx$ = sqrt_a$ ' Whole number.
END IF
h&& = LEN(decx$) ' The number of digits of the whole number (portion).
IF h&& / 2 = h&& \ 2 THEN i&& = 1 ELSE i&& = 0 ' 0 = Odd vs 1 = even number of digits in whole number (portion).
' Varible review:
' sqrt_a$ is the entire number with any decimal point removed.
' decx$ is the whole number (portion) of the number.
' h&& is number of digits in the whole number (portion).
' i&& index for parsing sqrt_a$ for drop-down division. If hh& is odd index starts at zero. Index at 1 if h&& is even.
' sqrt$ is the root answer per each iteration.
' custom_limit&& is the max. number of digits to evaluate the root number.
DO
IF betatest% < -1 THEN PRINT "Combine remainder ";: COLOR 12: PRINT sm_sub$(z$, k$);: COLOR 7: PRINT " with ";
z$ = sm_sub$(z$, k$) + (MID$(sqrt_a$, i&&, 2))
IF LEFT$(z$, 1) = "0" THEN z$ = MID$(z$, 2) ' Remove any leading zero.
IF betatest% < -1 THEN PRINT MID$(sqrt_a$, 1, i&& - 1);: COLOR 9: PRINT MID$(sqrt_a$, i&&, 2);: COLOR 7: PRINT MID$(sqrt_a$, i&& + 2); " and remove any leading zero to = ";: COLOR 6: PRINT z$;: COLOR 7: PRINT " as the target."
FOR j&& = 1 TO 10
IF i&& > 1 THEN
y$ = sm_mult$(sqrt$, "2") + LTRIM$(STR$(j&&))
COLOR 2: PRINT "y$ = (sqrt$ * 2 ~ j&& * J&&): "; sqrt$; " * 2 = "; sm_mult$(sqrt$, "2"); " ~"; j&&; "= "; y$; " *"; j&&; "= "; sm_mult$(y$, LTRIM$(STR$(j&&))): COLOR 7
ELSE
y$ = LTRIM$(STR$(j&&)) ' 1st loop starts here as there is no sqrt$ yet.
COLOR 2: PRINT "y$: j&& ="; j&&; "y$ is same as j&& = "; y$; " *"; j&&; " = "; sm_mult$(y$, LTRIM$(STR$(j&&))): COLOR 7
END IF
sm_greater_lesser sm_mult$(y$, LTRIM$(STR$(j&&))), z$, gl%
IF gl% > -1 THEN ' 0 = perfect square division. 1 = Too big so use the previous iteration.
IF gl% = 0 THEN
IF betatest% < -1 THEN PRINT "gl% ="; gl%; " Perfect square division. j&& =";: COLOR 14: PRINT j&&: COLOR 7
oldy$ = y$ ' Perfect square division. Not over, so do not subtract 1. j&& =";: COLOR 14: PRINT j&& - 1: COLOR 7
ELSE
IF betatest% < -1 THEN PRINT "gl% ="; gl%; " Estimate over target: "; sm_mult$(y$, LTRIM$(STR$(j&&))); " > "; z$; " so j&& =";: COLOR 14: PRINT j&& - 1: COLOR 7
j&& = j&& - 1 ' Adjusts size to the previous iteration.
END IF
IF STRING$(LEN(z$), "0") = z$ AND i&& >= h&& THEN
IF betatest% < -1 THEN PRINT "Checking for exit as z$ is null and i&&"; i&&; ">= h&&"; h&&; " so if oldy$ "; oldy$; " = zero then exit."
IF sm_mult$(oldy$, LTRIM$(STR$(j&&))) = "0" THEN EXIT DO
END IF
IF dpx&& = 0 THEN ' Decimal point relocator. Limited to && size unless converted to string.
IF i&& >= h&& THEN
dpx&& = INT(h&& / 2 + .5)
IF dpx&& = 0 THEN dpx&& = -1 ' Do not set to zero as -1 accomplishes the same thing and prevents ongoing loops here.
END IF
END IF
IF betatest% < -1 THEN PRINT "Sqrt combined with j&& * j&&: ";: temp$ = LTRIM$(STR$(VAL(MID$(sqrt$, 1, 1)))) + MID$(sqrt$, 2): PRINT temp$; " * 2 = "; LTRIM$(STR$(VAL(sqrt$) * 2));: COLOR 14: PRINT LTRIM$(STR$(j&&));: COLOR 7: PRINT " * ";: COLOR 14: PRINT LTRIM$(STR$(j&&));: COLOR 7: PRINT " =";: COLOR 5: PRINT VAL(oldy$) * j&&;: COLOR 7: PRINT " This is k$."
sqrt$ = sqrt$ + LTRIM$(STR$(j&&))
k$ = sm_mult$(oldy$, LTRIM$(STR$(j&&)))
IF betatest% < -1 THEN PRINT "Remainder ";: COLOR 6: PRINT z$;: COLOR 7: PRINT " - ";: COLOR 5: PRINT k$;: COLOR 7: PRINT " = ";: COLOR 12: PRINT sm_sub$(z$, k$): COLOR 7
EXIT FOR
END IF
oldy$ = y$
NEXT
oldy$ = ""
IF betatest% < -1 THEN PRINT "Increase index by 2 so i&& ="; i&& + 2
i&& = i&& + 2
IF betatest% <= -1 THEN IF LEN(z$) >= custom_limit&& THEN PRINT "Max. digits reached. Exiting..."
IF LEN(z$) >= custom_limit&& THEN EXIT DO
sqrt_a$ = sqrt_a$ + "00"
IF betatest% < -1 THEN PRINT "Multiply sqrt_a$ "; MID$(sqrt_a$, 1, LEN(sqrt_a$) - 2); " * 100 = "; sqrt_a$: SLEEP
LOOP
IF dpx&& THEN
sm_sqrt$ = MID$(sqrt$, 0, dpx&& + 1) + "." + MID$(sqrt$, dpx&& + 1)
ELSE
sm_sqrt$ = sqrt$
END IF
END FUNCTION
''----------------------------------------
SUB sm_greater_lesser (stringmatha$, stringmathb$, gl%)
compa$ = stringmatha$: compb$ = stringmathb$ ' So original variables do not get changed.
DO
WHILE -1 ' Falx loop.
IF gl% = 2 THEN EXIT WHILE ' For bypassing sign and decimal adjustments when only positive non-decimal numbers are being evaluated.
' Remove trailing zeros after a decimal point.
IF INSTR(compa$, ".") THEN
DO UNTIL RIGHT$(compa$, 1) <> "0" AND RIGHT$(compa$, 1) <> "." AND RIGHT$(compa$, 1) <> "-"
compa$ = MID$(compa$, 1, LEN(compa$) - 1)
LOOP
END IF
IF INSTR(compb$, ".") THEN
DO UNTIL RIGHT$(compb$, 1) <> "0" AND RIGHT$(compb$, 1) <> "." AND RIGHT$(compb$, 1) <> "-"
compb$ = MID$(compb$, 1, LEN(compb$) - 1)
LOOP
END IF
IF MID$(compa$, 1, 2) = "-0" OR compa$ = "" OR compa$ = "-" THEN compa$ = "0"
IF MID$(compb$, 1, 2) = "-0" OR compb$ = "" OR compb$ = "-" THEN compb$ = "0"
' A - and +
j% = 0: k% = 0
IF LEFT$(compa$, 1) = "-" THEN j% = -1
IF LEFT$(compb$, 1) = "-" THEN k% = -1
IF k% = 0 AND j% THEN gl% = -1: EXIT DO
IF j% = 0 AND k% THEN gl% = 1: EXIT DO
j&& = INSTR(compa$, ".")
k&& = INSTR(compb$, ".")
' A starting decimal and non-decimal.
IF j&& = 0 AND k&& = 1 THEN
IF compa$ = "0" THEN gl% = -1 ELSE gl% = 1
EXIT DO
END IF
IF k&& = 0 AND j&& = 1 THEN
IF compb$ = "0" THEN gl% = 1 ELSE gl% = -1
EXIT DO
END IF
' remove decimals and align.
j2&& = 0: k2&& = 0
IF j&& <> 0 OR k&& <> 0 THEN
IF j&& THEN compa$ = MID$(compa$, 1, INSTR(compa$, ".") - 1) + MID$(compa$, INSTR(compa$, ".") + 1): j2&& = LEN(compa$) - j&& + 1
IF k&& THEN compb$ = MID$(compb$, 1, INSTR(compb$, ".") - 1) + MID$(compb$, INSTR(compb$, ".") + 1): k2&& = LEN(compb$) - k&& + 1
compa$ = compa$ + STRING$(k2&& - j2&&, "0")
compb$ = compb$ + STRING$(j2&& - k2&&, "0")
END IF
EXIT WHILE
WEND
' Remove leading zeros if any.
DO UNTIL LEFT$(compa$, 1) <> "0"
compa$ = MID$(compa$, 2)
LOOP
IF compa$ = "" THEN compa$ = "0"
DO UNTIL LEFT$(compb$, 1) <> "0"
compb$ = MID$(compb$, 2)
LOOP
IF compb$ = "" THEN compb$ = "0"
' Both positive or both negative whole numbers.
SELECT CASE LEN(compa$)
CASE IS < LEN(compb$)
gl% = -1
CASE IS = LEN(compb$)
IF compa$ = compb$ THEN
gl% = 0
ELSEIF compa$ > compb$ THEN gl% = 1
ELSEIF compa$ < compb$ THEN gl% = -1
END IF
CASE IS > LEN(compb$)
gl% = 1
END SELECT
EXIT DO
LOOP
END SUB
SUB sm_add_subtract_router (stringmatha$, operator$, stringmathb$, runningtotal$)
DIM AS _INTEGER64 a, c, s
a1$ = stringmatha$: b1$ = stringmathb$
s = 18: i&& = 0: c = 0
a$ = stringmatha$: b$ = stringmathb$: op$ = operator$
IF op$ = "-" THEN
IF LEFT$(b$, 1) = "-" THEN b$ = MID$(b$, 2) ELSE b$ = "-" + b$
END IF
IF INSTR(a$, ".") <> 0 OR INSTR(b$, ".") <> 0 THEN
decimal% = -1
IF INSTR(a$, ".") <> 0 THEN
dec_a&& = LEN(MID$(a$, INSTR(a$, ".") + 1))
a$ = MID$(a$, 1, INSTR(a$, ".") - 1) + MID$(a$, INSTR(a$, ".") + 1)
END IF
IF INSTR(b$, ".") <> 0 THEN
dec_b&& = LEN(MID$(b$, INSTR(b$, ".") + 1))
b$ = MID$(b$, 1, INSTR(b$, ".") - 1) + MID$(b$, INSTR(b$, ".") + 1)
END IF
' Line up decimal places by inserting trailing zeros.
IF dec_b&& > dec_a&& THEN
j&& = dec_b&&
a$ = a$ + STRING$(dec_b&& - dec_a&&, "0")
ELSE
j&& = dec_a&&
b$ = b$ + STRING$(dec_a&& - dec_b&&, "0")
END IF
END IF
IF LEFT$(a$, 1) = "-" OR LEFT$(b$, 1) = "-" THEN
IF LEFT$(a$, 1) = "-" AND LEFT$(b$, 1) = "-" THEN
sign$ = "": a$ = MID$(a$, 2): b$ = MID$(b$, 2)
ELSE
IF LEFT$(a$, 1) = "-" THEN a$ = MID$(a$, 2): sign_a$ = "-"
IF LEFT$(b$, 1) = "-" THEN b$ = MID$(b$, 2): sign_b$ = "-"
IF LEFT$(a1$, 1) = "-" THEN a1_x$ = MID$(a1$, 2) ELSE a1_x$ = a1$
IF LEFT$(b1$, 1) = "-" THEN b1_x$ = MID$(b1$, 2) ELSE b1_x$ = b1$
sm_greater_lesser a1_x$, b1_x$, gl%
IF gl% < 0 THEN
IF LEN(sign_b$) THEN sign$ = "-": SWAP a$, b$
ELSE
IF LEN(sign_a$) THEN sign$ = "-": SWAP sign_a$, sign_b$
END IF
END IF
END IF
z$ = ""
' Addition and subtraction of digits.
DO
i&& = i&& + s
x1$ = MID$(a$, LEN(a$) - i&& + 1, s)
x2$ = MID$(b$, LEN(b$) - i&& + 1, s)
IF LEN(x2$) > LEN(x1$) THEN SWAP x1$, x2$
a = VAL(sign_a$ + x1$) + VAL(sign_b$ + x2$) + c
IF x1$ + x2$ = "" AND c = 0 THEN EXIT DO
c = 0
IF a > VAL(STRING$(s, "9")) THEN a = a - 10 ^ s: c = 1
IF a < 0 THEN a = a + 10 ^ s: c = -1 ' a will never be less than 0.
tmp$ = LTRIM$(STR$(a))
z$ = STRING$(LEN(x1$) - LEN(tmp$), "0") + tmp$ + z$
LOOP
IF decimal% THEN
z$ = MID$(z$, 1, LEN(z$) - j&&) + "." + MID$(z$, LEN(z$) - j&& + 1)
END IF
' Remove any leading zeros.
DO
IF LEFT$(z$, 1) = "0" THEN z$ = MID$(z$, 2) ELSE EXIT DO
LOOP
IF z$ = "" OR z$ = "0" THEN z$ = "0" ELSE z$ = LEFT$(sign$, 1) + z$
runningtotal$ = z$
sign$ = "": sign_a$ = "": sign_b$ = "": i&& = 0: j&& = 0: decimal% = 0: c = 0
END SUB
FUNCTION sm_add$ (stringmatha$, stringmathb$)
operator$ = "+"
sm_add_subtract_router stringmatha$, operator$, stringmathb$, runningtotal$
sm_add$ = runningtotal$
END FUNCTION
FUNCTION sm_sub$ (stringmatha$, stringmathb$)
operator$ = "-"
sm_add_subtract_router stringmatha$, operator$, stringmathb$, runningtotal$
sm_sub$ = runningtotal$
END FUNCTION
FUNCTION sm_mult$ (stringmatha$, stringmathb$)
DIM AS _INTEGER64 a, c, aa, cc, s, ss
z$ = "": sign$ = "": mult&& = 0: h&& = 0: i&& = 0: j&& = 0: c = 0: decimal% = 0
zz$ = "": ii&& = 0: jj&& = 0
s = 8: ss = 18
a$ = stringmatha$: b$ = stringmathb$
IF INSTR(a$, "-") <> 0 OR INSTR(b$, "-") <> 0 THEN
IF INSTR(a$, "-") <> 0 AND INSTR(b$, "-") <> 0 THEN
a$ = MID$(a$, 2): b$ = MID$(b$, 2)
ELSE
IF INSTR(a$, "-") <> 0 THEN a$ = MID$(a$, 2) ELSE b$ = MID$(b$, 2)
sign$ = "-"
END IF
END IF
IF INSTR(a$, ".") <> 0 OR INSTR(b$, ".") <> 0 THEN
decimal% = -1
IF INSTR(a$, ".") <> 0 THEN
dec_a&& = LEN(MID$(a$, INSTR(a$, ".") + 1))
a$ = MID$(a$, 1, INSTR(a$, ".") - 1) + MID$(a$, INSTR(a$, ".") + 1)
END IF
IF INSTR(b$, ".") <> 0 THEN
dec_b&& = LEN(MID$(b$, INSTR(b$, ".") + 1))
b$ = MID$(b$, 1, INSTR(b$, ".") - 1) + MID$(b$, INSTR(b$, ".") + 1)
END IF
END IF
IF LEN(a$) < LEN(b$) THEN SWAP a$, b$ ' Needed so x1$ is always the largest for leading zero replacements.
' Multiplication of digits.
DO
h&& = h&& + s: i&& = 0
x2$ = MID$(b$, LEN(b$) - h&& + 1, s)
DO
i&& = i&& + s
x1$ = MID$(a$, LEN(a$) - i&& + 1, s)
a = VAL(x1$) * VAL(x2$) + c
c = 0
tmp$ = LTRIM$(STR$(a))
IF LEN(tmp$) > s THEN c = VAL(MID$(tmp$, 1, LEN(tmp$) - s)): tmp$ = MID$(tmp$, LEN(tmp$) - s + 1)
z$ = STRING$(LEN(x1$) - LEN(tmp$), "0") + tmp$ + z$
LOOP UNTIL i&& >= LEN(a$) AND c = 0
jj&& = jj&& + 1
IF jj&& > 1 THEN
ii&& = 0: cc = 0
aa$ = holdaa$
bb$ = z$ + STRING$((jj&& - 1) * s, "0")
' Addition only of digits.
DO
ii&& = ii&& + ss
xx1$ = MID$(aa$, LEN(aa$) - ii&& + 1, ss)
xx2$ = MID$(bb$, LEN(bb$) - ii&& + 1, ss)
IF LEN(xx1$) < LEN(xx2$) THEN SWAP xx1$, xx2$
aa = VAL(xx1$) + VAL(xx2$) + cc
IF xx1$ + xx2$ = "" AND cc = 0 THEN EXIT DO ' Prevents leading zeros.
cc = 0
IF aa > VAL(STRING$(ss, "9")) THEN aa = aa - 10 ^ ss: cc = 1
tmp$ = LTRIM$(STR$(aa))
zz$ = STRING$(LEN(xx1$) - LEN(tmp$), "0") + tmp$ + zz$
LOOP
DO WHILE LEFT$(zz$, 1) = "0"
IF LEFT$(zz$, 1) = "0" THEN zz$ = MID$(zz$, 2)
LOOP
IF zz$ = "" THEN zz$ = "0"
holdaa$ = zz$
ELSE
holdaa$ = z$ + STRING$(jj&& - 1, "0")
END IF
z$ = "": zz$ = ""
LOOP UNTIL h&& >= LEN(b$)
z$ = holdaa$
IF decimal% THEN
DO UNTIL LEN(z$) >= dec_a&& + dec_b&&
z$ = "0" + z$
LOOP
z$ = MID$(z$, 0, LEN(z$) - (dec_a&& + dec_b&& - 1)) + "." + MID$(z$, LEN(z$) - (dec_a&& + dec_b&&) + 1)
DO UNTIL RIGHT$(z$, 1) <> "0" AND RIGHT$(z$, 1) <> "."
z$ = MID$(z$, 1, LEN(z$) - 1)
LOOP
END IF
IF STRING$(LEN(z$), "0") = z$ OR z$ = "" OR z$ = "0" THEN z$ = "0" ELSE z$ = sign$ + z$
decimal% = 0: sign$ = ""
runningtotal$ = z$
sm_mult$ = z$
END FUNCTION
FUNCTION sm_div$ (stringmatha$, stringmathb$)
hold_stringmatha$ = stringmatha$: hold_stringmathb$ = stringmathb$
q$ = "": divisor$ = stringmathb$: dividend$ = stringmatha$
DO ' Falx loop.
'Strip off neg(s) and determine quotent sign.
IF LEFT$(divisor$, 1) = "-" THEN divisor$ = MID$(divisor$, 2): q$ = "-"
IF LEFT$(dividend$, 1) = "-" THEN dividend$ = MID$(dividend$, 2): IF q$ = "-" THEN q$ = "" ELSE q$ = "-"
' Quick results for divisor 1 or 0.
IF dividend$ = "0" THEN q$ = "0": EXIT DO
IF divisor$ = "1" THEN q$ = dividend$: EXIT DO
IF divisor$ = "0" THEN q$ = "Division by zero not possible.": EXIT DO
' Determine decimal direction. -1 to left, +1 to right.
gl% = 0: sm_greater_lesser divisor$, dividend$, gl%
IF betatest% AND gl% = 1 THEN PRINT divisor$; " > "; dividend$; " Move decimal to the left"
IF betatest% AND gl% = 0 THEN PRINT divisor$; " = "; dividend$
IF betatest% AND gl% = -1 THEN PRINT divisor$; " < "; dividend$; " Move deciml to the right."
IF gl% = 1 THEN ' Divisor is larger than dividend so decimal moves to the left.
div_decimal% = -1 ' Move decimal point to the left.
ELSEIF gl% = -1 THEN
div_decimal% = 1 ' Move decimal point to the right.
ELSE
' Divisor and dividend are the same number.
q$ = q$ + "1": EXIT DO
END IF
divisor_ratio_dividend% = gl%
' Strip off decimal point(s) and determine places in these next 2 routines.
dp&& = 0: dp2&& = 0: j2&& = 0
temp&& = INSTR(divisor$, ".")
IF temp&& THEN
divisor$ = MID$(divisor$, 1, temp&& - 1) + MID$(divisor$, temp&& + 1)
IF temp&& = 1 THEN
DO UNTIL LEFT$(divisor$, 1) <> "0" ' Strip off any leading zeros on divisor only.
divisor$ = MID$(divisor$, 2)
dp&& = dp&& + 1
LOOP
dp&& = dp&& + 1
ELSE
dp&& = -(temp&& - 2)
END IF
ELSE
dp&& = -(LEN(divisor$) - 1)
END IF
temp&& = INSTR(dividend$, ".")
IF temp&& THEN
dividend$ = MID$(dividend$, 1, temp&& - 1) + MID$(dividend$, temp&& + 1)
IF temp&& = 1 THEN
DO UNTIL LEFT$(dividend$, 1) <> "0" ' Strip off any leading zeros on divisor only.
dividend$ = MID$(dividend$, 2)
dp2&& = dp2&& + 1
LOOP
dp2&& = dp2&& + 1
ELSE
dp2&& = -(temp&& - 2)
END IF
ELSE
dp2&& = -(LEN(dividend$) - 1)
END IF
IF betatest% THEN COLOR 11: PRINT "Divisor decimal moves "; LTRIM$(STR$(dp&&)); ". Dividend decimal moves"; LTRIM$(STR$(dp2&&)); ". Quotent decimal ABS("; LTRIM$(STR$(dp&&)); " - "; LTRIM$(STR$(dp2&&)); ") =";: COLOR 14: PRINT ABS(dp&& - dp2&&);: COLOR 11: PRINT "+ any adjustment.": COLOR 7
dp&& = ABS(dp&& - dp2&&)
IF betatest% THEN PRINT "Divisor 1st# = "; MID$(divisor$, 1, 1); " Remainder 1st# = "; MID$(dividend$, 1, 1)
' Adjust decimal place for instances when divisor is larger than remainder the length of the divisor.
j% = 0
IF MID$(divisor$, 1, 1) > MID$(dividend$, 1, 1) THEN
j% = 1
IF betatest% THEN PRINT "Larger divisor, so move quotent decimal one place back to: ";: COLOR 14: PRINT LTRIM$(STR$(dp&&)): COLOR 7
ELSEIF MID$(divisor$, 1, 1) = MID$(dividend$, 1, 1) THEN
IF LEN(divisor$) = LEN(dividend$) THEN
IF divisor$ > dividend$ THEN j% = 1
ELSE
IF LEN(divisor$) > LEN(dividend$) THEN
temp$ = dividend$ + STRING$(LEN(divisor$) - LEN(dividend$), "0")
ELSE
temp$ = MID$(dividend$, 1, LEN(divisor$))
END IF
IF divisor$ > temp$ THEN j% = 1
END IF
IF j% THEN
dp&& = dp&& - div_decimal%
IF betatest% THEN PRINT "Larger divisor than dividend at LEN(divisor$), so move quotent decimal one place back to: ";: COLOR 14: PRINT LTRIM$(STR$(dp&&)): COLOR 7
ELSE
IF betatest% THEN PRINT "Smaller divisor than dividend at LEN(divisor$), so no quotent decimal place adjustment needed. Quotent decimal place = ";: COLOR 14: PRINT LTRIM$(STR$(dp&&)): COLOR 7
END IF
ELSE
j% = 0
IF betatest% THEN PRINT "Smaller divisor, so no quotent decimal place adjustment needed. Quotent decimal place = ";: COLOR 14: PRINT LTRIM$(STR$(dp&&)): COLOR 7
END IF
IF j% THEN dp&& = dp&& - div_decimal%
origdividend$ = dividend$
' Determine length of divisor and dividend to begin initial long divison step.
gl% = 2: sm_greater_lesser divisor$, MID$(dividend$, 1, LEN(divisor$)) + STRING$(LEN(divisor$) - LEN(dividend$), "0"), gl%
divisor_ratio_dividend% = gl%
IF gl% = 1 AND MID$(dividend$, 1, 1) <> "0" THEN
dividend$ = MID$(dividend$, 1, LEN(divisor$) + 1) + STRING$(LEN(divisor$) + 1 - LEN(dividend$), "0")
ELSE
dividend$ = MID$(dividend$, 1, LEN(divisor$)) + STRING$(LEN(divisor$) - LEN(dividend$), "0")
END IF
' Long divison loop. Mult and subtraction of dividend and remainder.
k&& = 0
IF betatest% THEN PRINT "Begin long divison loop..."
DO
SELECT CASE MID$(divisor$, 1, 1)
CASE IS < MID$(dividend$, 1, 1)
adj_rem_len% = 0
CASE IS = MID$(dividend$, 1, 1)
gl% = 2: sm_greater_lesser divisor$, MID$(dividend$, 1, LEN(divisor$)), gl%
IF gl% = 1 THEN adj_rem_len% = 1 ELSE adj_rem_len% = 0
CASE IS > MID$(dividend$, 1, 1)
adj_rem_len% = 1
END SELECT
IF j2&& = 0 THEN j2&& = LEN(divisor$) + adj_rem_len%
DO
IF LEN(divisor$) > LEN(dividend$) THEN
w3&& = 0: runningtotal$ = dividend$: stringmathb$ = "0"
IF betatest% THEN PRINT: COLOR 3: PRINT "Divisor is larger so "; dividend$; " \ "; divisor$; " =";: COLOR 5: PRINT w3&&: COLOR 7
EXIT DO
END IF
IF LEN(divisor$) = LEN(dividend$) THEN
gl% = 2: sm_greater_lesser divisor$, dividend$, gl%
IF gl% = 1 THEN
w3&& = 0: runningtotal$ = dividend$: stringmathb$ = "0"
IF betatest% THEN COLOR 9: PRINT "Length of divisor is the same as remainder but remainder is smaller so w3&& = ";: COLOR 5: PRINT "0": COLOR 7
EXIT DO
END IF
END IF
SELECT CASE LEN(dividend$)
CASE IS > 2
w3&& = VAL(MID$(dividend$, 1, 2 + adj_rem_len%)) \ VAL(MID$(divisor$, 1, 2))
IF betatest% THEN PRINT MID$(dividend$, 1, 2 + adj_rem_len%); " \ "; MID$(divisor$, 1, 2); " =";
CASE ELSE
w3&& = VAL(MID$(dividend$, 1, 1 + adj_rem_len%)) \ VAL(MID$(divisor$, 1, 1))
IF betatest% THEN PRINT MID$(dividend$, 1, 1 + adj_rem_len%); " \ "; MID$(divisor$, 1, 1); " =";
END SELECT
IF betatest% THEN COLOR 5: PRINT " " + LTRIM$(STR$(w3&&));: COLOR 7: PRINT ". Begin mult est. at or one above this number."
IF w3&& < 9 THEN w3&& = w3&& + 1 ELSE IF w3&& = 10 THEN w3&& = 9
DO
stringmatha$ = divisor$: stringmathb$ = LTRIM$(STR$(w3&&))
runningtotal$ = sm_mult$(divisor$, LTRIM$(STR$(w3&&)))
gl% = 2: sm_greater_lesser runningtotal$, dividend$, gl%
IF gl% <= 0 OR w3&& = 0 THEN EXIT DO
IF betatest% THEN COLOR 8: PRINT "Mult loop:"; w3&&; "* "; divisor$; " = "; runningtotal$: COLOR 7
w3&& = w3&& - 1
LOOP
stringmatha$ = dividend$: stringmathb$ = runningtotal$
sm_add_subtract_router dividend$, "-", stringmathb$, runningtotal$
EXIT DO
LOOP
IF betatest% THEN PRINT LTRIM$(STR$(w3&&)); " * "; divisor$; " = "; stringmathb$; " | "; stringmatha$; " - "; stringmathb$; " = "; runningtotal$; " Remainder and drop-down = ";
j2&& = j2&& + 1
drop$ = "0": MID$(drop$, 1, 1) = MID$(origdividend$, j2&&, 1)
IF runningtotal$ <> "0" THEN remainder$ = runningtotal$ ELSE remainder$ = ""
dividend$ = remainder$ + drop$
w3$ = LTRIM$(STR$(w3&&))
temp$ = ""
IF div_decimal% = -1 THEN
IF dp&& AND k&& = 0 THEN
q$ = q$ + "." + STRING$(dp&& - 1, "0")
IF w3&& = 0 THEN w3$ = ""
END IF
END IF
IF div_decimal% >= 0 THEN
IF dp&& = k&& THEN
temp$ = "."
END IF
END IF
q$ = q$ + w3$ + temp$
IF betatest% AND remainder$ = "" THEN betatemp$ = CHR$(34) + CHR$(34) ELSE IF betatest% THEN betatemp$ = remainder$
IF betatest% AND MID$(origdividend$, j2&&, 1) = "" THEN betatemp2$ = CHR$(34) + CHR$(34) ELSE IF betatest% THEN betatemp2$ = MID$(origdividend$, j2&&, 1)
IF betatest% THEN PRINT dividend$; " ("; betatemp$; " + "; drop$; ") at:"; j2&&; "of "; origdividend$; " Loop"; k&& + 1; "Quotent = ";: COLOR 14, 4: PRINT q$;: COLOR 7, 0: PRINT: SLEEP
' Check to terminate
IF div_decimal% = -1 THEN
' Decimal to left.
IF remainder$ = "" AND MID$(origdividend$, j2&&, 1) = "" OR LEN(q$) >= limit&& THEN EXIT DO
ELSE
' Decimal to right.
IF remainder$ = "" AND MID$(origdividend$, j2&&, 1) = "" AND k&& >= dp&& OR LEN(q$) >= limit&& THEN EXIT DO
END IF
IF INKEY$ = " " THEN EXIT DO
k&& = k&& + 1
LOOP
EXIT DO
LOOP
IF RIGHT$(q$, 1) = "." AND divisor$ <> "0" THEN runningtotal$ = MID$(q$, 1, LEN(q$) - 1) ELSE runningtotal$ = q$
sm_div$ = runningtotal$
stringmatha$ = hold_stringmatha$: stringmathb$ = hold_stringmathb$
END FUNCTIONPete
RE: Newton had a fun way to approximate general roots... - Kernelpanic - 09-21-2022
@Pete - Why don't you buy the damn book? I'm sure this would be a treasure trove for such a math freak like you.
Formelsammlung zur numerischen Mathematik mit Quick Basic-Programmen
Description
RE: Newton had a fun way to approximate general roots... - Jack - 09-22-2022
Kernelpanic, here's an interesting book NUMERICAL MATHEMATICS AND COMPUTING by Ward Cheney and David Kincaid
RE: Newton had a fun way to approximate general roots... - Pete - 09-22-2022
German math? No thanks. It's like the old saying goes... There are only two types of German cars. The ones in the shop, and the ones going to the shop.
Other than that, it isn't available for shipping in my region, the center of the universe.
Pete
