Multiple nichtlineare Regression
=======================
Multipel bedeutet, daß im folgenden Programm bis zu 9 Einflußgrössen X1...X9 auf eine Ergebnisvariable Y wirken können. Weiters wird dadurch in der jeweiligen Dimension eine Parabel bestimmten Grades (je Einflußgröße verschieden) nach dem Gauss-Kriterium der kleinsten quadratischen Abweichungssumme angepasst. Das klappt allerdings nur, wenn die Einflußgrößen X voneinander unabhängig sind, was z.B. in den Sozialwissenschaften nur sehr schwer zu beurteilen ist. In der Technik finden sich oft gute Argumente, warum ein Einfluß z.B. kubisch in das Ergebnis eingeht: Masse geht natürlich mit der 3. Potenz der Längenabmessungen etc.
Funktionsweise: Nach Eingabe des jeweiligen Ergebniswertes werden die zugehörigen Einflußgrößen eingegeben (Dabei sollten genügend Daten unter unterschiedlichsten Bedingungen vorliegen, etwa aus Laborversuchen). Das Programm errechnet die (bis zu 9-)dimensionale Vandermonde-Matrix und passt iterativ-zyklich alle Parabeln so lange an, bis die Gesamtvarianz nicht weiter verbessert werden kann.
Ergebnis ist dann der Vektor der Koeffizienten der Einflußgrößen-Kombinationen.
Nicht ausführlich geprüft - private Demo ohne jede Gewähr!
Beispiel für eine Vandermonde-Matrix für 2 Einflußgrößen x1 und x2, die hier bis zu quadratisch in das Ergebnis eingehen können:
1 ______ X2______ x2^2
x1 ____ x1*x2___ x1*x2^2
X1^2__ x1^2*x2_ x1^2*x2^2
Anzumerken ist noch, daß mit dem Programm tatsächlich nur Parabeln angepasst werden können. Bei Wachstumsvorgängen etc. sind aber oft Exponentialfunktionen, Logarithmusfunktionen, Sinusschwingungen u.v.a.m. im Spiel. Dafür gibt es dann auch andere Methoden und Herangehensweisen!
WindowTitle "Mehrdimensionale nichtlineare Regression"
'Q: https://jean-pierre.moreau.pagesperso-orange.fr/Basic/regiter_bas.txt
'XProfan-11.2a-Demo (D) transponiert 2017-02 by P.Specht, Vienna/Austria
WindowStyle 24:window 0,0-%maxx,%maxy:font 2:CLS:set("decimals",17)
'*******************************************************
'* Program to demonstrate multidimensional operation *
'* of the multi-nonlinear regression subroutine with *
'* iterative error reduction *
'* --------------------------------------------------- *
'* Ref.: BASIC Scientific Subroutines Vol. II, By *
'* F.R. Ruckdeschel, Byte/McGRAW-HILL, 1981 *
'* --------------------------------------------------- *
'* SAMPLE RUN: *
'* MULTI-DIMENSIONAL AND MULTI-NONLINEAR REGRESSION *
'* *
'* How many data points ? 10 *
'* *
'* How many dimensions ? 2 *
'* *
'* What is the fit for dimension 1 ? 2 *
'* What is the fit for dimension 2 ? 1 *
'* *
'* Input the data as prompted: *
'* *
'* Y( 1) = ? 7 *
'* X( 1, 1) = ? 1 *
'* X( 1, 2) = ? 6 *
'* *
'* Y( 2) = ? 7 *
'* X( 2, 1) = ? 6 *
'* X( 2, 2) = ? 1 *
'* *
'* Y( 3) = ? 6 *
'* X( 3, 1) = ? 3 *
'* X( 3, 2) = ? 3 *
'* *
'* Y( 4) = ? 8 *
'* X( 4, 1) = ? 2 *
'* X( 4, 2) = ? 6 *
'* *
'* Y( 5) = ? 9 *
'* X( 5, 1) = ? 1 *
'* X( 5, 2) = ? 8 *
'* *
'* Y( 6) = ? 9 *
'* X( 6, 1) = ? 7 *
'* X( 6, 2) = ? 2 *
'* *
'* Y( 7) = ? 6 *
'* X( 7, 1) = ? 3 *
'* X( 7, 2) = ? 3 *
'* *
'* Y( 8) = ? 7 *
'* X( 8, 1) = ? 3 *
'* X( 8, 2) = ? 4 *
'* *
'* Y( 9) = ? 7 *
'* X( 9, 1) = ? 4 *
'* X( 9, 2) = ? 3 *
'* *
'* Y( 10) = ? 2 *
'* X( 10, 1) = ? 0 *
'* X( 10, 2) = ? 2 *
'* *
'* The calculated coefficients are: *
'* *
'* 1 0 *
'* 2 .999999 *
'* 3 0 *
'* 4 .999999 *
'* 5 0 *
'* 6 0 *
'* *
'* Standard deviation: 0 *
'* *
'* Number of iterations: 4 *
'* *
'*******************************************************
'DEFINT I-N
'DEFDBL A-H, O-Z
declare i&,j&,k&,L&,M&,n&,n1&,n2&,n3&,n4&
declare i1&,i2&,i3&,i4&,i5&,i6&,i7&,i8&,i9&
declare m&[9],m1&,m2&,m3&,m4&,L1&,sw&
declare b!,c!,d!,d1!,y!
Print "\n Frage: Programmiertes Beispiel rechnen [blank oder 0 = Nein]? Beispl.Nr.= ";
input sw&
case sw&<>0:sw&=1' 0:Eigene Eingabe. 1:Obiges Beispiel rechnen
print
PRINT " MULTIDIMENSIONAL NONLINEAR REGRESSION"
PRINT
print " How many data points ? ";
case sw&=1:m&=10
case sw&=0:INPUT m&
PRINT
print " How many dimensions ? ";
case sw&=1:L&=2
case sw&=0:Input L&
PRINT
if sw&=0
whileloop l&:i&=&Loop'=FOR i& = 1 TO l&
PRINT " What is the grade of the fit for dimension Nbr."; i&;": x^";
INPUT m&[i&]
endwhile'=NEXT i
elseif sw&=1
m&[1]=2
m&[2]=1
else
Print "\n\n Beispiel noch nicht eingegeben!":sound 200,10:waitinput:end
endif
n& = 1
Whileloop l&:i&=&Loop'=FOR i = 1 TO l
n&=n&*(m&[i&]+1)
endwhile'=NEXT i
case m&<n&:m&=n&'=IF m < n THEN m = n
Declare x![m&,l&],y![m&],z![m&,n&],d![n&],A![m&,m&],B![m&,2*m&],C![m&,m&],d1![n&],y1![m&]
'=DIM x(m, l), y(m), z(m, n), d(n), A(m, m), b(m, 2 * m), c(m, m), d1(n), y1(m)
PRINT
if sw&=0
PRINT " Input the data as prompted:"
PRINT
Whileloop m&:i&=&Loop'=FOR i = 1 TO m
PRINT " Y("; i&; ") = "; : INPUT y![i&]
whileloop l&:j&=&Loop'=FOR j = 1 TO l
PRINT " X("; i&; ","; j&; ") = "; : INPUT x![i&, j&]
endwhile'=NEXT j
PRINT
endwhile'=NEXT i
else
y![1]=7:x![1,1]=1:x![1,2]=6
y![2]=7:x![2,1]=6:x![2,2]=1
y![3]=6:x![3,1]=3:x![3,2]=3
y![4]=8:x![4,1]=2:x![4,2]=6
y![5]=9:x![5,1]=1:x![5,2]=8
y![6]=9:x![6,1]=7:x![6,2]=2
y![7]=6:x![7,1]=3:x![7,2]=3
y![8]=7:x![8,1]=3:x![8,2]=4
y![9]=7:x![9,1]=4:x![9,2]=3
y![10]=2:x![10,1]=0:x![10,2]=2
endif
'Call iteration supervisor
GOSUB "S2000"
PRINT
PRINT " The calculated coefficients are:"
PRINT
Whileloop n&:i&=&Loop'= FOR i = 1 TO n
PRINT " "; i&; " "; INT(1000000 * d![i&]) / 1000000
endwhile'=NEXT i
PRINT
PRINT " Standard deviation: "; INT(1000000 * d!) / 1000000
PRINT
PRINT " Number of iterations: "; l1&
PRINT
print "============================================================="
sound 2000,200
waitinput
beep:cls:Print "\n\n\n\n\n BYE!"
waitinput 1000
END
'*************************************************************
'* Coefficient matrix generation subroutine *
'* for multiple non-linear regression. *
'* --------------------------------------------------------- *
'* Also calculates the standard deviation d, even though *
'* there is some redundant computing. *
'* The maximum number of dimensions is 9. *
'* The input data set consists of m data sets of the form: *
'* Y(i),X(i,1),X(i,2) ... X(i,l) *
'* The number of dimensions is l. *
'* The order of the fit to each dimension is M(j). *
'* The result is an (m1+1)(m2+1)...(ml+1)+1 column by m row *
'* matrix, Z. This matrix is arranged as follows *
'* (Ex.:l=2,M(1)=2,M(2)=2): *
'* 1 X1 X1*X1 X2 X2*X1 X2*X1*X1 X2*X2 X2*X2*X1 X2*X2*X1*X1 *
'* This matrix should be dimensioned in the calling program *
'* as should also the X(i,j) matrix of data values. *
'*************************************************************
'Calculate the total number of dimensions
S1000:
n& = 1
Whileloop l&:i&=&Loop'= FOR i = 1 TO l
n& = n& * (m&[i&]+1)
endwhile'= NEXT
d! = 0
Whileloop m&:i&=&Loop'= FOR i = 1 TO m
'Branch according to dimension l (return if l > 9)
case l&>0:GOTO "G10"
l& = 0: RETURN
G10:
case l&<=9 : GOTO "G15"
l& = 0: RETURN
G15:
j& = 0
case l& = 1:GOSUB "S40"
case l& = 2:GOSUB "S50"
case l& = 3:GOSUB "S60"
case l& = 4:GOSUB "S70"
case l& = 5:GOSUB "S80"
case l& = 6:GOSUB "S90"
case l& = 7:GOSUB "S100"
case l& = 8:GOSUB "S110"
case l& = 9:GOSUB "S120"
y! = 0
Whileloop n&:k&=&Loop'= FOR k& = 1 TO n
y! = y! + d![k&] * z![i&, k&]
endwhile'= NEXT k
d! = d! + (y![i&] - y!) * (y![i&] - y!)
endwhile'= NEXT i
'Calculate standard deviation (if m > n)
G30:
case (m&-n&)>0:GOTO "G35"
d! = 0: RETURN
G35:
d!=d!/(m&-n&)
d!=SQRT(d!)'Quickbasic: sqr
RETURN
'Subroutines used by subroutine 1000
S40:
b! = 1
S41:
c! = b!
Whileloop 0,m&[1]:i1&=&Loop'= FOR i1& = 0 TO m&[1]
j&=j&+1: z![i&,j&] = b!: b! = b! * x![i&,1]
endwhile'=NEXT i1
b! = c!
RETURN
S50:
b!= 1
S51:
c!= b!
Whileloop 0,m&[2]:i2&=&Loop'= FOR i2 = 0 TO m(2)
GOSUB "S41"
b!=b!*x![i&,2]
endwhile'= NEXT i2
b!= c!
RETURN
S60:
b!= 1
S61:
c!= b!
Whileloop 0,m&[3]:i3&=&Loop'= FOR i3 = 0 TO m(3)
GOSUB "S51"
b!= b!* x![i&,3]
endwhile'= NEXT i3
b! = c!
RETURN
S70:
b!= 1
S71:
c!= b!
Whileloop 0,m&[4]:i4&=&Loop'= FOR i4 = 0 TO m(4)
GOSUB "S61"
b!= b!*x![i&,4]
endwhile'=NEXT i4
b!= c!
RETURN
S80:
b!= 1
S81:
c! = b!
whileloop 0,m&[5]:i5&=&Loop'=FOR i5 = 0 TO m(5)
GOSUB "S71"
b!= b!*x![i&,5]
endwhile'=NEXT i5
b!= c!
RETURN
S90:
b!= 1
S91:
c!= b!
Whileloop 0,m&[6]:i6&=&Loop'= FOR i6 = 0 TO m(6)
GOSUB "S81"
b! = b!* x![i&,6]
endwhile'=NEXT i6
b!= c!
RETURN
S100:
b!= 1
S101:
c!= b!
whileloop 0,m&[7]'= FOR i7 = 0 TO m(7)
GOSUB "S91"
b!= b!* x![i&, 7]
endwhile'= NEXT i7
b!=c!
RETURN
S110:
b!= 1
S111:
c!= b!
whileloop 0,m&[8]:i8&=&Loop'= FOR i8 = 0 TO m(8)
GOSUB "S101"
b! = b! * x![i&,8]
endhwile'= NEXT i8
b!= c!
RETURN
S120:
b!= 1
S121:
c!= b!
whileloop 0,m&[9]'= FOR i9 = 0 TO m(9)
GOSUB "S111"
b! = b! * x![i&,9]
endwhile'= NEXT i9
b!= c!
RETURN
'**********************************************************
'* Least squares fitting subroutine, general purpose *
'* subroutine for multidimensional, nonlinear regression *
'* ------------------------------------------------------ *
'* The equation fitted has the form: *
'* Y = D(1)X1 + D(2)X2 + ... + D(n)Xn *
'* The coefficients are returned by the program in D(i). *
'* The X(i) can be simple powers of x, or functions. *
'* Note that the X(i) are assumed to be independent. *
'* The measured responses are Y(i), there are m of them. *
'* Y is a m row column vector, Z(i,j) is a m by n matrix. *
'* m must be >= n+2. The subroutine inputs are m, n, Y(i) *
'* and Z(i,j) previously calculated. The subroutine calls *
'* several other matrix routines during the calculation. *
'**********************************************************
S1200:
m4& = m&
n4& = n&
whileloop m&:i&=&Loop'= FOR i = 1 TO m
Whileloop n&:j&=&Loop'= FOR j = 1 TO n
A![i&,j&] = z![i&, j&]
endwhile'= NEXT j
endwhile'= NEXT i
GOSUB "S5100"'b=Transpose(a)
n1&= m&: n2& = n&: GOSUB "S5400"'move A to C
n1&= n&: n2& = m&: GOSUB "S5200"'move B to A
n1&= m&: n2& = n&: GOSUB "S5300"'move C to B
m1&= n&: n1& = m&: n2& = n&: GOSUB "S5000"'multiply A and B
n1&= n&: GOSUB "S5500"'move C to A
GOSUB "S6000"'b=Inverse(a)
m& = m4&'restore m
GOSUB "S5200"'move B to A
Whileloop m&:i&=&Loop'= FOR i = 1 TO m
Whileloop n&:j&=&Loop'= FOR j = 1 TO n
b![j&,i&] = z![i&,j&]
endwhile'=NEXT j
endwhile'= NEXT i
m2& = n&: n2& = m&: GOSUB "S5000"'multiply A and B
n1& = n&: n2& = m&: GOSUB "S5500"'move C to A
whileloop m&:i&=&Loop'=FOR i = 1 TO m
b![i&,1] = y![i&]
endwhile'=NEXT
m1& = n&: n2& = 1: n1& = m&: GOSUB "S5000"'multiply A and B
'Product C is N by 1 - Regression coefficients are in C(I,1)
whileloop n&:i&=&Loop'= FOR i = 1 TO n
d![i&] = c![i&,1]
endwhile'= NEXT
RETURN
S5000:
'Matrix multiplication
whileloop m1&:i&=&Loop'= FOR i = 1 TO m1
whileloop n2&:j&=&Loop'= FOR j = 1 TO n2
c![i&,j&] = 0
whileloop n1&:k&=&Loop'= FOR k = 1 TO n1
c![i&,j&] = c![i&,j&] + A![i&,k&] * B![k&,j&]
endwhile'= NEXT k
endwhile'= NEXT j
endwhile'= NEXT i
RETURN
S5100:
'Matrix transpose
whileloop n&:i&=&Loop'= FOR i = 1 TO n
whileloop m&:j&=&Loop'= FOR j = 1 TO m
b![i&,j&] = A![j&,i&]
endwhile'= NEXT j
endwhile'= NEXT i
RETURN
S5200:
'Matrix save (B in A)
case (n1&*n2&)=0:RETURN
Whileloop n1&:i1&=&Loop'= FOR i1 = 1 TO n1
whileloop n2&:i2&=&Loop'= FOR i2 = 1 TO n2
A![i1&,i2&] = b![i1&,i2&]
endwhile'= NEXT i2
endwhile'= NEXT i1
RETURN
S5300:
'Matrix save (C in B)
case (n1& * n2&)=0:RETURN
Whileloop n1&:i1&=&Loop'= FOR i1 = 1 TO n1
whileloop n2&:i2&=&Loop'= FOR i2 = 1 TO n2
b![i1&, i2&] = c![i1&, i2&]
endwhile'= NEXT i2
endwhile'= NEXT i1
RETURN
S5400:
'Matrix save (A in C)
case (n1& * n2&)=0:RETURN
whileloop n1&:i1&=&Loop'= FOR i1 = 1 TO n1
whileloop n2&:i2&=&Loop'= FOR i2 = 1 TO n2
c![i1&,i2&] = A![i1&,i2&]
endwhile'NEXT i2
endwhile'NEXT i1
RETURN
S5500:
'Matrix save (C in A)
case (n1&*n2&)=0:RETURN
whileloop n1&:i1&=&Loop'= FOR i1 = 1 TO n1
whileloop n2&:i2&=&Loop'= FOR i2 = 1 TO n2
A![i1&,i2&] = c![i1&,i2&]
endwhile'=NEXT i2
endwhile'=NEXT i1
RETURN
S6000:
'Matrix inversion
Whileloop n&:i&=&Loop'= FOR i = 1 TO n
whileloop n&:j&=&Loop'= FOR j = 1 TO n
b![i&, j& + n&] = 0
b![i&, j&] = A![i&, j&]
endwhile'= NEXT j
b![i&, i& + n&] = 1
endwhile'= NEXT i
whileloop n&:k&=&Loop'= FOR k = 1 TO n
case k& = n&:GOTO "G6010"
m& = k&
whileloop k&+1,n&:i&=&Loop'= FOR i = k + 1 TO n
case ABS(b![i&, k&]) > ABS(b![m&, k&]):m& = i&
endwhile'= NEXT i
case m& = k&:GOTO "G6010"
whileloop k&,2*n&:j&=&Loop'= FOR j = k TO 2 * n
b! = b![k&, j&]
b![k&, j&] = b![m&, j&]
b![m&, j&] = b!
endwhile'= NEXT j
G6010:
whileloop k&+1,2*n&:j&=&Loop'= FOR j = k + 1 TO 2 * n
b![k&, j&] = b![k&, j&] / b![k&, k&]
endwhile'= NEXT j
case k& = 1:GOTO "G6020"
whileloop k&-1:i&=&Loop'= FOR i = 1 TO k - 1
whileloop k&+1,2*n&:j&=&Loop'= FOR j = k + 1 TO 2 * n
b![i&,j&] = b![i&,j&] - b![i&,k&] * b![k&,j&]
endwhile'= NEXT j
endwhile'= NEXT i
Case k& = n&:GOTO "G6030"
G6020:
whileloop k&+1,n&:i&=&Loop'= FOR i = k + 1 TO n
whileloop k&+1,2*n&:j&=&Loop'= FOR j = k + 1 TO 2 * n
b![i&,j&] = b![i&,j&] - b![i&,k&] * b![k&,j&]
endwhile'= NEXT j
endwhile'=NEXT i
endwhile'= NEXT k
G6030:
Whileloop n&:i&=&Loop'= FOR i = 1 TO n
whileloop n&:j&=&Loop'= FOR j = 1 TO n
b![i&,j&] = b![i&,j&+n&]
endwhile'= NEXT j
endwhile'= NEXT i
RETURN
'********************************************************************
'* Multi-dimensional polynomial regression iteration subroutine *
'* ---------------------------------------------------------------- *
'* This routine supervises the calling of several other subroutines *
'* in order to iteratively fit least squares polynomials in more *
'* than one dimension. *
'* The routine repeatedly calculates improved coefficients until *
'* the standard deviation is no longer reduced. The inputs to the *
'* subroutine are the number of dimensions l&, the degree of fit *
'* for each dimension m(i), and the input data, x(i) and y(i). *
'* The coefficients are returned in d(i), with the standard devia- *
'* tion in d. Also returned is the number of iterations tried, l1&. *
'* y1(i), d1(i) and d1 are used respectively to save the original *
'* values of y(i) and the current values of d(i) and d. *
'********************************************************************
S2000:
l1& = 0
'Save the y![i&]
whileloop m&:i&=&Loop'= FOR i = 1 TO m
y1![i&] = y![i&]
endwhile'= NEXT
'Zero d1(i)
whileloop n&:i&=&Loop'= FOR i = 1 TO n
d1![i&] = 0
endwhile'= NEXT
'Set the initial standard deviation high
d1! = 10000000
'Call coefficients subroutine
G2050:
GOSUB "S1000"
'Call regression subroutine
GOSUB "S1200"
'Get standard deviation
GOSUB "S1000"
'If standard deviation is decreasing, continue
case d1! > d!:GOTO "G2100"
'Terminate iteration
whileloop n&:i&=&Loop'= FOR i = 1 TO n
d![i&] = d1![i&]
endwhile'= NEXT
' Restore y![i&]
Whileloop m&:i&=&Loop'=FOR i = 1 TO m
y![i&] = y1![i&]
endwhile'= NEXT
'Get the final standard deviation
GOSUB "S1000"
RETURN
'Save the standard deviation
G2100:
d1! = d!: l1& = l1& + 1
'Augment coefficient matrix
whileloop n&:i&=&Loop'= FOR i = 1 TO n
d![i&] = d1![i&] + d![i&]
d1![i&] = d![i&]
endwhile'=NEXT
'Restore y![i&]
whileloop m&:i&=&Loop'=FOR i = 1 TO m
y![i&] = y1![i&]
endwhile'= NEXT
'Reduce y![i&] according to the d(i)
GOSUB "S2150"
'We now have a set of error values
GOTO "G2050"
'End subroutine 2000
'Subroutine 2150
S2150:
whileloop m&:i&=&Loop'= FOR i = 1 TO m
j& = 0
Case l& = 1:GOSUB "S2160"
Case l& = 2:GOSUB "S2170"
Case l& = 3:GOSUB "S2180"
Case l& = 4:GOSUB "S2190"
Case l& = 5:GOSUB "S2200"
Case l& = 6:GOSUB "S2210"
Case l& = 8:GOSUB "S2230"
Case l& = 9:GOSUB "S2240"
'Array generated for row i
y! = 0
whileloop n&:k&=&Loop'= FOR k = 1 TO n
y! = y! + d![k&] * z![i&,k&]
endwhile'= NEXT k
y![i&] = y![i&] - y!
endwhile'= NEXT i
RETURN
'End subroutine S2150
S2160:
b!=1
S2161:
c!=b!
whileloop 0,m&[1]:i1&=&Loop'FOR i1 = 0 TO m(1)
j&= j& + 1
z![i&,j&] = b!: b! = b! * x![i&,1]
endwhile'NEXT i1
b! = c!
RETURN
S2170:
b!= 1
S2171:
c!= b!
whileloop 0,m&[2]:i2&=&Loop'= FOR i2 = 0 TO m(2)
GOSUB "S2161"
b!= b!* x![i&,2]
endwhile'NEXT i2
b!=c!
RETURN
S2180:
b! = 1
S2181:
c! = b!
whileloop 0,m&[3]:i3&=&Loop'= FOR i3 = 0 TO m(3)
GOSUB "S2171"
b = b * x(i, 3)
endwhile'= NEXT i3
b!= c!
RETURN
S2190:
b!= 1
S2191:
c!= b!
whileloop 0,m&[4]:i4&=&Loop'= FOR i4 = 0 TO m(4)
GOSUB "S2181"
b!= b!* x![i&,4]
endwhile'NEXT i4
b! = c!
RETURN
S2200:
b! = 1
S2201:
c!= b!
whileloop 0,m&[5]:i5&=&Loop'= FOR i5 = 0 TO m(5)
GOSUB "S2191"
b! = b! * x![i&,5]
endwhile'NEXT i5
b! = c!
RETURN
S2210:
b! = 1
S2211:
c! = b!
whileloop 0,m&[6]:i6&=&Loop'= FOR i6 = 0 TO m(6)
GOSUB "S2201"
b! = b! * x![i&,6]
endwhile'NEXT i6
b! = c!
RETURN
S2220:
b! = 1
S2221:
c! = b!
whileloop 0,m&[7]:i7&=&Loop'= FOR i7 = 0 TO m(7)
GOSUB "S2211"
b!= b!* x![i&,7]
endwhile'= NEXT i7
b! = c!
RETURN
S2230:
b! = 1
S2231:
c! = b!
whileloop 0,m&[8]:i8&=&Loop'= FOR i8 = 0 TO m(7)
GOSUB "S2221"
b! = b! * x![i,8]' war 7 ! Fehler?
endwhile'NEXT i8
b! = c!
RETURN
S2240:
b! = 1
whileloop 0,m&[9]:i9&=&Loop'= FOR i9 = 0 TO m(9)
GOSUB "S2231"
b! = b! * x!(i&,9)
endwhile'NEXT i9
RETURN
'End Pgm regiter.prf
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