An algebra problem by Rocco Dalto

After doing the problem I supply the program I wrote for the general case which generated the output for this specific problem. I added to the program written from the previous problem. Refer to previous problem. . .

$$

Since $f: \mathbb C^4 \rightarrow \mathbb C^4$ is a linear transform, for each integer $j \ni (1 <= j <= 4)$

$$

$f \left( \begin{array}{ccc} z_{1j} \\ z_{2j} \\ z_{3j} \\ z_{4j} \\ \end{array} \right) = \alpha_{1j} * \left( \begin{array}{cccc} 1 + 2i\\ 3 + 4i \\ 5 + 6i \\ 7 + 8i \end{array} \right) + \alpha_{2j} * \left( \begin{array}{cccc} -7 + i \\ 2 - 3i \\ 11 + 7i \\ 6 + 5i \end{array} \right) + \alpha_{3j} * \left( \begin{array}{cccc} 8 - 7i \\ 5 - 6i \\ 4 + 2i \\ 1 - 11i \end{array} \right) +\alpha_{4j} * \left( \begin{array}{cccc} 7 - i \\ 3 + 2i \\ -4 + 8i \\ 17 - 11i \end{array} \right)$

$= \begin{bmatrix}{1 + 2i} && {-7 + i} && {8 - 7i} && {7 - i}\\ {3 + 4i} && {2 - 3i} && {5 - 6i} && {3 + 2i}\\ {5 + 6i} && {11 + 7i} && {4 +2i} && {-4 + 8i}\\ {7 + 8i} && {6 + 5i} && {1 - 11i} && {17 - 11i} \end{bmatrix} * \begin{vmatrix}{\alpha_{1j}} \\{\alpha_{2j}} \\ {\alpha_{3j}} \\ {\alpha_{4j}}\end{vmatrix}$

where $\alpha_{1j}, \alpha_{2j}, \alpha_{3j}, \alpha_{4j} \in \mathbb C$

and $f \left( \begin{array}{cccc} 1 - 2i \\ 3 - 4i \\ 5 - 6i \\ 7 - 8i \\ \end{array} \right) = \left( \begin{array}{cccc} 37 + 63i \\ -83 + 127i \\ 40 + 26i \\ 15 - 14i \\ \end{array} \right)$

$f \left( \begin{array}{cccc} 2 - 5i \\ 6 + 11i \\ 8 - 7i \\ 3 + 2i \\ \end{array} \right) = \left( \begin{array}{cccc} 127 - 52i \\ 56 - 7i \\ 131 - 110i \\ 125 - 63i \end{array} \right)$

$f \left( \begin{array}{cccc} 8 - 7i \\ 5 - 6i \\ 3 - 4i \\ 2 + i \\ \end{array} \right) = \left( \begin{array}{cccc} 95 - 74i \\ -23 - 92i \\ 21 + 20i \\ 116 + 66i \\ \end{array} \right)$

$f \left( \begin{array}{cccc} 9 - 8i \\ 7 - 5i \\ 6 + 4i \\ 3 -2i \\ \end{array} \right) = \left( \begin{array}{cccc} 130 - 43i \\ -59 + 132i \\ 125 - 30i \\ 159 - 4i \\ \end{array} \right)$

$$

Using the above we can set up the augmented matrix below to solve for the $$

four 4 X 4 systems of equations.

$$

$\left[ \begin{array}{cccc|cccc} 1 + 2i & -7 + i & 8 - 7i & 7 - i & 37 + 63i & -83 + 127i & 40 + 26i & 15 + - 14i\\ 3 + 4i & 2 - 3i & 5 - 6i & 3 + 2i & 127 - 52i & 56 - 7i & 131 - 110i & 125 - 63i \\ 5 + 6i & 11 + 7i & 4 + 2i & -4. + 8i & 95 - 74i & -23 i- 92i & 21 +20i & 116 + 66i \\ 7 + 8i & 6 + 5i & 1 - 11i & 17 - 11i & 130 - 43i & -59 + 132i & 125 - 30i & 159 - 4i \\ \ \end{array} \right]$

$$

Refer to previous problem for row operations. . .

Using row operation we solve for matrix $M$ obtaining:

$$ $M = \begin{bmatrix}{-9.5092 - 17.3602i} && {-19.5531 - 11.4032i} && {-22.3483 - 8.2119i} && {-3.7798 - 11.1438i}\\ {5.8734 + 3.3746i} && {7.8190 + 7.8931i} && {14.6273 + 7.3921i} && {10.2018 + 5.9260i}\\ {1.4659 + 13.7409i} && { -7.1655 + 18.5113i} && {9.4243 + 16.1723i} && { 5.2315 + 6.5938i}\\ {-7.5783 + 1.4119i} && {-13.9574 +1.8195i} && {-5.6573 +6.7085i} && {-2.3819 + 2.4960i}\end{bmatrix}$

$$

To find the eigenvalues we calculate $\det(M - \lambda * I) = 0$ obtaining:

$$

$\lambda^4 + (-5.3522 -9.2013i) * \lambda^3 + (585.2473 -102.4948i) * \lambda^2 + (-1686.5838 + 3479.3919i) * \lambda$ $+ (7356.6016 + 9105.8547i) = 0$

$\implies$

$\lambda_{1} = 4.0062 + 31.0478i$

$\lambda_{2} = -4.3122 -17.1633i$

$\lambda_{3} = 6.6889 - 6.6688i$

$\lambda_{4} = -1.0307 + 1.9855i$

$where, \lambda_{j} = \alpha_{j} + \beta_{j}i$ for $(1 <= j <= 4)$

$\implies$

$\sum_{j = 1}^{4} \alpha_{j} + \sum_{j = 1}^{4} \beta{j} = \boxed{14.5535}$

$$

I wrote the program in Free Pascal:

$$

program matrixrepresentaionoflineartransform complex extension;

{special case - restricted to f:R4 --> R4}

uses crt;

                                {self contained for brillant}

const maxnum = 100;

type matrix = array[1 .. maxnum,1 .. maxnum] of real;

var coeffa,coeffb,reb,imgb,rea,imga,resol,imgsol:matrix;

n,m,l,p,v:integer;

mywork:text;

RECOEFF 1,RECOEFF 2,RECOEFF 3,RECOEFF 4,RECOEFF 5,IMGCO EFF 1,IMGCOEFF 2,IMGCOEFF 3,IMGCOEFF 4,IMGCOEFF 5,REW, IMGW,REU1,IMGU1,REU2,IMGU2,REU3,IMGU3,REU4,IMGU4,REZ1,IMGZ1,REZ2,IMGZ2,REZ3,IMGZ3,REZ4,IMGZ4:EXTENDED;

$$

function power(base:real; exponent:integer):real;

var n:integer;

product:real;

begin

product:= 1;

for n:= 1 to exponent do

product:= product * base;

power:= product;

end;

$$

function nroot(x:real; n:integer):real;

const error = 0.0000000001;

var right,left,midpt,diff:real;

  count:integer;

begin{nroot}

IF (ABS(X) >= 1) THEN

BEGIN

left:= 1; right:= ABS(x);

END;

IF (ABS(X) < 1) THEN

BEGIN

LEFT:= ABS(X);    RIGHT:= 1;

END;

    midpt:= (left + right)/2;

count:= 1;

 repeat

if power(midpt,n) > ABS(x) then {X CAN < 0 FOR N ODD}

    right:= midpt                {ASSUMES X >= 0 FOR N EVEN}

else

    left:= midpt;


    count:= count + 1;

midpt:= (left + right)/2;

diff:= abs(ABS(x) - power(midpt,n));

  until((diff < error) or (count = 100));

IF (X < 0) AND (N MOD 2 = 1) THEN

nroot:= -1 * midpt;

IF (X >= 0) THEN

NROOT:= MIDPT;

IF (X < 0) AND (N MOD 2 = 0) THEN

HALT;

 end;{nroot}

$$

procedure getsize; {modified previous program}

begin

writeln('Let f:R4 ---> R4 be linear transform ');

n:= 4;

m:= 4;

end;

$$

procedure getcoefficients;

var j,k:integer;

begin

for j:= 1 to m do

begin

writeln('For row,', j, ', enter each coefficient of Xj for Rm');

for k:= 1 to n do

begin

read(coeffa[j,k]);

read(coeffb[j,k]);

end;

end;

end;

$$

procedure complexinverse(var reinvz,imginvz:real; rez,imgz:real);

var lengthsqr:real;

begin

lengthsqr:= power(rez,2) + power(imgz,2);

reinvz:= rez/lengthsqr;

imginvz:= -imgz/lengthsqr;

end;

$$

function reproduct(rez1,imgz1,rez2,imgz2:real):real;

begin

reproduct:= rez1 * rez2 - imgz1 * imgz2;

end;

$$

function imgproduct(rez1,imgz1,rez2,imgz2:real):real;

begin

imgproduct:= rez2 * imgz1 + rez1 * imgz2;

end;

$$

procedure getaugmentedmatrix;

var j,k,q,r,s:integer;

revec,imgvec:matrix;

resum,imgsum:real;

myfile:text;

begin

assign(myfile,'holdcmatrix.txt');

rewrite(myfile);

writeln('To Enter ', n, ' basis vectors for Rn: ');

for q:= 1 to n do

begin

write('Enter elements of vector ', q, ' : ');

for k:= 1 to n do

begin

read(reb[k,q]);

read(imgb[k,q]);

end;

for j:= 1 to m do

begin

resum:= 0;

imgsum:= 0;

for k:= 1 to n do

begin

resum:= resum + reproduct(coeffa[j,k],coeffb[j,k],reb[k,q],imgb[k,q]);

imgsum:= imgsum + imgproduct(coeffa[j,k],coeffb[j,k],reb[k,q],imgb[k,q]);

end;

revec[j,q]:= resum;

imgvec[j,q]:= imgsum;

end;

end;

writeln('To Enter ', m, ' basis vectors for Rm: ');

for q:= 1 to m do

begin

write('Enter elements of vector ', q, ' : ');

for k:= 1 to m do

begin

read(rea[k,q]);

read(imga[k,q]);

end;

end;

for q:= 1 to m do

begin

for k:= 1 to m do

begin

write(myfile,rea[q,k]:0:4,' ',imga[q,k],' ');

end;

for r:= 1 to n do

begin

write(myfile,revec[q,r]:0:4,' ',imgvec[q,r],' ');

end;

writeln(myfile);

end;

reset(myfile);

for j:= 1 to m do

begin

for k:= 1 to m + n do

begin

read(myfile,rea[j,k]);

read(myfile,imga[j,k]);

end;

end;

close(myfile);

{To show augmented matrix - for show work}

writeln(mywork,' Augmented Matrix');

writeln(mywork);

for q:= 1 to m do

begin

for k:= 1 to m do

begin

write(mywork,rea[q,k]:0:4,' + i', imga[q,k]:0:4,' ');

end;

for r:= 1 to n do

begin

write(mywork,revec[q,r]:0:4,' + i', imgvec[q,r]:0:4,' ');

end;

writeln(mywork);

end;

writeln(mywork);

writeln(mywork);

end;

$$

procedure switch(var c,d:matrix; l,p:integer);

var q:integer;

z,w:real;

begin{switch}

for q:= 1 to m + n do

begin{m}

   z:= c[l,q];

   c[l,q]:= c[p,q];

   c[p,q]:= z;

   w:= d[l,q];

   d[l,q]:= d[p,q];

   d[p,q]:= w;

end;{m}

end;{switch}

$$

procedure safeguard(var c,d:matrix; l:integer; var p:integer; var v:integer);

var q:integer;

begin{safeguard} {Still to do!}

v:=0;

if (c[l,l] = 0) and (d[l,l] = 0) then

begin{then}

v:= 1;

q:= l;

while ((q + 1) <= n + m) and (v = 1) do

begin{loop}

if (c[q + 1,l] <> 0) or (d[q + 1,l] <> 0) then

begin{then}

        p:= q + 1;

        switch(c,d,l,p);

        v:= 0;

end;{then}

q:= q + 1;

end;{loop}

end;{then}

end;{safeguard}

$$

procedure operation1(var a,b:matrix; l:integer);

var q,k:integer;

res,imgs:matrix;

reinvz,imginvz:real;

begin{op1}

for q:= 1 to m + n do

begin{q}

complexinverse(reinvz,imginvz,a[l,l],b[l,l]);

res[l,q]:= reproduct(a[l,q],b[l,q],reinvz,imginvz);

imgs[l,q]:= imgproduct(a[l,q],b[l,q],reinvz,imginvz);

end;{q}

for q:= 1 to m + n do

begin{q}

a[l,q]:= res[l,q];

b[l,q]:= imgs[l,q];

end;{q}

writeln(mywork,'(',reinvz:0:4,' + i', imginvz:0:4,') * ROW ', l);

writeln(mywork);

writeln(mywork);

for q:= 1 to m do

begin

for k:= 1 to m + n do

begin

write(mywork,a[q,k]:0:4,' + i', b[q,k]:0:4,' ');

end;

writeln(mywork);

end;

writeln(mywork);

writeln(mywork);

end;{op1}

$$

procedure operation2(var a,b:matrix; l:integer);

var q,r,k:integer;

x,y:real;

res,imgs:matrix;

begin{op2}

for q:= 1 to m do

begin{q}

if q <> l then

begin{then}

x:= -1 * a[q,l];

y:= -1 * b[q,l];

for r:= 1 to m + n do

begin{r}

res[q,r]:= reproduct(x,y,a[l,r],b[l,r]) + a[q,r];

imgs[q,r]:= imgproduct(x,y,a[l,r],b[l,r]) + b[q,r];

end;{r}

writeln(mywork,'(',x:0:4, ' + i',y:0:4,' ) * ROW ', l,' + ROW ', q);

for r:= 1 to m + n do

begin{r}

a[q,r]:= res[q,r];

b[q,r]:= imgs[q,r];

end;{r}

end;{then}

end;{q}

writeln(mywork);

for q:= 1 to m do

begin

for k:= 1 to m + n do

begin

write(mywork,a[q,k]:0:4,' + i ',b[q,k]:0:4,' ');

end;

writeln(mywork);

end;

writeln(mywork);

writeln(mywork);

end;{op2}

$$

PROCEDURE SHOWDATA;

VAR J,k:INTEGER;

myfile2:text;

BEGIN

assign(myfile2,'holdsolution6.txt');

rewrite(myfile2);

writeln;

writeln('Matrix Representation of linear transformation f: Rn ---> Rm:');

writeln('-------------------------------------------------------------');

writeln;

for j:= 1 to m do

begin

for k:= m + 1 to m + n do

begin

write(myfile2,rea[j,k]:0:4,' ',imga[j,k],' ');

end;

writeln(myfile2);

end;

reset(myfile2);

for j:= 1 to m do

begin

for k:= 1 to n do

begin

read(myfile2,resol[j,k]);

read(myfile2,imgsol[j,k]);

end;

end;

close(myfile2);

for j:= 1 to m do

begin

for k:= 1 to n do

begin

write(resol[j,k]:0:4,' + i', imgsol[j,k]:0:4,' ');

end;

writeln;

end;

END;

$$

function reproduct2(rez1,imgz1,rez2,imgz2,rez3,imgz3:extended):extended;

begin

reproduct2:= rez3 * reproduct(rez1,imgz1,rez2,imgz2) - imgz3 *

imgproduct(rez1,imgz1,rez2,imgz2);

end;

$$

function imgproduct2(rez1,imgz1,rez2,imgz2,rez3,imgz3:extended):extended;

begin

imgproduct2:= rez3 * imgproduct(rez1,imgz1,rez2,imgz2) + imgz3 *

reproduct(rez1,imgz1,rez2,imgz2);

end;

$$

function reproduct3(rez1,imgz1,rez2,imgz2,rez3,imgz3,rez4,imgz4:extended):extended;

begin

reproduct3:= rez4 * reproduct2(rez1,imgz1,rez2,imgz2,rez3,imgz3) - imgz4 *

imgproduct2(rez1,imgz1,rez2,imgz2,rez3,imgz3);

end;

$$

function imgproduct3(rez1,imgz1,rez2,imgz2,rez3,imgz3,rez4,imgz4:extended):extended;

begin

imgproduct3:= rez4 * imgproduct2(rez1,imgz1,rez2,imgz2,rez3,imgz3) + imgz4 *

reproduct2(rez1,imgz1,rez2,imgz2,rez3,imgz3);

end;

$$

PROCEDURE COMPLEX POLY COEFFS(VAR RECOEFF 1,RECOEFF 2,RECOEFF 3,RECOEFF 4,RECOEFF 5, IMGCOEFF 1,IMGCOEFF 2,IMGCOEFF 3,IMGCOEFF 4,IMGCOEFF 5:extended; A,B:MATRIX);

BEGIN

RECOEFF_1:= 1;

IMGCOEFF_1:= 0;

RECOEFF_2:= -(A[1,1] + A[2,2] + A[3,3] + A[4,4]);

IMGCOEFF_2:= -(B[1,1] + B[2,2] + B[3,3] + B[4,4]);

RECOEFF_3:= reproduct(A[1,1],B[1,1],A[2,2],B[2,2]) + reproduct(A[1,1],B[1,1],A[3,3],B[3,3]) + reproduct(A[1,1],B[1,1],A[4,4],B[4,4]) + reproduct(A[2,2],B[2,2],A[3,3],B[3,3]) + reproduct(A[2,2],B[2,2],A[4,4],B[4,4]) + reproduct(A[3,3],B[3,3],A[4,4],B[4,4]) - reproduct(A[1,2],B[1,2],A[2,1],B[2,1]) - reproduct(A[3,4],B[3,4],A[4,3],B[4,3] - reproduct(A[2,3],B[2,3],A[3,2],B[3,2]) - reproduct(A[2,4],B[2,4],A[4,2],B[4,2] - reproduct(A[1,3],B[1,3],A[3,1],B[3,1]) - reproduct(A[1,4],B[1,4],A[4,1],B[4,1]);

IMGCOEFF_3:= imgproduct(A[1,1],B[1,1],A[2,2],B[2,2]) + imgproduct(A[1,1],B[1,1],A[3,3],B[3,3]) + imgproduct(A[1,1],B[1,1],A[4,4],B[4,4]) + imgproduct(A[2,2],B[2,2],A[3,3],B[3,3]) + imgproduct(A[2,2],B[2,2],A[4,4],B[4,4]) + imgproduct(A[3,3],B[3,3],A[4,4],B[4,4]) - imgproduct(A[1,2],B[1,2],A[2,1],B[2,1]) - imgproduct(A[3,4],B[3,4],A[4,3],B[4,3]) - imgproduct(A[2,3],B[2,3],A[3,2],B[3,2]) - imgproduct(A[2,4],B[2,4],A[4,2],B[4,2]) - imgproduct(A[1,3],B[1,3],A[3,1],B[3,1]) - imgproduct(A[1,4],B[1,4],A[4,1],B[4,1]);

RECOEFF_4:= -(reproduct2(A[1,1],B[1,1],A[2,2],B[2,2],A[3,3],B[3,3]) + reproduct2(A[1,1],B[1,1],A[2,2],B[2,2],A[4,4],B[4,4]) + reproduct2(A[1,1],B[1,1],A[3,3],B[3,3],A[4,4],B[4,4]) + reproduct2(A[2,2],B[2,2],A[3,3],B[3,3],A[4,4],B[4,4]) + reproduct2(A[3,4],B[3,4],A[4,2],B[4,2],A[2,3],B[2,3]) + reproduct2(A[2,4],B[2,4],A[3,2],B[3,2],A[4,3],B[4,3])+ reproduct2(A[1,2],B[1,2],A[2,3],B[2,3],A[3,1],B[3,1]) + reproduct2(A[1,2],B[1,2],A[2,4],B[2,4],A[4,1],B[4,1])+ reproduct2(A[1,3],B[1,3],A[2,1],B[2,1],A[3,2],B[3,2]) + reproduct2(A[1,3],B[1,3],A[3,4],B[3,4],A[4,1],B[4,1]) + reproduct2(A[1,4],B[1,4],A[4,2],B[4,2],A[2,1],B[2,1]) + reproduct2(A[1,4],B[1,4],A[3,1],B[3,1],A[4,3],B[4,3]) - reproduct2(A[3,4],B[3,4],A[4,3],B[4,3],A[1,1],B[1,1]) - reproduct2(A[3,4],B[3,4],A[4,3],B[4,3],A[2,2],B[2,2]) - reproduct2(A[1,1],B[1,1],A[2,3],B[2,3],A[3,2],B[3,2]) - reproduct2(A[1,1],B[1,1],A[2,4],B[2,4],A[4,2],B[4,2]) - reproduct2(A[2,3],B[2,3],A[3,2],B[3,2],A[4,4],B[4,4]) - reproduct2(A[3,3],B[3,3],A[4,2],B[4,2],A[2,4],B[2,4]) - reproduct2(A[1,2],B[1,2],A[2,1],B[2,1],A[3,3],B[3,3]) - reproduct2(A[1,2],B[1,2],A[2,1],B[2,1],A[4,4],B[4,4]) - reproduct2(A[1,3],B[1,3],A[3,1],B[3,1],A[4,4],B[4,4]) - reproduct2(A[1,3],B[1,3],A[3,1],B[3,1],A[2,2],B[2,2]) - reproduct2(A[1,4],B[1,4],A[4,1],B[4,1],A[2,2],B[2,2]) - reproduct2(A[1,4],B[1,4],A[4,1],B[4,1],A[3,3],B[3,3]));

IMGCOEFF_4:= -(imgproduct2(A[1,1],B[1,1],A[2,2],B[2,2],A[3,3],B[3,3]) + imgproduct2(A[1,1],B[1,1],A[2,2],B[2,2],A[4,4],B[4,4]) + imgproduct2(A[1,1],B[1,1],A[3,3],B[3,3],A[4,4],B[4,4]) + imgproduct2(A[2,2],B[2,2],A[3,3],B[3,3],A[4,4],B[4,4]) + imgproduct2(A[3,4],B[3,4],A[4,2],B[4,2],A[2,3],B[2,3]) + imgproduct2(A[2,4],B[2,4],A[3,2],B[3,2],A[4,3],B[4,3]) + imgproduct2(A[1,2],B[1,2],A[2,3],B[2,3],A[3,1],B[3,1]) + imgproduct2(A[1,2],B[1,2],A[2,4],B[2,4],A[4,1],B[4,1]) + imgproduct2(A[1,3],B[1,3],A[2,1],B[2,1],A[3,2],B[3,2]) + imgproduct2(A[1,3],B[1,3],A[3,4],B[3,4],A[4,1],B[4,1]) + imgproduct2(A[1,4],B[1,4],A[4,2],B[4,2],A[2,1],B[2,1]) + imgproduct2(A[1,4],B[1,4],A[3,1],B[3,1],A[4,3],B[4,3]) - imgproduct2(A[3,4],B[3,4],A[4,3],B[4,3],A[1,1],B[1,1]) - imgproduct2(A[3,4],B[3,4],A[4,3],B[4,3],A[2,2],B[2,2]) - imgproduct2(A[1,1],B[1,1],A[2,3],B[2,3],A[3,2],B[3,2]) - imgproduct2(A[1,1],B[1,1],A[2,4],B[2,4],A[4,2],B[4,2]) - imgproduct2(A[2,3],B[2,3],A[3,2],B[3,2],A[4,4],B[4,4]) - imgproduct2(A[3,3],B[3,3],A[4,2],B[4,2],A[2,4],B[2,4]) - imgproduct2(A[1,2],B[1,2],A[2,1],B[2,1],A[3,3],B[3,3]) - imgproduct2(A[1,2],B[1,2],A[2,1],B[2,1],A[4,4],B[4,4]) - imgproduct2(A[1,3],B[1,3],A[3,1],B[3,1],A[4,4],B[4,4]) - imgproduct2(A[1,3],B[1,3],A[3,1],B[3,1],A[2,2],B[2,2]) - imgproduct2(A[1,4],B[1,4],A[4,1],B[4,1],A[2,2],B[2,2]) - imgproduct2(A[1,4],B[1,4],A[4,1],B[4,1],A[3,3],B[3,3]));

RECOEFF_5:= reproduct3(A[1,1],B[1,1],A[2,2],B[2,2],A[3,3],B[3,3],A[4,4],B[4,4]) + reproduct3(A[1,1],B[1,1],A[3,4],B[3,4],A[4,2],B[4,2],A[2,3],B[2,3]) + reproduct3(A[1,1],B[1,1],A[2,4],B[2,4],A[3,2],B[3,2],A[4,3],B[4,3]) + reproduct3(A[1,2],B[1,2],A[2,1],B[2,1],A[3,4],B[3,4],A[4,3],B[4,3]) + reproduct3(A[1,2],B[1,2],A[2,3],B[2,3],A[3,1],B[3,1],A[4,4],B[4,4]) + reproduct3(A[1,2],B[1,2],A[2,4],B[2,4],A[3,3],B[3,3],A[4,1],B[4,1]) + reproduct3(A[1,3],B[1,3],A[2,1],B[2,1],A[3,2],B[3,2],A[4,4],B[4,4]) + reproduct3(A[1,3],B[1,3],A[3,1],B[3,1],A[2,4],B[2,4],A[4,2],B[4,2]) + reproduct3(A[1,3],B[1,3],A[2,2],B[2,2],A[3,4],B[3,4],A[4,1],B[4,1]) + reproduct3(A[1,4],B[1,4],A[4,1],B[4,1],A[2,3],B[2,3],A[3,2],B[3,2]) + reproduct3(A[1,4],B[1,4],A[2,1],B[2,1],A[4,2],B[4,2],A[3,3],B[3,3]) + reproduct3(A[1,4],B[1,4],A[2,2],B[2,2],A[3,1],B[3,1],A[4,3],B[4,3]) - reproduct3(A[1,1],B[1,1],A[2,3],B[2,3],A[3,2],B[3,2],A[4,4],B[4,4]) - reproduct3(A[1,1],B[1,1],A[3,3],B[3,3],A[4,2],B[4,2],A[2,4],B[2,4]) - reproduct3(A[1,1],B[1,1],A[2,2],B[2,2],A[3,4],B[3,4],A[4,3],B[4,3]) - reproduct3(A[1,2],B[1,2],A[2,4],B[2,4],A[3,1],B[3,1],A[4,3],B[4,3]) - reproduct3(A[1,2],B[1,2],A[2,1],B[2,1],A[3,3],B[3,3],A[4,4],B[4,4]) - reproduct3(A[1,2],B[1,2],A[2,3],B[2,3],A[3,4],B[3,4],A[4,1],B[4,1]) - reproduct3(A[1,3],B[1,3],A[2,1],B[2,1],A[3,4],B[3,4],A[4,2],B[4,2]) - reproduct3(A[1,3],B[1,3],A[3,1],B[3,1],A[2,2],B[2,2],A[4,4],B[4,4]) - reproduct3(A[1,3],B[1,3],A[2,4],B[2,4],A[3,2],B[3,2],A[4,1],B[4,1]) - reproduct3(A[1,4],B[1,4],A[2,1],B[2,1],A[3,2],B[3,2],A[4,3],B[4,3]) - reproduct3(A[1,4],B[1,4],A[4,1],B[4,1],A[2,2],B[2,2],A[3,3],B[3,3]) - reproduct3(A[1,4],B[1,4],A[2,3],B[2,3],A[3,1],B[3,1],A[4,2],B[4,2]);

IMGCOEFF_5:= imgproduct3(A[1,1],B[1,1],A[2,2],B[2,2],A[3,3],B[3,3],A[4,4],B[4,4]) + imgproduct3(A[1,1],B[1,1],A[3,4],B[3,4],A[4,2],B[4,2],A[2,3],B[2,3]) + imgproduct3(A[1,1],B[1,1],A[2,4],B[2,4],A[3,2],B[3,2],A[4,3],B[4,3]) + imgproduct3(A[1,2],B[1,2],A[2,1],B[2,1],A[3,4],B[3,4],A[4,3],B[4,3]) + imgproduct3(A[1,2],B[1,2],A[2,3],B[2,3],A[3,1],B[3,1],A[4,4],B[4,4]) + imgproduct3(A[1,2],B[1,2],A[2,4],B[2,4],A[3,3],B[3,3],A[4,1],B[4,1]) + imgproduct3(A[1,3],B[1,3],A[2,1],B[2,1],A[3,2],B[3,2],A[4,4],B[4,4]) + imgproduct3(A[1,3],B[1,3],A[3,1],B[3,1],A[2,4],B[2,4],A[4,2],B[4,2]) + imgproduct3(A[1,3],B[1,3],A[2,2],B[2,2],A[3,4],B[3,4],A[4,1],B[4,1]) + imgproduct3(A[1,4],B[1,4],A[4,1],B[4,1],A[2,3],B[2,3],A[3,2],B[3,2]) + imgproduct3(A[1,4],B[1,4],A[2,1],B[2,1],A[4,2],B[4,2],A[3,3],B[3,3]) + imgproduct3(A[1,4],B[1,4],A[2,2],B[2,2],A[3,1],B[3,1],A[4,3],B[4,3]) - imgproduct3(A[1,1],B[1,1],A[2,3],B[2,3],A[3,2],B[3,2],A[4,4],B[4,4]) - imgproduct3(A[1,1],B[1,1],A[3,3],B[3,3],A[4,2],B[4,2],A[2,4],B[2,4]) - imgproduct3(A[1,1],B[1,1],A[2,2],B[2,2],A[3,4],B[3,4],A[4,3],B[4,3]) - imgproduct3(A[1,2],B[1,2],A[2,4],B[2,4],A[3,1],B[3,1],A[4,3],B[4,3]) - imgproduct3(A[1,2],B[1,2],A[2,1],B[2,1],A[3,3],B[3,3],A[4,4],B[4,4]) - imgproduct3(A[1,2],B[1,2],A[2,3],B[2,3],A[3,4],B[3,4],A[4,1],B[4,1]) - imgproduct3(A[1,3],B[1,3],A[2,1],B[2,1],A[3,4],B[3,4],A[4,2],B[4,2]) - imgproduct3(A[1,3],B[1,3],A[3,1],B[3,1],A[2,2],B[2,2],A[4,4],B[4,4]) - imgproduct3(A[1,3],B[1,3],A[2,4],B[2,4],A[3,2],B[3,2],A[4,1],B[4,1]) - imgproduct3(A[1,4],B[1,4],A[2,1],B[2,1],A[3,2],B[3,2],A[4,3],B[4,3]) - imgproduct3(A[1,4],B[1,4],A[4,1],B[4,1],A[2,2],B[2,2],A[3,3],B[3,3]) - imgproduct3(A[1,4],B[1,4],A[2,3],B[2,3],A[3,1],B[3,1],A[4,2],B[4,2]);

END;

$$

procedure COMPLEX_QUADRADTIC(A1,B1,A2,B2,A3,B3:EXTENDED; VAR reu1,imgu1,reu2,imgu2:EXTENDED);

VAR R,ALPHA,BETA,R1,C,S:EXTENDED;

BEGIN

ALPHA:= POWER(A2,2) - POWER(B2,2) - 4 * (A1 * A3 - B1 * B3);

BETA:= 2 * A2 * B2 - 4 * (A1 * B3 + B1 * A3);

R:= SQRT(POWER(ALPHA,2) + POWER(BETA,2));

R1:= SQRT(POWER(A1,2) + POWER(B1,2));

C:= SQRT(ABS((R + ALPHA))/2);

S:= SQRT(ABS((R - ALPHA))/2);

IF BETA < 0 THEN

S:= -S;

REU1:= (1/(2 * R1)) * ((A1/R1) * (-A2 + C) + (B1/R1) *

    (-B2 + S));

IMGU1:= (1/(2 * R1)) * ((A1/R1) * (-B2 + S) - (B1/R1) *

     (-A2 + C));

REU2:= (1/(2 * R1)) * ((A1/R1) * (-A2 - C) + (B1/R1) *

    (-B2 - S));

IMGU2:= (1/(2 * R1)) * ((A1/R1) * (-B2 - S) - (B1/R1) *

     (-A2 - C));

END;

$$

PROCEDURE CUBICCOMPLEX2(A3,B3,A2,B2,A1,B1,A0,B0:EXTENDED; VAR REW,IMGW:EXTENDED);

VAR M1,M2,N1,N2,S1,S2,T1,T2,REP,IMGP,COEFFP,ALPHA,BETA,COEFF2,R,C,S, LAMDA,V,RAD,ANGLE,REW1,REW2,REW3,IMGW1,IMGW2,IMGW3,REU1,REU2,REU3, IMGU1,IMGU2,IMGU3,REV1,REV2,REV3,IMGV1,IMGV2,IMGV3,POW,POW2, REfZ1,REfZ2,REfZ3,IMGfZ1,IMGfZ2,IMGfZ3,THETA:EXTENDED;

BEGIN

COEFFP:= 1/(3 * (POWER(POWER(A3,2) + POWER(B3,2),2)));

M1:= 3 * (A3 * A1 - B3 * B1) - (POWER(A2,2) - POWER(B2,2));

M2:= 3 * (A1 * B3 + A3 * B1) - 2 * A2 * B2;

N1:= (POWER(A3,2) - POWER(B3,2)) * M1 + 2 * A3 * B3 * M2;

N2:= (POWER(A3,2) - POWER(B3,2)) * M2 - 2 * A3 * B3 * M1;

REP:= COEFFP * ((POWER(A3,2) - POWER(B3,2)) * M1 + 2 * A3 * B3 * M2);

IMGP:= COEFFP * ((POWER(A3,2) - POWER(B3,2)) * M2 - 2 * A3 * B3 * M1);

S1:= 2 * (POWER(A2,3) - 3 * A2 * POWER(B2,2)) - 9 * (A1 * (A3 * A2 - B3 * B2) -

 B1 * (B3 * A2 + A3 * B2)) + 27 * (A0 * (POWER(A3,2) - POWER(B3,2)) -

 2 * A3 * B3 * B0);   {RETOPQ}

S2:= 2 * (-POWER(B2,3) + 3 * B2 * POWER(A2,2)) - 9 * (B1 * (A3 * A2 - B3 * B2) +

 A1 * (B3 * A2 + A3 * B2)) + 27 * (B0 * (POWER(A3,2) - POWER(B3,2)) +

 2 * A3 * B3 * A0);   {IMGTOPQ}

T1:= POWER(A3,3) - 3 * A3 * POWER(B3,2);{REBOTTOMQ}

T2:= 3 * POWER(A3,2) * B3 - POWER(B3,3); {IMGBOTTOMQ}

{NEED TO ADD FOR ALL Q = 0 + 0i}

{I.E; FOR Q <> 0 + 0i THEN DO BELOW ... ELSE Q = 0 + 0i.}

COEFF2:= 1/(2916 * POWER(POWER(T1,2) + POWER(T2,2),2));

ALPHA:= COEFF2 * (POWER(S1 * T1 + S2 * T2,2) - POWER(S2 * T1 - T2 * S1,2)) +

(POWER(N1,3) - 3 * N1 * POWER(N2,2))/(POWER(3 * (POWER(A3,2) + POWER(B3,2)),6));

{ALPHA = REDISC}

BETA:= COEFF2 * (2 * (S1 * T1 + S2 * T2) * (S2 * T1 - T2 * S1)) +

(-POWER(N2,3) + 3 * N2 * POWER(N1,2))/(POWER(3 * (POWER(A3,2) + POWER(B3,2)),6));

{BETA = IMGDISC}

R:= SQRT(POWER(ALPHA,2) + POWER(BETA,2));

C:= SQRT(ABS((R + ALPHA))/2); {C = SQRT(R) * COS(THETA/2)}

S:= SQRT(ABS((R - ALPHA))/2); {S = SQRT(R) * SIN(THETA/2)}

{DISC^(1/2) = C + iS}

IF BETA < 0 THEN

S:= -S;

LAMDA:= -(1/(54 * (POWER(T1,2) + POWER(T2,2)))) * (S1 * T1 + S2 * T2) + C;

V:= -(1/(54 * (POWER(T1,2) + POWER(T2,2)))) * (S2 * T1 - T2 * S1) + S;

RAD:= SQRT(POWER(LAMDA,2) + POWER(V,2));

IF (LAMDA = 0) AND (V > 0) THEN {NEED |LAMDA| < 1/10^(N) FOR LARGE N}

ANGLE:= PI/2;

IF (LAMDA = 0) AND (V < 0) THEN

ANGLE:= 3 * PI/2;

IF LAMDA <> 0 THEN

BEGIN

ANGLE:= ARCTAN(ABS(V/LAMDA)); {INCLUDES V = 0}

{NEXT DO ALL 4 CASES FOR LAMDA AND V}

IF (LAMDA < 0) AND (V > 0) THEN

ANGLE:= PI - ANGLE;

IF (LAMDA < 0) AND (V < 0) THEN

ANGLE:= PI + ANGLE;

IF (LAMDA > 0) AND (V < 0) THEN

ANGLE:= 2 * PI - ANGLE;

IF (LAMDA > 0) AND (V > 0) THEN

ANGLE:= ANGLE;

END;

REW1:= NROOT(RAD,3) * COS(ANGLE/3);

REW2:= NROOT(RAD,3) * COS(ANGLE/3 + 2 * PI/3);

REW3:= NROOT(RAD,3) * COS(ANGLE/3 + 4 * PI/3);

IMGW1:= NROOT(RAD,3) * SIN(ANGLE/3);

IMGW2:= NROOT(RAD,3) * SIN(ANGLE/3 + 2 * PI/3);

IMGW3:= NROOT(RAD,3) * SIN(ANGLE/3 + 4 * PI/3);

POW:= POWER(NROOT(RAD,3),2);

REU1:= (1/(3 * POW)) * ((3 * POW - REP) * REW1 - IMGP * IMGW1);

REU2:= (1/(3 * POW)) * ((3 * POW - REP) * REW2 - IMGP * IMGW2);

REU3:= (1/(3 * POW)) * ((3 * POW - REP) * REW3 - IMGP * IMGW3);

IMGU1:= (1/(3 * POW)) * ((3 * POW + REP) * IMGW1 - IMGP * REW1);

IMGU2:= (1/(3 * POW)) * ((3 * POW + REP) * IMGW2 - IMGP * REW2);

IMGU3:= (1/(3 * POW)) * ((3 * POW + REP) * IMGW3 - IMGP * REW3);

POW2:= POWER(A3,2) + POWER(B3,2);

REV1:= (1/(3 * POW2)) * (3 * POW2 * REU1 - (A2 * A3 + B2 * B3));

REV2:= (1/(3 * POW2)) * (3 * POW2 * REU2 - (A2 * A3 + B2 * B3));

REV3:= (1/(3 * POW2)) * (3 * POW2 * REU3 - (A2 * A3 + B2 * B3));

IMGV1:= (1/(3 * POW2)) * (3 * POW2 * IMGU1 - (B2 * A3 - A2 * B3));

IMGV2:= (1/(3 * POW2)) * (3 * POW2 * IMGU2 - (B2 * A3 - A2 * B3));

IMGV3:= (1/(3 * POW2)) * (3 * POW2 * IMGU3 - (B2 * A3 - A2 * B3));

REW:= REV1; {NEED ONE ROOT FOR DEGREE4_COMPLEX}

IMGW:= IMGV1;

END;

$$

PROCEDURE DEGREE4_COMPLEX(VAR A4,B4,A3,B3,A2,B2,A1,B1,A0,B0:EXTENDED; VAR REALX1,IMGX1,REALX2,IMGX2,REALX3,IMGX3,REALX4,IMGX4:EXTENDED);

VAR REP,IMGP,REQ,IMGQ,RER,IMGR,RE1,IMG1,RE2,IMG2,RE3,IMG3,R,THETA,REC,IMGC, TREA,TIMGA,REB,IMGB,RED,IMGD,ABSC2:EXTENDED; DP,DQ,DR,DZ,A33,B33:EXTENDED; STRREALX,STRIMGX:STRING; Sum:extended;

BEGIN

COMPLEX POLY COEFFS(RECOEFF 1,RECOEFF 2,RECOEFF 3,RECO EFF 4,RECOEFF_5,

IMGCOEFF 1,IMGCOEFF 2,IMGCOEFF 3,IMGCOEFF 4,IMGCOEFF_5,RE SOL,IMGSOL);

WRITELN(RECOEFF 1:0:4,' + i',IMGCOEFF 1:0:4,' * z^4 + ',RECOEFF 2:0:4,' + i',IMGCOEFF 2:0:4,' * z^3 + ');

WRITELN(RECOEFF 3:0:4,' + i',IMGCOEFF 3:0:4,' * z^2 + ',RECOEFF 4:0:4,' + i',IMGCOEFF 4:0:4,' * z + ');

WRITELN(RECOEFF 5:0:4,' + i',IMGCOEFF 5:0:4,' = 0');

writeln(mywork);

WRITELN(mywork,RECOEFF 1:0:4,' + i',IMGCOEFF 1:0:4,' * z^4 + ',RECOEFF 2:0:4,' + i',IMGCOEFF 2:0:4,' * z^3 + ');

WRITELN(mywork,RECOEFF 3:0:4,' + i',IMGCOEFF 3:0:4,' * z^2 + ',RECOEFF 4:0:4,' + i',IMGCOEFF 4:0:4,' * z + ');

WRITELN(mywork,RECOEFF 5:0:4,' + i',IMGCOEFF 5:0:4);

writeln(mywork);

DP:= 8 * POWER(POWER(A4,2) + POWER(B4,2),2);

REP:= (8 * (POWER(A4,2) + POWER(B4,2)) * (A2 * A4 + B2 * B4) - 3 * (POWER(A3 * A4 + B3 * B4,2)

  - POWER(B3 * A4 - A3 * B4,2)))/DP;

IMGP:= (8 * (POWER(A4,2) + POWER(B4,2)) * (B2 * A4 - A2 * B4) - 6 * (A4 * B3 - B4 * A3) * (A3 * A4 + B3 * B4))/DP;

DQ:= 16 * POWER(POWER(A4,2) + POWER(B4,2),3);

A33:= POWER(A3,3) - 3 * A3 * POWER(B3,2);

B33:= 3 * POWER(A3,2) * B3 - POWER(B3,3);

REQ:= (16 * (POWER(POWER(A4,2) + POWER(B4,2),2) * (A1 * A4 + B1 * B4)) -

  8 * (POWER(A4,2) + POWER(B4,2)) * ((A2 * A3 - B2 * B3) *

(POWER(A4,2) - POWER(B4,2)) +

  2 * (B2 * A3 + A2 * B3) * (A4 * B4)) + 2 * (A33 * (POWER(A4,3) - 3 * A4 *

POWER(B4,2)) -

  B33 * (POWER(B4,3) - 3 * POWER(A4,2) * B4)))/DQ;

IMGQ:= (16 * (POWER(POWER(A4,2) + POWER(B4,2),2) * (A4 * B1 - B4 * A1)) -

  8 * (POWER(A4,2) + POWER(B4,2)) * ((B2 * A3 + A2 * B3) *

(POWER(A4,2) - POWER(B4,2)) -

  2 * (A2 * A3 - B2 * B3) * (A4 * B4)) + 2 * (B33 * (POWER(A4,3) - 3 * A4 *

POWER(B4,2)) +

  A33 * (POWER(B4,3) - 3 * POWER(A4,2) * B4)))/DQ;

DR:= 256 * POWER(POWER(A4,2) + POWER(B4,2),4);

RER:= (256 * POWER(POWER(A4,2) + POWER(B4,2),3) * (A4 * A0 + B4 * B0) - 64 * POWER(POWER(A4,2) + POWER(B4,2),2)

  * ((A1 * A3 - B1 * B3) * (POWER(A4,2) - POWER(B4,2)) + 2 * (B1 * A3 + B3

* A1) * (A4 * B4))

  + 16 * (POWER(A4,2) + POWER(B4,2)) * ((A2 * (POWER(A3,2) -

POWER(B3,2)) - 2 * A3 * B2 * B3)

  * (POWER(A4,3) - 3 * A4 * POWER(B4,2)) - (B2 * (POWER(A3,2) -

POWER(B3,2)) + 2 * A2 * A3 * B3)

  * (POWER(B4,3) - 3 * B4 * POWER(A4,2))) - 3 * (POWER(A3 * A4 + B3 *

B4,4) - 6 * (POWER(A3 * A4 + B3 * B4,2))

  * POWER(B3 * A4 - A3 * B4,2) + POWER(B3 * A4 - A3 * B4,4)))/DR;

IMGR:= (-256 * POWER(POWER(A4,2) + POWER(B4,2),3) * (B4 * A0 - A4 * B0) - 64 * POWER(POWER(A4,2) + POWER(B4,2),2)

  * ((B1 * A3 + B3 * A1) * (POWER(A4,2) - POWER(B4,2)) - 2 * (A1 * A3 - B1

* B3) * (A4 * B4))

  + 16 * (POWER(A4,2) + POWER(B4,2)) * ((B2 * (POWER(A3,2) -

POWER(B3,2)) + 2 * A2 * A3 * B3)

  * (POWER(A4,3) - 3 * A4 * POWER(B4,2)) + (A2 * (POWER(A3,2) -

POWER(B3,2)) - 2 * A3 * B2 * B3)

  * (POWER(B4,3) - 3 * B4 * POWER(A4,2))) - 3 * (4 * POWER(A3 * A4 + B3

* B4,3) * (B3 * A4 - A3 * B4)

  - 4 * (A3 * A4 + B3 * B4) * POWER(B3 * A4 - A3 * B4,3)))/DR;

RE1:= 2 * REP;

IMG1:= 2 * IMGP;

RE2:= POWER(REP,2) - POWER(IMGP,2) - 4 * RER;

IMG2:= 2 * REP * IMGP - 4 * IMGR;

RE3:= -(POWER(REQ,2) - POWER(IMGQ,2));

IMG3:= -(2 * REQ * IMGQ);

CUBICCOMPLEX2(1,0,RE1,IMG1,RE2,IMG2,RE3,IMG3,REW,IMGW);

R:= SQRT(POWER(REW,2) + POWER(IMGW,2));

R:= SQRT(R);

if rew = 0 then

begin

if imgw > 0 then

theta:= pi/2;

if imgw < 0 then

theta:= 3 * pi/2;

end;

if rew <> 0 then

begin

theta:= arctan(abs(imgw/rew));

if (imgw < 0) and (rew > 0) then

theta:= 2 * pi - theta;

if (imgw > 0) and (rew < 0) then

theta:= pi - theta;

if (imgw < 0) and (rew < 0) then

theta:= pi + theta;

if (imgw = 0) and (rew > 0) then

theta:= 0;

if (imgw = 0) and (rew < 0) then

theta:= pi;

end;

REC:= R * COS(THETA/2);

IMGC:= R * SIN(THETA/2);

TREA:= -REC;

TIMGA:= -IMGC;

ABSC2:= POWER(REC,2) + POWER(IMGC,2);

REB:= (ABSC2 * (POWER(REC,2) - POWER(IMGC,2)) + ABSC2 * REP + REQ * REC + IMGQ * IMGC)/(2 * ABSC2);

IMGB:= (2 * ABSC2 * REC * IMGC + ABSC2 * IMGP + IMGQ * REC - REQ * IMGC)/(2 * ABSC2);

RED:= (ABSC2 * (POWER(REC,2) - POWER(IMGC,2)) + ABSC2 * REP - (REQ * REC + IMGQ * IMGC))/(2 * ABSC2);

IMGD:= (2 * ABSC2 * REC * IMGC + ABSC2 * IMGP - (IMGQ * REC - REQ * IMGC))/(2 * ABSC2);

COMPLEX_QUADRADTIC(1,0,TREA,TIMGA,REB,IMGB,REU1,IMGU1,REU2, IMGU2);

COMPLEX_QUADRADTIC(1,0,REC,IMGC,RED,IMGD,REU3,IMGU3,REU4,I MGU4);

DZ:= 4 * (POWER(A4,2) + POWER(B4,2));

REALX1:= (DZ * REU1 - (A3 * A4 + B3 * B4))/DZ;

IMGX1:= (DZ * IMGU1 - (B3 * A4 - A3 * B4))/DZ;

REALX2:= (DZ * REU2 - (A3 * A4 + B3 * B4))/DZ;

IMGX2:= (DZ * IMGU2 - (B3 * A4 - A3 * B4))/DZ;

REALX3:= (DZ * REU3 - (A3 * A4 + B3 * B4))/DZ;

IMGX3:= (DZ * IMGU3 - (B3 * A4 - A3 * B4))/DZ;

REALX4:= (DZ * REU4 - (A3 * A4 + B3 * B4))/DZ;

IMGX4:= (DZ * IMGU4 - (B3 * A4 - A3 * B4))/DZ;

STR(REALX1:0:4,STRREALX);

STR(IMGX1:0:4,STRIMGX);

WRITELN(STRREALX + ' + i' + STRIMGX);

writeln(mywork,STRREALX + ' + i' + STRIMGX);

STR(REALX2:0:4,STRREALX);

STR(IMGX2:0:4,STRIMGX);

WRITELN(STRREALX + ' + i' + STRIMGX);

writeln(mywork,STRREALX + ' + i' + STRIMGX);

STR(REALX3:0:4,STRREALX);

STR(IMGX3:0:4,STRIMGX);

WRITELN(STRREALX + ' + i' + STRIMGX);

WRITELN(mywork,STRREALX + ' + i' + STRIMGX);

STR(REALX4:0:4,STRREALX);

STR(IMGX4:0:4,STRIMGX);

WRITELN(STRREALX + ' + i' + STRIMGX);

writeln(mywork,STRREALX + ' + i' + STRIMGX);

{For brillant only}

sum:= realx1 + realx2 + realx3+ realx4 + imgx1 + imgx2 + imgx3 + imgx4;

writeln(mywork);

writeln(mywork,'Sum = ',sum:0:4);

writeln;

writeln('Sum = ',sum:0:4);

END;

$$

PROCEDURE GETCOMPLEXROOTS(VAR REZ1,IMGZ1,REZ2,IMGZ2,REZ3,IMGZ3,REZ4,IMGZ4:EXTENDED);

BEGIN

DEGREE4 COMPLEX(RECOEFF 1,IMGCOEFF 1,RECOEFF 2,IMGCOEFF 2,RECOEFF 3,IMGCOEFF 3,RECOEFF 4,IMGCOEFF 4, RECOEFF 5,IMGCOEFF_5,REZ1,IMGZ1,REZ2,IMGZ2,REZ3,IMGZ3,REZ4,IMGZ4);

END;

$$

begin

assign(mywork,'cmywork.txt');

rewrite(mywork);

getsize;

getcoefficients;

getaugmentedmatrix;

for l:= 1 to m do

begin{l}

safeguard(rea,imga,l,p,v);

if v = 0 then

begin{then}

operation1(rea,imga,l);

operation2(rea,imga,l);

end;{then}

end;{l}

showdata;

GETCOMPLEXROOTS(REZ1,IMGZ1,REZ2,IMGZ2,REZ3,IMGZ3,REZ4,IMGZ4);

close(mywork);

readln;

readln;

end.