Task

Display a finite linear combination in an infinite vector basis 



(

e

1


,

e

2


,
…
)


{\displaystyle (e_{1},e_{2},\ldots )}

.

Write a function that, when given a finite list of scalars 



(

α

1


,

α

2


,
…
)


{\displaystyle (\alpha ^{1},\alpha ^{2},\ldots )}

, 
creates a string representing the linear combination 




∑

i



α

i



e

i




{\displaystyle \sum _{i}\alpha ^{i}e_{i}}

 in an explicit format often used in mathematics, that is:






α


i

1





e


i

1




±

|


α


i

2





|


e


i

2




±

|


α


i

3





|


e


i

3




±
…


{\displaystyle \alpha ^{i_{1}}e_{i_{1}}\pm |\alpha ^{i_{2}}|e_{i_{2}}\pm |\alpha ^{i_{3}}|e_{i_{3}}\pm \ldots }


where 




α


i

k




≠
0


{\displaystyle \alpha ^{i_{k}}\neq 0}







The output must comply to the following rules:

   don't show null terms, unless the whole combination is null. 
 e(1)     is fine,     e(1) + 0*e(3)     or     e(1) + 0     is wrong.
   don't show scalars when they are equal to one or minus one. 
 e(3)     is fine,     1*e(3)     is wrong.
   don't prefix by a minus sign if it follows a preceding term.   Instead you use subtraction. 
 e(4) - e(5)     is fine,     e(4) + -e(5)     is wrong.


Show here output for the following lists of scalars:

 1)    1,  2,  3
 2)    0,  1,  2,  3
 3)    1,  0,  3,  4
 4)    1,  2,  0
 5)    0,  0,  0
 6)    0
 7)    1,  1,  1
 8)   -1, -1, -1
 9)   -1, -2,  0, -3
10)   -1



MODULE Linear;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
 
PROCEDURE WriteInt(n : INTEGER);
VAR buf : ARRAY[0..15] OF CHAR;
BEGIN
    FormatString("%i", buf, n);
    WriteString(buf)
END WriteInt;
 
PROCEDURE WriteLinear(c : ARRAY OF INTEGER);
VAR
    buf : ARRAY[0..15] OF CHAR;
    i,j : CARDINAL;
    b : BOOLEAN;
BEGIN
    b := TRUE;
    j := 0;
 
    FOR i:=0 TO HIGH(c) DO
        IF c[i]=0 THEN CONTINUE END;
 
        IF c[i]<0 THEN
            IF b THEN WriteString("-")
            ELSE      WriteString(" - ") END;
        ELSIF c[i]>0 THEN
            IF NOT b THEN WriteString(" + ") END;
        END;
 
        IF c[i] > 1 THEN
            WriteInt(c[i]);
            WriteString("*")
        ELSIF c[i] < -1 THEN
            WriteInt(-c[i]);
            WriteString("*")
        END;
 
        FormatString("e(%i)", buf, i+1);
        WriteString(buf);
 
        b := FALSE;
        INC(j)
    END;
 
    IF j=0 THEN WriteString("0") END;
    WriteLn
END WriteLinear;
 
TYPE
    Array1 = ARRAY[0..0] OF INTEGER;
    Array3 = ARRAY[0..2] OF INTEGER;
    Array4 = ARRAY[0..3] OF INTEGER;
BEGIN
    WriteLinear(Array3{1,2,3});
    WriteLinear(Array4{0,1,2,3});
    WriteLinear(Array4{1,0,3,4});
    WriteLinear(Array3{1,2,0});
    WriteLinear(Array3{0,0,0});
    WriteLinear(Array1{0});
    WriteLinear(Array3{1,1,1});
    WriteLinear(Array3{-1,-1,-1});
    WriteLinear(Array4{-1,-2,0,-3});
    WriteLinear(Array1{-1});
 
    ReadChar
END Linear.