Cheking a Zero Polynomial

Question

Can anyone explain me in below algorithm how the "ISZERO" function checking whether the polynomial is zero or not. Here "REM(P,e)" function removes all the values with exponent "e". what i don't able to understand is the significance of "if COEF(P,e) = - c". And also what is this "SMULT" function.

 structure POLYNOMIAL
    declare ZERO( ) poly; ISZERO(poly) Boolean
    COEF(poly,exp) coef;
    ATTACH(poly,coef,exp) poly
    REM(poly,exp) poly
    SMULT(poly,coef,exp) poly
    ADD(poly,poly) poly; MULT(poly,poly) poly;
    for all P,Q, poly c,d, coef e,f exp let
    REM(ZERO,f) :: = ZERO
    REM(ATTACH(P,c,e),f) :: =
    if e = f then REM(P,f) else ATTACH(REM(P,f),c,e)
    ***ISZERO(ZERO) :: = true
    ISZERO(ATTACH(P,c,e)):: =
    if COEF(P,e) = - c then ISZERO(REM(P,e)) else false***
    COEF(ZERO,e) :: = 0
    COEF(ATTACH(P,c,e),f) :: =
    if e = f then c + COEF(P,f) else COEF(P,f)
    SMULT(ZERO,d,f) :: = ZERO
    SMULT(ATTACH(P,c,e),d,f) :: =
    ATTACH(SMULT(P,d,f),c d,e + f)
    ADD(P,ZERO):: = P
    ADD(P,ATTACH(Q,d,f)) :: = ATTACH(ADD(P,Q),d,f)
    MULT(P,ZERO) :: = ZERO
    MULT(P,ATTACH(Q,d,f)) :: =
    ADD(MULT(P,Q),SMULT(P,d,f))
    end
    end POLYNOMIAL

*"..the significance of "if COEF(P,e) = - c.". Thats not the only thing not understood here. Why this is tagged with C? — WhozCraig, Aug 17 '13 at 03:51
Source of this specification [here](http://www.icodeguru.com/vc/10book/books/book1/chap02.htm). There is apparently no complete description of this "language", but is partially described [here](http://www.icodeguru.com/vc/10book/books/book1/appa.htm). — jxh, Aug 17 '13 at 05:55

score 1 · Accepted Answer · answered Aug 17 '13 at 05:32

1

Without knowing what language this is, it looks like this line

ISZERO(ATTACH(P,c,e)):: =
if COEF(P,e) = - c then ISZERO(REM(P,e)) else false

is specifying ISZERO recursively. We are trying to determine whether ATTACH(P, c, e), otherwise known as P(x) + cx^e, is zero. It first checks whether the x^e coefficient of P is -c. If not, then P(x) + cx^e is definitely not zero, and you can return false immediately. Otherwise, P(x) + cx^e = REM(P, e), so you have to check ISZERO(REM(P, e)).

I believe SMULT is multiplication, so SMULT(P, a, b) is equivalent to a * x^b * P(x).

answered Aug 17 '13 at 05:32

arghbleargh

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the language used in this algo is sparks. The problem is i am not able to understand the significance of checking the "negative co-efficient" . Negative or positive polynomial is not zero if it has any co-efficient. Please explain. – Abhishek Aug 17 '13 at 06:57
For example, suppose P is x^2 - 3x + 2, c is 3, and e is 1. Then the algorithm is checking whether P + 3x is zero. It first checks if COEF(P, 1) is -3. This is true, so it checks whether REM(P, 1) = x^2 + 2 is zero. It's not zero, so it returns false (well, it has to do further reductions, like x^2 + 2 = ATTACH(x^2, 2, 0), but it eventually finds that it's false). – arghbleargh Aug 17 '13 at 07:31
If we changed P to x^2 - 2x + 2 in the above example, then we would first check that COEF(P, 1) is not -3. Then we know that P + 3x cannot possibly be zero because the x coefficient is not zero. So then we can return false immediately. – arghbleargh Aug 17 '13 at 07:32
sorry for asking again but plz explain. why do we need to remove negative coefficient using REM(P, 1) for checking a polynomial is zero or not. An addition of polynomial algorithm is using ISZERO function before addition, is it invalid to add two negative co-efficient polynomial. – Abhishek Aug 17 '13 at 07:45
Also if u think that SMULT function is for multiplication then what is the purpose of MUL function. – Abhishek Aug 17 '13 at 07:56
The SMULT function is for multiplying with a polynomial with a single term (known as a "monomial"). It's being used as a building block for the MULT function, which multiplies any two polynomials. As for the ISZERO thing, I'm afraid I can't explain it much more simply than I already have. I'd suggest you try to work through a few examples to see what's going on, or re-read my comment. – arghbleargh Aug 17 '13 at 08:08

Cheking a Zero Polynomial

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