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The Maxima on-line user's manual

Algebra Calculator

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Gcdex Calculator

Gcdex

Function: gcdex (<f>, <g>)

load(gcdex);
igcdex(9292162750094637473537,13029506445834204798802);

Function: gcdex (<f>, <g>, <x>) Returns a list [<a>, <b>, <u>] where <u> is the greatest common divisor (gcd) of <f> and <g>, and <u> is equal to <a> <f> + <b> <g>. The arguments <f> and <g> should be univariate polynomials, or else polynomials in <x> a supplied main variable since we need to be in a principal ideal domain for this to work. The gcd means the gcd regarding <f> and <g> as univariate polynomials with coefficients being rational functions in the other variables.

gcdex implements the Euclidean algorithm, where we have a sequence of L[i]: [a[i], b[i], r[i]] which are all perpendicular to [f, g, -1] and the next one is built as if q = quotient(r[i]/r[i+1]) then L[i+2]: L[i] - q L[i+1], and it terminates at L[i+1] when the remainder r[i+2] is zero.

          (%i1) gcdex (x^2 + 1, x^3 + 4);
                                 2
                                x  + 4 x - 1  x + 4
          (%o1)/R/           [- ------------, -----, 1]
                                     17        17
          (%i2) % . [x^2 + 1, x^3 + 4, -1];
          (%o2)/R/                        0

Note that the gcd in the following is 1 since we work in k(y)[x], not the y+1 we would expect in k[y, x].

          (%i1) gcdex (x*(y + 1), y^2 - 1, x);
                                         1
          (%o1)/R/                 [0, ------, 1]
                                        2
                                       y  - 1

(%o1)                                true
(%i2) 

Gcdex Example

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f:x*y^2-y;

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gcdex(98,-199);

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gcdex-load

load(gcdex);

igcdex(0,5);

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gcdex-load

load (gcdex);

igcdex(2,5);

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