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15.1.4 Toom 4-Way Multiplication

Karatsuba and Toom-3 split the operands into 2 and 3 coefficients, respectively. Toom-4 analogously splits the operands into 4 coefficients. Using the notation from the section on Toom-3 multiplication, we form two polynomials:

X(t) = x3*t^3 + x2*t^2 + x1*t + x0
Y(t) = y3*t^3 + y2*t^2 + y1*t + y0

X(t) and Y(t) are evaluated and multiplied at 7 points, giving values of W(t) at those points. In GMP the following points are used,

t=0x0 * y0, which gives w0 immediately
t=1/2(x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0)
t=-1/2(-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0)
t=1(x3+x2+x1+x0) * (y3+y2+y1+y0)
t=-1(-x3+x2-x1+x0) * (-y3+y2-y1+y0)
t=2(8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0)
t=infx3 * y3, which gives w6 immediately

The number of additions and subtractions for Toom-4 is much larger than for Toom-3. But several subexpressions occur multiple times, for example x2+x0, occurs for both t=1 and t=-1.

Toom-4 is asymptotically O(N^1.404), the exponent being log(7)/log(4), representing 7 recursive multiplies of 1/4 the original size each.