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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="_monomial_sporderings.html" title="">monomial orderings</a> > <a href="_packing_spmonomials_spfor_spefficiency.html" title="">packing monomials for efficiency</a></div>
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<div><h1>packing monomials for efficiency</h1>
<div>Sometimes for efficiency reasons, it is important to pack exponent vectors several exponents per machine word. Polynomials take less space, and monomial operations such as comparison and multiplication become faster.<p/>
The monomial order keys <a href="___Lex.html" title="lexicographical monomial order.">Lex</a> and <a href="___G__Rev__Lex.html" title="graded reverse lexicographical monomial order.">GRevLex</a> allow packing. The <tt>MonomialSize => n</tt> option allows one to set the minimum packing size, in number of bits. Monomials are stored as signed exponent vectors, so maximum exponents of 2^(n-1)-1 are possible for packed variables. Useful values include 8, 16, 32, and (on 64-bit machines) 64. The default monomial size is 32.<table class="examples"><tr><td><pre>i1 : A = QQ[a..d,MonomialSize=>8]
o1 = A
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : B = QQ[x,y,z,w,MonomialSize=>16,MonomialOrder=>Lex]
o2 = B
o2 : PolynomialRing</pre>
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The maximum degree for monomials in A is 127. Monomials of higher degree will encounter a monomial overflow. In the second example, the maximum exponent is 32767 (2^15-1).<p/>
It is possible to pack different parts of the monomial with different sizes. For example, the following order has two blocks: a graded reverse lexicographic block of 3 variables, packed into one 32-bit word, and a second lexicographic block for 4 variables, taking 4 32-bit words. Each monomial will be packed into 5 32-bit words (on a computer with a 32-bit word size).<table class="examples"><tr><td><pre>i3 : C = QQ[a,b,c,x,y,z,w,MonomialOrder=>{MonomialSize=>8,3,MonomialSize=>32,Lex=>4}];</pre>
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<table class="examples"><tr><td><pre>i4 : D = QQ[a..d,MonomialOrder=>Lex];</pre>
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<tr><td><pre>i5 : a^1000000000
1000000000
o5 = a
o5 : D</pre>
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<p/>
This exponent would give a monomial overflow error in the next two rings.<table class="examples"><tr><td><pre>i6 : E = QQ[a..d,MonomialSize=>16,MonomialOrder=>Lex];</pre>
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<tr><td><pre>i7 : F = QQ[a..d,MonomialSize=>8,MonomialOrder=>Lex];</pre>
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