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78 lines
2.4 KiB
Plaintext
78 lines
2.4 KiB
Plaintext
Multiplication
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This describes the multiplication algorithm used by the MPI library.
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This is basically a standard "schoolbook" algorithm. It is slow --
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O(mn) for m = #a, n = #b -- but easy to implement and verify.
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Basically, we run two nested loops, as illustrated here (R is the
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radix):
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k = 0
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for j <- 0 to (#b - 1)
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for i <- 0 to (#a - 1)
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w = (a[j] * b[i]) + k + c[i+j]
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c[i+j] = w mod R
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k = w div R
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endfor
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c[i+j] = k;
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k = 0;
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endfor
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It is necessary that 'w' have room for at least two radix R digits.
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The product of any two digits in radix R is at most:
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(R - 1)(R - 1) = R^2 - 2R + 1
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Since a two-digit radix-R number can hold R^2 - 1 distinct values,
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this insures that the product will fit into the two-digit register.
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To insure that two digits is enough for w, we must also show that
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there is room for the carry-in from the previous multiplication, and
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the current value of the product digit that is being recomputed.
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Assuming each of these may be as big as R - 1 (and no larger,
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certainly), two digits will be enough if and only if:
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(R^2 - 2R + 1) + 2(R - 1) <= R^2 - 1
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Solving this equation shows that, indeed, this is the case:
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R^2 - 2R + 1 + 2R - 2 <= R^2 - 1
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R^2 - 1 <= R^2 - 1
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This suggests that a good radix would be one more than the largest
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value that can be held in half a machine word -- so, for example, as
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in this implementation, where we used a radix of 65536 on a machine
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with 4-byte words. Another advantage of a radix of this sort is that
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binary-level operations are easy on numbers in this representation.
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Here's an example multiplication worked out longhand in radix-10,
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using the above algorithm:
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a = 999
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b = x 999
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-------------
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p = 98001
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w = (a[jx] * b[ix]) + kin + c[ix + jx]
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c[ix+jx] = w % RADIX
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k = w / RADIX
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product
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ix jx a[jx] b[ix] kin w c[i+j] kout 000000
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0 0 9 9 0 81+0+0 1 8 000001
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0 1 9 9 8 81+8+0 9 8 000091
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0 2 9 9 8 81+8+0 9 8 000991
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8 0 008991
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1 0 9 9 0 81+0+9 0 9 008901
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1 1 9 9 9 81+9+9 9 9 008901
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1 2 9 9 9 81+9+8 8 9 008901
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9 0 098901
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2 0 9 9 0 81+0+9 0 9 098001
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2 1 9 9 9 81+9+8 8 9 098001
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2 2 9 9 9 81+9+9 9 9 098001
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------------------------------------------------------------------
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This Source Code Form is subject to the terms of the Mozilla Public
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# License, v. 2.0. If a copy of the MPL was not distributed with this
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# file, You can obtain one at http://mozilla.org/MPL/2.0/.
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