Annotation of OpenXM/src/kan96xx/gmp-2.0.2-ssh-2/PROJECTS, Revision 1.1.1.1
1.1 takayama 1: IDEAS ABOUT THINGS TO WORK ON
2:
3: * mpq_cmp: Maybe the most sensible thing to do would be to multiply the, say,
4: 4 most significant limbs of each operand and compare them. If that is not
5: sufficient, do the same for 8 limbs, etc.
6:
7: * Write mpi, the Multiple Precision Interval Arithmetic layer.
8:
9: * Write `mpX_eval' that take lambda-like expressions and a list of operands.
10:
11: * As a general rule, recognize special operand values in mpz and mpf, and
12: use shortcuts for speed. Examples: Recognize (small or all) 2^n in
13: multiplication and division. Recognize small bases in mpz_pow_ui.
14:
15: * Implement lazy allocation? mpz->d == 0 would mean no allocation made yet.
16:
17: * Maybe store one-limb numbers according to Per Bothner's idea:
18: struct {
19: mp_ptr d;
20: union {
21: mp_limb val; /* if (d == NULL). */
22: mp_size size; /* Length of data array, if (d != NULL). */
23: } u;
24: };
25: Problem: We can't normalize to that format unless we free the space
26: pointed to by d, and therefore small values will not be stored in a
27: canonical way.
28:
29: * Document complexity of all functions.
30:
31: * Add predicate functions mpz_fits_signedlong_p, mpz_fits_unsignedlong_p,
32: mpz_fits_signedint_p, etc.
33:
34: mpz_floor (mpz, mpq), mpz_trunc (mpz, mpq), mpz_round (mpz, mpq).
35:
36: * Better random number generators. There should be fast (like mpz_random),
37: very good (mpz_veryrandom), and special purpose (like mpz_random2). Sizes
38: in *bits*, not in limbs.
39:
40: * It'd be possible to have an interface "s = add(a,b)" with automatic GC.
41: If the mpz_xinit routine remembers the address of the variable we could
42: walk-and-mark the list of remembered variables, and free the space
43: occupied by the remembered variables that didn't get marked. Fairly
44: standard.
45:
46: * Improve speed for non-gcc compilers by defining umul_ppmm, udiv_qrnnd,
47: etc, to call __umul_ppmm, __udiv_qrnnd. A typical definition for
48: umul_ppmm would be
49: #define umul_ppmm(ph,pl,m0,m1) \
50: {unsigned long __ph; (pl) = __umul_ppmm (&__ph, (m0), (m1)); (ph) = __ph;}
51: In order to maintain just one version of longlong.h (gmp and gcc), this
52: has to be done outside of longlong.h.
53:
54: Bennet Yee at CMU proposes:
55: * mpz_{put,get}_raw for memory oriented I/O like other *_raw functions.
56: * A function mpfatal that is called for exceptions. Let the user override
57: a default definition.
58:
59: * Make all computation mpz_* functions return a signed int indicating if the
60: result was zero, positive, or negative?
61:
62: * Implement mpz_cmpabs, mpz_xor, mpz_to_double, mpz_to_si, mpz_lcm, mpz_dpb,
63: mpz_ldb, various bit string operations. Also mpz_@_si for most @??
64:
65: * Add macros for looping efficiently over a number's limbs:
66: MPZ_LOOP_OVER_LIMBS_INCREASING(num,limb)
67: { user code manipulating limb}
68: MPZ_LOOP_OVER_LIMBS_DECREASING(num,limb)
69: { user code manipulating limb}
70:
71: Brian Beuning proposes:
72: 1. An array of small primes
73: 3. A function to factor a mpz_t. [How do we return the factors? Maybe
74: we just return one arbitrary factor? In the latter case, we have to
75: use a data structure that records the state of the factoring routine.]
76: 4. A routine to look for "small" divisors of an mpz_t
77: 5. A 'multiply mod n' routine based on Montgomery's algorithm.
78:
79: Dough Lea proposes:
80: 1. A way to find out if an integer fits into a signed int, and if so, a
81: way to convert it out.
82: 2. Similarly for double precision float conversion.
83: 3. A function to convert the ratio of two integers to a double. This
84: can be useful for mixed mode operations with integers, rationals, and
85: doubles.
86:
87: Elliptic curve method description in the Chapter `Algorithms in Number
88: Theory' in the Handbook of Theoretical Computer Science, Elsevier,
89: Amsterdam, 1990. Also in Carl Pomerance's lecture notes on Cryptology and
90: Computational Number Theory, 1990.
91:
92: * Harald Kirsh suggests:
93: mpq_set_str (MP_RAT *r, char *numerator, char *denominator).
94:
95: * New function: mpq_get_ifstr (int_str, frac_str, base,
96: precision_in_som_way, rational_number). Convert RATIONAL_NUMBER to a
97: string in BASE and put the integer part in INT_STR and the fraction part
98: in FRAC_STR. (This function would do a division of the numerator and the
99: denominator.)
100:
101: * Should mpz_powm* handle negative exponents?
102:
103: * udiv_qrnnd: If the denominator is normalized, the n0 argument has very
104: little effect on the quotient. Maybe we can assume it is 0, and
105: compensate at a later stage?
106:
107: * Better sqrt: First calculate the reciprocal square root, then multiply by
108: the operand to get the square root. The reciprocal square root can be
109: obtained through Newton-Raphson without division. To compute sqrt(A), the
110: iteration is,
111:
112: 2
113: x = x (3 - A x )/2.
114: i+1 i i
115:
116: The final result can be computed without division using,
117:
118: sqrt(A) = A x .
119: n
120:
121: * Newton-Raphson using multiplication: We get twice as many correct digits
122: in each iteration. So if we square x(k) as part of the iteration, the
123: result will have the leading digits in common with the entire result from
124: iteration k-1. A _mpn_mul_lowpart could help us take advantage of this.
125:
126: * Peter Montgomery: If 0 <= a, b < p < 2^31 and I want a modular product
127: a*b modulo p and the long long type is unavailable, then I can write
128:
129: typedef signed long slong;
130: typedef unsigned long ulong;
131: slong a, b, p, quot, rem;
132:
133: quot = (slong) (0.5 + (double)a * (double)b / (double)p);
134: rem = (slong)((ulong)a * (ulong)b - (ulong)p * (ulong)quot);
135: if (rem < 0} {rem += p; quot--;}
136:
137: * Speed modulo arithmetic, using Montgomery's method or my pre-inversion
138: method. In either case, special arithmetic calls would be needed,
139: mpz_mmmul, mpz_mmadd, mpz_mmsub, plus some kind of initialization
140: functions. Better yet: Write a new mpr layer.
141:
142: * mpz_powm* should not use division to reduce the result in the loop, but
143: instead pre-compute the reciprocal of the MOD argument and do reduced_val
144: = val-val*reciprocal(MOD)*MOD, or use Montgomery's method.
145:
146: * mpz_mod_2expplussi -- to reduce a bignum modulo (2**n)+s
147:
148: * It would be a quite important feature never to allocate more memory than
149: really necessary for a result. Sometimes we can achieve this cheaply, by
150: deferring reallocation until the result size is known.
151:
152: * New macro in longlong.h: shift_rhl that extracts a word by shifting two
153: words as a unit. (Supported by i386, i860, HP-PA, POWER, 29k.) Useful
154: for shifting multiple precision numbers.
155:
156: * The installation procedure should make a test run of multiplication to
157: decide the threshold values for algorithm switching between the available
158: methods.
159:
160: * Fast output conversion of x to base B:
161: 1. Find n, such that (B^n > x).
162: 2. Set y to (x*2^m)/(B^n), where m large enough to make 2^n ~~ B^n
163: 3. Multiply the low half of y by B^(n/2), and recursively convert the
164: result. Truncate the low half of y and convert that recursively.
165: Complexity: O(M(n)log(n))+O(D(n))!
166:
167: * Improve division using Newton-Raphson. Check out "Newton Iteration and
168: Integer Division" by Stephen Tate in "Synthesis of Parallel Algorithms",
169: Morgan Kaufmann, 1993 ("beware of some errors"...)
170:
171: * Improve implementation of Karatsuba's algorithm. For most operand sizes,
172: we can reduce the number of operations by splitting differently.
173:
174: * Faster multiplication: The best approach is to first implement Toom-Cook.
175: People report that it beats Karatsuba's algorithm already at about 100
176: limbs. FFT would probably never beat a well-written Toom-Cook (not even for
177: millions of bits).
178:
179: FFT:
180: {
181: * Multiplication could be done with Montgomery's method combined with
182: the "three primes" method described in Lipson. Maybe this would be
183: faster than to Nussbaumer's method with 3 (simple) moduli?
184:
185: * Maybe the modular tricks below are not needed: We are using very
186: special numbers, Fermat numbers with a small base and a large exponent,
187: and maybe it's possible to just subtract and add?
188:
189: * Modify Nussbaumer's convolution algorithm, to use 3 words for each
190: coefficient, calculating in 3 relatively prime moduli (e.g.
191: 0xffffffff, 0x100000000, and 0x7fff on a 32-bit computer). Both all
192: operations and CRR would be very fast with such numbers.
193:
194: * Optimize the Schoenhage-Stassen multiplication algorithm. Take advantage
195: of the real valued input to save half of the operations and half of the
196: memory. Use recursive FFT with large base cases, since recursive FFT has
197: better memory locality. A normal FFT get 100% cache misses for large
198: enough operands.
199:
200: * In the 3-prime convolution method, it might sometimes be a win to use 2,
201: 3, or 5 primes. Imagine that using 3 primes would require a transform
202: length of 2^n. But 2 primes might still sometimes give us correct
203: results with that same transform length, or 5 primes might allow us to
204: decrease the transform size to 2^(n-1).
205:
206: To optimize floating-point based complex FFT we have to think of:
207:
208: 1. The normal implementation accesses all input exactly once for each of
209: the log(n) passes. This means that we will get 0% cache hit when n >
210: our cache. Remedy: Reorganize computation to compute partial passes,
211: maybe similar to a standard recursive FFT implementation. Use a large
212: `base case' to make any extra overhead of this organization negligible.
213:
214: 2. Use base-4, base-8 and base-16 FFT instead of just radix-2. This can
215: reduce the number of operations by 2x.
216:
217: 3. Inputs are real-valued. According to Knuth's "Seminumerical
218: Algorithms", exercise 4.6.4-14, we can save half the memory and half
219: the operations if we take advantage of that.
220:
221: 4. Maybe make it possible to write the innermost loop in assembly, since
222: that could win us another 2x speedup. (If we write our FFT to avoid
223: cache-miss (see #1 above) it might be logical to write the `base case'
224: in assembly.)
225:
226: 5. Avoid multiplication by 1, i, -1, -i. Similarly, optimize
227: multiplication by (+-\/2 +- i\/2).
228:
229: 6. Put as many bits as possible in each double (but don't waste time if
230: that doesn't make the transform size become smaller).
231:
232: 7. For n > some large number, we will get accuracy problems because of the
233: limited precision of our floating point arithmetic. This can easily be
234: solved by using the Karatsuba trick a few times until our operands
235: become small enough.
236:
237: 8. Precompute the roots-of-unity and store them in a vector.
238: }
239:
240: * When a division result is going to be just one limb, (i.e. nsize-dsize is
241: small) normalization could be done in the division loop.
242:
243: * Never allocate temporary space for a source param that overlaps with a
244: destination param needing reallocation. Instead malloc a new block for
245: the destination (and free the source before returning to the caller).
246:
247: * Parallel addition. Since each processors have to tell it is ready to the
248: next processor, we can use simplified synchronization, and actually write
249: it in C: For each processor (apart from the least significant):
250:
251: while (*svar != my_number)
252: ;
253: *svar = my_number + 1;
254:
255: The least significant processor does this:
256:
257: *svar = my_number + 1; /* i.e., *svar = 1 */
258:
259: Before starting the addition, one processor has to store 0 in *svar.
260:
261: Other things to think about for parallel addition: To avoid false
262: (cache-line) sharing, allocate blocks on cache-line boundaries.
263:
264:
265: Local Variables:
266: mode: text
267: fill-column: 77
268: fill-prefix: " "
269: version-control: never
270: End:
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