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535 lines
16 KiB
Java
535 lines
16 KiB
Java
/*
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* Copyright 2009 Google Inc.
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*
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* Licensed under the Apache License, Version 2.0 (the "License"); you may not
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* use this file except in compliance with the License. You may obtain a copy of
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* the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
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* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the
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* License for the specific language governing permissions and limitations under
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* the License.
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*/
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/*
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* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with this
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* work for additional information regarding copyright ownership. The ASF
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* licenses this file to You under the Apache License, Version 2.0 (the
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* "License"); you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
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* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the
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* License for the specific language governing permissions and limitations under
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* the License.
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*
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* INCLUDES MODIFICATIONS BY RICHARD ZSCHECH AS WELL AS GOOGLE.
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*/
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package java.math;
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/**
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* Static library that provides all multiplication of {@link BigInteger}
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* methods.
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*/
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class Multiplication {
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/**
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* An array with the first powers of five in {@code BigInteger} version. (
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* {@code 5^0,5^1,...,5^31})
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*/
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static final BigInteger bigFivePows[] = new BigInteger[32];
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/**
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* An array with the first powers of ten in {@code BigInteger} version. (
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* {@code 10^0,10^1,...,10^31})
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*/
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static final BigInteger[] bigTenPows = new BigInteger[32];
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/**
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* An array with powers of five that fit in the type {@code int}. ({@code
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* 5^0,5^1,...,5^13})
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*/
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static final int fivePows[] = {
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1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625,
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48828125, 244140625, 1220703125};
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/**
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* An array with powers of ten that fit in the type {@code int}. ({@code
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* 10^0,10^1,...,10^9})
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*/
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static final int tenPows[] = {
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1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000};
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/**
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* Break point in digits (number of {@code int} elements) between Karatsuba
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* and Pencil and Paper multiply.
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*/
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static final int whenUseKaratsuba = 63; // an heuristic value
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static {
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int i;
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long fivePow = 1L;
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for (i = 0; i <= 18; i++) {
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bigFivePows[i] = BigInteger.valueOf(fivePow);
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bigTenPows[i] = BigInteger.valueOf(fivePow << i);
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fivePow *= 5;
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}
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for (; i < bigTenPows.length; i++) {
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bigFivePows[i] = bigFivePows[i - 1].multiply(bigFivePows[1]);
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bigTenPows[i] = bigTenPows[i - 1].multiply(BigInteger.TEN);
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}
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}
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/**
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* Performs the multiplication with the Karatsuba's algorithm. <b>Karatsuba's
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* algorithm:</b> <tt>
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* u = u<sub>1</sub> * B + u<sub>0</sub><br>
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* v = v<sub>1</sub> * B + v<sub>0</sub><br>
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*
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*
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* u*v = (u<sub>1</sub> * v<sub>1</sub>) * B<sub>2</sub> + ((u<sub>1</sub> - u<sub>0</sub>) * (v<sub>0</sub> - v<sub>1</sub>) + u<sub>1</sub> * v<sub>1</sub> +
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* u<sub>0</sub> * v<sub>0</sub> ) * B + u<sub>0</sub> * v<sub>0</sub><br>
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*</tt>
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*
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* @param op1 first factor of the product
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* @param op2 second factor of the product
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* @return {@code op1 * op2}
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* @see #multiply(BigInteger, BigInteger)
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*/
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static BigInteger karatsuba(BigInteger op1, BigInteger op2) {
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BigInteger temp;
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if (op2.numberLength > op1.numberLength) {
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temp = op1;
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op1 = op2;
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op2 = temp;
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}
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if (op2.numberLength < whenUseKaratsuba) {
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return multiplyPAP(op1, op2);
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}
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/*
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* Karatsuba: u = u1*B + u0 v = v1*B + v0 u*v = (u1*v1)*B^2 +
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* ((u1-u0)*(v0-v1) + u1*v1 + u0*v0)*B + u0*v0
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*/
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// ndiv2 = (op1.numberLength / 2) * 32
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int ndiv2 = (op1.numberLength & 0xFFFFFFFE) << 4;
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BigInteger upperOp1 = op1.shiftRight(ndiv2);
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BigInteger upperOp2 = op2.shiftRight(ndiv2);
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BigInteger lowerOp1 = op1.subtract(upperOp1.shiftLeft(ndiv2));
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BigInteger lowerOp2 = op2.subtract(upperOp2.shiftLeft(ndiv2));
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BigInteger upper = karatsuba(upperOp1, upperOp2);
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BigInteger lower = karatsuba(lowerOp1, lowerOp2);
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BigInteger middle = karatsuba(upperOp1.subtract(lowerOp1),
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lowerOp2.subtract(upperOp2));
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middle = middle.add(upper).add(lower);
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middle = middle.shiftLeft(ndiv2);
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upper = upper.shiftLeft(ndiv2 << 1);
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return upper.add(middle).add(lower);
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}
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static void multArraysPAP(int[] aDigits, int aLen, int[] bDigits, int bLen,
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int[] resDigits) {
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if (aLen == 0 || bLen == 0) {
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return;
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}
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if (aLen == 1) {
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resDigits[bLen] = multiplyByInt(resDigits, bDigits, bLen, aDigits[0]);
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} else if (bLen == 1) {
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resDigits[aLen] = multiplyByInt(resDigits, aDigits, aLen, bDigits[0]);
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} else {
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multPAP(aDigits, bDigits, resDigits, aLen, bLen);
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}
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}
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/**
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* Performs a multiplication of two BigInteger and hides the algorithm used.
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*
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* @see BigInteger#multiply(BigInteger)
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*/
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static BigInteger multiply(BigInteger x, BigInteger y) {
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return karatsuba(x, y);
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}
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/**
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* Multiplies a number by a power of five. This method is used in {@code
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* BigDecimal} class.
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*
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* @param val the number to be multiplied
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* @param exp a positive {@code int} exponent
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* @return {@code val * 5<sup>exp</sup>}
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*/
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static BigInteger multiplyByFivePow(BigInteger val, int exp) {
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// PRE: exp >= 0
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if (exp < fivePows.length) {
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return multiplyByPositiveInt(val, fivePows[exp]);
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} else if (exp < bigFivePows.length) {
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return val.multiply(bigFivePows[exp]);
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} else {
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// Large powers of five
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return val.multiply(bigFivePows[1].pow(exp));
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}
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}
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/**
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* Multiplies an array of integers by an integer value.
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*
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* @param a the array of integers
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* @param aSize the number of elements of intArray to be multiplied
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* @param factor the multiplier
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* @return the top digit of production
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*/
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static int multiplyByInt(int a[], final int aSize, final int factor) {
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return multiplyByInt(a, a, aSize, factor);
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}
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/**
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* Multiplies a number by a positive integer.
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*
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* @param val an arbitrary {@code BigInteger}
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* @param factor a positive {@code int} number
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* @return {@code val * factor}
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*/
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static BigInteger multiplyByPositiveInt(BigInteger val, int factor) {
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int resSign = val.sign;
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if (resSign == 0) {
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return BigInteger.ZERO;
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}
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int aNumberLength = val.numberLength;
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int[] aDigits = val.digits;
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if (aNumberLength == 1) {
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long res = unsignedMultAddAdd(aDigits[0], factor, 0, 0);
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int resLo = (int) res;
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int resHi = (int) (res >>> 32);
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return ((resHi == 0) ? new BigInteger(resSign, resLo) : new BigInteger(
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resSign, 2, new int[] {resLo, resHi}));
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}
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// Common case
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int resLength = aNumberLength + 1;
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int resDigits[] = new int[resLength];
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resDigits[aNumberLength] = multiplyByInt(resDigits, aDigits, aNumberLength,
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factor);
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BigInteger result = new BigInteger(resSign, resLength, resDigits);
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result.cutOffLeadingZeroes();
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return result;
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}
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/**
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* Multiplies a number by a power of ten. This method is used in {@code
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* BigDecimal} class.
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*
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* @param val the number to be multiplied
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* @param exp a positive {@code long} exponent
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* @return {@code val * 10<sup>exp</sup>}
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*/
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static BigInteger multiplyByTenPow(BigInteger val, int exp) {
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// PRE: exp >= 0
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return ((exp < tenPows.length) ? multiplyByPositiveInt(val,
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tenPows[(int) exp]) : val.multiply(powerOf10(exp)));
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}
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/**
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* Multiplies two BigIntegers. Implements traditional scholar algorithm
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* described by Knuth.
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*
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* <br>
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* <tt>
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* <table border="0">
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* <tbody>
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*
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*
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* <tr>
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* <td align="center">A=</td>
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* <td>a<sub>3</sub></td>
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* <td>a<sub>2</sub></td>
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* <td>a<sub>1</sub></td>
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* <td>a<sub>0</sub></td>
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* <td></td>
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* <td></td>
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* </tr>
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*
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* <tr>
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* <td align="center">B=</td>
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* <td></td>
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* <td>b<sub>2</sub></td>
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* <td>b<sub>1</sub></td>
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* <td>b<sub>1</sub></td>
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* <td></td>
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* <td></td>
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* </tr>
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*
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* <tr>
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* <td></td>
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* <td></td>
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* <td></td>
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* <td>b<sub>0</sub>*a<sub>3</sub></td>
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* <td>b<sub>0</sub>*a<sub>2</sub></td>
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* <td>b<sub>0</sub>*a<sub>1</sub></td>
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* <td>b<sub>0</sub>*a<sub>0</sub></td>
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* </tr>
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*
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* <tr>
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* <td></td>
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* <td></td>
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* <td>b<sub>1</sub>*a<sub>3</sub></td>
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* <td>b<sub>1</sub>*a<sub>2</sub></td>
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* <td>b<sub>1</sub>*a1</td>
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* <td>b<sub>1</sub>*a0</td>
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* </tr>
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*
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* <tr>
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* <td>+</td>
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* <td>b<sub>2</sub>*a<sub>3</sub></td>
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* <td>b<sub>2</sub>*a<sub>2</sub></td>
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* <td>b<sub>2</sub>*a<sub>1</sub></td>
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* <td>b<sub>2</sub>*a<sub>0</sub></td>
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* </tr>
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*
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* <tr>
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* <td></td>
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* <td>______</td>
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* <td>______</td>
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* <td>______</td>
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* <td>______</td>
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* <td>______</td>
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* <td>______</td>
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* </tr>
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*
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* <tr>
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*
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* <td align="center">A*B=R=</td>
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* <td align="center">r<sub>5</sub></td>
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* <td align="center">r<sub>4</sub></td>
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* <td align="center">r<sub>3</sub></td>
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* <td align="center">r<sub>2</sub></td>
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* <td align="center">r<sub>1</sub></td>
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* <td align="center">r<sub>0</sub></td>
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* <td></td>
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* </tr>
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*
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* </tbody>
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* </table>
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*
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*</tt>
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*
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* @param op1 first factor of the multiplication {@code op1 >= 0}
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* @param op2 second factor of the multiplication {@code op2 >= 0}
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* @return a {@code BigInteger} of value {@code op1 * op2}
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*/
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static BigInteger multiplyPAP(BigInteger a, BigInteger b) {
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// PRE: a >= b
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int aLen = a.numberLength;
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int bLen = b.numberLength;
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int resLength = aLen + bLen;
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int resSign = (a.sign != b.sign) ? -1 : 1;
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// A special case when both numbers don't exceed int
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if (resLength == 2) {
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long val = unsignedMultAddAdd(a.digits[0], b.digits[0], 0, 0);
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int valueLo = (int) val;
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int valueHi = (int) (val >>> 32);
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return ((valueHi == 0) ? new BigInteger(resSign, valueLo)
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: new BigInteger(resSign, 2, new int[] {valueLo, valueHi}));
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}
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int[] aDigits = a.digits;
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int[] bDigits = b.digits;
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int resDigits[] = new int[resLength];
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// Common case
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multArraysPAP(aDigits, aLen, bDigits, bLen, resDigits);
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BigInteger result = new BigInteger(resSign, resLength, resDigits);
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result.cutOffLeadingZeroes();
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return result;
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}
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static void multPAP(int a[], int b[], int t[], int aLen, int bLen) {
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if (a == b && aLen == bLen) {
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square(a, aLen, t);
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return;
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}
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for (int i = 0; i < aLen; i++) {
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long carry = 0;
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int aI = a[i];
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for (int j = 0; j < bLen; j++) {
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carry = unsignedMultAddAdd(aI, b[j], t[i + j], (int) carry);
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t[i + j] = (int) carry;
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carry >>>= 32;
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}
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t[i + bLen] = (int) carry;
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}
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}
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static BigInteger pow(BigInteger base, int exponent) {
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// PRE: exp > 0
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BigInteger res = BigInteger.ONE;
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BigInteger acc = base;
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for (; exponent > 1; exponent >>= 1) {
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if ((exponent & 1) != 0) {
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// if odd, multiply one more time by acc
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res = res.multiply(acc);
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}
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// acc = base^(2^i)
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// a limit where karatsuba performs a faster square than the square
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// algorithm
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if (acc.numberLength == 1) {
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acc = acc.multiply(acc); // square
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} else {
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acc = new BigInteger(1, square(acc.digits, acc.numberLength,
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new int[acc.numberLength << 1]));
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}
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}
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// exponent == 1, multiply one more time
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res = res.multiply(acc);
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return res;
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}
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/**
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* It calculates a power of ten, which exponent could be out of 32-bit range.
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* Note that internally this method will be used in the worst case with an
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* exponent equals to: {@code Integer.MAX_VALUE - Integer.MIN_VALUE}.
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*
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* @param exp the exponent of power of ten, it must be positive.
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* @return a {@code BigInteger} with value {@code 10<sup>exp</sup>}.
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*/
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static BigInteger powerOf10(double exp) {
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// PRE: exp >= 0
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int intExp = (int) exp;
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// "SMALL POWERS"
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if (exp < bigTenPows.length) {
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// The largest power that fit in 'long' type
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return bigTenPows[intExp];
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} else if (exp <= 50) {
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// To calculate: 10^exp
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return BigInteger.TEN.pow(intExp);
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} else if (exp <= 1000) {
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// To calculate: 5^exp * 2^exp
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return bigFivePows[1].pow(intExp).shiftLeft(intExp);
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}
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// "LARGE POWERS"
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/*
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* To check if there is free memory to allocate a BigInteger of the
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* estimated size, measured in bytes: 1 + [exp / log10(2)]
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*/
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if (exp > 1000000) {
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throw new ArithmeticException("power of ten too big"); //$NON-NLS-1$
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}
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if (exp <= Integer.MAX_VALUE) {
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// To calculate: 5^exp * 2^exp
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return bigFivePows[1].pow(intExp).shiftLeft(intExp);
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}
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/*
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* "HUGE POWERS"
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*
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* This branch probably won't be executed since the power of ten is too big.
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*/
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// To calculate: 5^exp
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BigInteger powerOfFive = bigFivePows[1].pow(Integer.MAX_VALUE);
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BigInteger res = powerOfFive;
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long longExp = (long) (exp - Integer.MAX_VALUE);
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intExp = (int) (exp % Integer.MAX_VALUE);
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while (longExp > Integer.MAX_VALUE) {
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res = res.multiply(powerOfFive);
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longExp -= Integer.MAX_VALUE;
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}
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res = res.multiply(bigFivePows[1].pow(intExp));
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// To calculate: 5^exp << exp
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res = res.shiftLeft(Integer.MAX_VALUE);
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longExp = (long) (exp - Integer.MAX_VALUE);
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while (longExp > Integer.MAX_VALUE) {
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res = res.shiftLeft(Integer.MAX_VALUE);
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longExp -= Integer.MAX_VALUE;
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}
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res = res.shiftLeft(intExp);
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return res;
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}
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/**
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* Performs a<sup>2</sup>.
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*
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* @param a The number to square.
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* @param aLen The length of the number to square.
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*/
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static int[] square(int[] a, int aLen, int[] res) {
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long carry;
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for (int i = 0; i < aLen; i++) {
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carry = 0;
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for (int j = i + 1; j < aLen; j++) {
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carry = unsignedMultAddAdd(a[i], a[j], res[i + j], (int) carry);
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res[i + j] = (int) carry;
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carry >>>= 32;
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|
}
|
|
res[i + aLen] = (int) carry;
|
|
}
|
|
|
|
BitLevel.shiftLeftOneBit(res, res, aLen << 1);
|
|
|
|
carry = 0;
|
|
for (int i = 0, index = 0; i < aLen; i++, index++) {
|
|
carry = unsignedMultAddAdd(a[i], a[i], res[index], (int) carry);
|
|
res[index] = (int) carry;
|
|
carry >>>= 32;
|
|
index++;
|
|
carry += res[index] & 0xFFFFFFFFL;
|
|
res[index] = (int) carry;
|
|
carry >>>= 32;
|
|
}
|
|
return res;
|
|
}
|
|
|
|
/**
|
|
* Computes the value unsigned ((uint)a*(uint)b + (uint)c + (uint)d). This
|
|
* method could improve the readability and performance of the code.
|
|
*
|
|
* @param a parameter 1
|
|
* @param b parameter 2
|
|
* @param c parameter 3
|
|
* @param d parameter 4
|
|
* @return value of expression
|
|
*/
|
|
static long unsignedMultAddAdd(int a, int b, int c, int d) {
|
|
return (a & 0xFFFFFFFFL) * (b & 0xFFFFFFFFL) + (c & 0xFFFFFFFFL)
|
|
+ (d & 0xFFFFFFFFL);
|
|
}
|
|
|
|
/**
|
|
* Multiplies an array of integers by an integer value and saves the result in
|
|
* {@code res}.
|
|
*
|
|
* @param a the array of integers
|
|
* @param aSize the number of elements of intArray to be multiplied
|
|
* @param factor the multiplier
|
|
* @return the top digit of production
|
|
*/
|
|
private static int multiplyByInt(int res[], int a[], final int aSize,
|
|
final int factor) {
|
|
long carry = 0;
|
|
for (int i = 0; i < aSize; i++) {
|
|
carry = unsignedMultAddAdd(a[i], factor, (int) carry, 0);
|
|
res[i] = (int) carry;
|
|
carry >>>= 32;
|
|
}
|
|
return (int) carry;
|
|
}
|
|
|
|
/**
|
|
* Just to denote that this class can't be instantiated.
|
|
*/
|
|
private Multiplication() {
|
|
}
|
|
|
|
}
|