Java tutorial
/* GeoGebra - Dynamic Mathematics for Everyone http://www.geogebra.org This file is part of GeoGebra. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. */ package geogebra.common.kernel.cas; import geogebra.common.euclidian.DrawEquation; import geogebra.common.kernel.Construction; import geogebra.common.kernel.Kernel; import geogebra.common.kernel.StringTemplate; import geogebra.common.kernel.algos.AlgoElement; import geogebra.common.kernel.arithmetic.ExpressionNode; import geogebra.common.kernel.arithmetic.ValidExpression; import geogebra.common.kernel.commands.Commands; import geogebra.common.kernel.geos.GeoElement; import geogebra.common.kernel.geos.GeoList; import geogebra.common.kernel.geos.GeoNumberValue; import geogebra.common.kernel.geos.GeoText; import geogebra.common.main.App; import geogebra.common.util.MyMathExact.MyDecimal; import geogebra.common.util.MyMathExact.MyDecimalMatrix; import geogebra.common.util.Unicode; import geogebra.common.util.debug.Log; import java.math.BigDecimal; import java.util.ArrayList; import org.apache.commons.math.util.MathUtils; /** * @author lightest * */ public class AlgoSurdText extends AlgoElement implements UsesCAS { // private DfpField decFull = new DfpField(64); // DfpField decLess = new DfpField(16); int fullScale = 64; int lessScale = 16; private GeoNumberValue num; // input private GeoList list; // input private GeoText text; // output private StringBuilder sb = new StringBuilder(); // double debug0, debug1, debug2; /** * @param cons * construction * @param label * label for output * @param num * number * @param list * list of hints */ public AlgoSurdText(Construction cons, String label, GeoNumberValue num, GeoList list) { this(cons, num, list); text.setLabel(label); } /** * @param cons * construction * @param num * number * @param list * list of hints */ AlgoSurdText(Construction cons, GeoNumberValue num, GeoList list) { super(cons); if (list != null) { cons.addCASAlgo(this); } this.num = num; this.list = list; text = new GeoText(cons); text.setFormulaType(kernel.getApplication().getPreferredFormulaRenderingType()); text.setLaTeX(true, false); text.setIsTextCommand(true); // stop editing as text setInputOutput(); compute(); // debug // IntRelationFinder irf = new IntRelationFinder(3, new // double[]{1.41421356,1,0.70710678}, decFull, decLess); } /** * @param cons * construction */ public AlgoSurdText(Construction cons) { super(cons); // not needed, only called from SurdTextPoint // cons.addCASAlgo(this); } @Override public Commands getClassName() { return Commands.SurdText; } @Override protected void setInputOutput() { input = new GeoElement[list == null ? 1 : 2]; input[0] = num.toGeoElement(); if (list != null) { input[1] = list; } setOutputLength(1); setOutput(0, text); setDependencies(); // done by AlgoElement } /** * Returns resulting text * * @return resulting text */ public GeoText getResult() { return text; } @Override public void compute() { // make sure answer is formatted as eg \sqrt not sqrt StringTemplate tpl = text.getStringTemplate(); if (input[0].isDefined()) { sb.setLength(0); /* * int[] primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, * 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, * * 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, * 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, * * 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, * 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, * * 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, * 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, * * 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, * 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, * * 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, * 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, * * 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997}; * * * debug0 = Double.MAX_VALUE; debug1 = Double.MAX_VALUE; debug2 = * Double.MAX_VALUE; double debug0max = Double.MAX_VALUE; double * debug1max = Double.MAX_VALUE; double debug2max = * Double.MAX_VALUE; * * * App.debug("START"); for (int p = 0 ; p < primes.length / 2; p++) * { for (int q = 1 ; q < 100 ; q++) { // up to 400 * * double num = q + Math.sqrt(primes[p]); * * sb.setLength(0); PSLQappendQuadratic(sb, num, tpl); * * * * if (!sb.toString().equals(q+"+\\sqrt{"+primes[p]+"}")) { * //App.debug * ("error:"+sb.toString()+" "+q+"+\\sqrt{"+primes[p]+"}"); * * if (Math.abs(debug0) > Math.abs(debug1) && Math.abs(debug0) > * Math.abs(debug2)) { if (Math.abs(debug0) < debug0max) { debug0max * = Math.abs(debug0); } } else if (Math.abs(debug1) > * Math.abs(debug2) && Math.abs(debug1) > Math.abs(debug0)) { if * (Math.abs(debug1) < debug1max) { debug1max = Math.abs(debug1); } * } else if (Math.abs(debug2) > Math.abs(debug1) && * Math.abs(debug2) > Math.abs(debug0)) { if (Math.abs(debug2) < * debug2max) { debug2max = Math.abs(debug2); } } } * * * for (int r = 2 ; r < 100 ; r++) { num = (q + * Math.sqrt(primes[p]))/r; * * sb.setLength(0); PSLQappendQuadratic(sb, num, tpl); * * if (sb.toString().indexOf("\\frac") > -1 && * !sb.toString().equals( * "\\frac{"+q+"+\\sqrt{"+primes[p]+"}}{"+r+"}")) { * //App.debug("error:" * +sb.toString()+" \\frac{"+q+"+\\sqrt{"+primes * [p]+"}}{"+r+"} "+sbDebug.toString()); * * * if (Math.abs(debug0) > Math.abs(debug1) && Math.abs(debug0) > * Math.abs(debug2)) { if (Math.abs(debug0) < debug0max) { debug0max * = Math.abs(debug0); } } else if (Math.abs(debug1) > * Math.abs(debug2) && Math.abs(debug1) > Math.abs(debug0)) { if * (Math.abs(debug1) < debug1max) { debug1max = Math.abs(debug1); } * } else if (Math.abs(debug2) > Math.abs(debug1) && * Math.abs(debug2) > Math.abs(debug0)) { if (Math.abs(debug2) < * debug2max) { debug2max = Math.abs(debug2); } } * * * } * * } * * } } App.debug("END "+debug0max+" "+debug1max+" "+debug2max); */ double decimal = num.getDouble(); if (Kernel.isEqual(decimal - Math.round(decimal), 0.0, Kernel.MAX_PRECISION)) { sb.append(kernel.format(Math.round(decimal), tpl)); } else { if (list == null) { PSLQappendQuadratic(sb, decimal, tpl); } else { PSLQappendGeneral(sb, decimal, tpl); } } text.setTextString(sb.toString()); text.setLaTeX(true, false); } else { // if the number is undefined, display a "?" text.setTextString("?"); text.setLaTeX(true, false); // old behaviour // text.setUndefined(); } } private void Fractionappend(StringBuilder sb, int numer, int denom, StringTemplate tpl) { if (denom < 0) { denom = -denom; numer = -numer; } // maybe we can simplify int gcdiv = (int) Kernel.gcd(Math.abs(numer), denom); if (gcdiv != 1) { denom = denom / gcdiv; numer = numer / gcdiv; } if (denom == 1) { // integer DrawEquation.appendNumber(sb, tpl, kernel.format(numer, tpl)); } else if (denom == 0) { // 1 / 0 or -1 / 0 if (numer < 0) { DrawEquation.appendMinusInfinity(sb, tpl); } else { DrawEquation.appendInfinity(sb, tpl); } } else { DrawEquation.appendFormulaStart(sb, tpl); boolean negative = numer < 0; if (negative) { numer = -numer; DrawEquation.appendNegation(sb, tpl); } DrawEquation.appendFractionStart(sb, tpl); sb.append(kernel.format(numer, tpl)); DrawEquation.appendFractionMiddle(sb, tpl); sb.append(kernel.format(denom, tpl)); DrawEquation.appendFractionEnd(sb, tpl); } } /** * Goal: modifies a StringBuilder object sb to be a radical up to quartic * roots The precision is adapted, according to setting * * @param sb * @param num * @param tpl */ protected void PSLQappendGeneral(StringBuilder sb, double num, StringTemplate tpl) { // Zero Test: Is num 0? if (Kernel.isZero(num)) { sb.append(kernel.format(0, tpl)); return; } // Rational Number Test. num is not 0. Is num rational (with small // denominator <= 1000) ? AlgebraicFit fitter = new AlgebraicFit(null, null, AlgebraicFittingType.RATIONAL_NUMBER, tpl); fitter.setCoeffBound(1000); fitter.compute(num); ValidExpression ve = sbToCAS(fitter.formalSolution); if (fitter.formalSolution.length() > 0 && Kernel.isEqual(ve.evaluateDouble(), num)) { sb.append(kernel.getGeoGebraCAS().evaluateGeoGebraCAS(ve, null, tpl, kernel)); return; } double[] testValues; String[] testNames; if (list != null) { ArrayList<Double> values = new ArrayList<Double>(); ArrayList<String> names = new ArrayList<String>(); for (int i = 0; i < list.size(); i++) { double x = list.get(i).evaluateDouble(); if (Kernel.isEqual(x, Math.PI)) { values.add(Math.PI); names.add("pi"); } else if (Kernel.isEqual(x, 1 / Math.PI)) { values.add(1 / Math.PI); names.add("1/pi"); } else if (Kernel.isEqual(x, Math.PI * Math.PI)) { values.add(Math.PI * Math.PI); names.add("pi^2"); } else if (Kernel.isEqual(x, Math.sqrt(Math.PI))) { values.add(Math.sqrt(Math.PI)); names.add("sqrt(pi)"); } else if (Kernel.isEqual(x, Math.E)) { values.add(Math.E); names.add(Unicode.EULER_STRING); } else if (Kernel.isEqual(x, 1 / Math.E)) { values.add(1 / Math.E); names.add("1/" + Unicode.EULER_STRING); } else if (Kernel.isEqual(x, Math.E * Math.E)) { values.add(Math.E * Math.PI); names.add(Unicode.EULER_STRING + "^2"); } else if (Kernel.isEqual(x, Math.sqrt(Math.E))) { values.add(Math.sqrt(Math.E)); names.add("sqrt(" + Unicode.EULER_STRING + ")"); } else { int j; for (j = 2; j < 100; j++) { double sqrt = Math.sqrt(j); if (!Kernel.isInteger(sqrt) && Kernel.isEqual(x, sqrt)) { values.add(sqrt); names.add("sqrt(" + j + ")"); break; } double ln = Math.log(j); if (Kernel.isEqual(x, ln)) { values.add(ln); names.add("ln(" + j + ")"); break; } } } } testValues = new double[values.size()]; testNames = new String[values.size()]; for (int i = 0; i < values.size(); i++) { testValues[i] = values.get(i); testNames[i] = names.get(i); // App.debug(testNames[i]); } } else { // default constants if none supplied testValues = new double[] { Math.sqrt(2.0), Math.sqrt(3.0), Math.sqrt(5.0), Math.sqrt(6.0), Math.sqrt(7.0), Math.sqrt(10.0), Math.PI }; testNames = new String[] { "sqrt(2)", "sqrt(3)", "sqrt(5)", "sqrt(6)", "sqrt(7)", "sqrt(10)", "pi" }; } boolean success = fitLinearComb(num, testNames, testValues, 100, sb, tpl); if (success) { return; } sb.append(kernel.format(num, StringTemplate.maxPrecision)); } private boolean fitLinearComb(double y, String[] constNameSet, double[] constValueSet, int coeffBound, StringBuilder sb1, StringTemplate tpl) { // long t1= System.currentTimeMillis(); // long t2; AlgebraicFit fitter0 = new AlgebraicFit(constNameSet, constValueSet, AlgebraicFittingType.LINEAR_COMBINATION, tpl); fitter0.setCoeffBound(coeffBound); fitter0.compute(y); // t2 = System.currentTimeMillis(); // System.out.println("time of algebraic fit compute: " + (t2-t1)); // t1 = t2; ValidExpression ve0 = sbToCAS(fitter0.formalSolution); // t2 = System.currentTimeMillis(); // System.out.println("time of sb to ve: " + (t2-t1)); // t1 = t2; if (fitter0.formalSolution.length() > 0 && Kernel.isEqual(ve0.evaluateDouble(), y)) { sb1.append(kernel.getGeoGebraCAS().evaluateGeoGebraCAS(ve0, null, tpl, kernel)); // t2 = System.currentTimeMillis(); // System.out.println("time of ve to CAS: " + (t2-t1)); // t1 = t2; return true; } return false; } private ValidExpression sbToCAS(StringBuilder sb) { if (sb != null) { return kernel.getGeoGebraCAS().getCASparser().parseGeoGebraCASInputAndResolveDummyVars(sb.toString(), getKernel()); } return null; } // returns the sum of constValue[j] * coeffs[offset+j*step] over j static double evaluateCombination(int n, double[] constValue, int[] coeffs, int offset, int step) { double sum = 0; for (int j = 0; j < n; j++) { sum += constValue[j] * coeffs[offset + j * step]; } return sum; } // append a linear combination coeffs[offset + j*step] * vars[j] to the // StringBuilder sb /** * @param sbToCAS * @param numOfTerms * @param vars * @param coeffs * @param offset * @param step * @param tpl */ public void appendCombination(StringBuilder sbToCAS, int numOfTerms, String[] vars, int[] coeffs, int offset, int step, StringTemplate tpl) { int numOfAllTerms = vars.length; if (numOfAllTerms - 1 > Math.floor((coeffs.length - 1 - step - offset) / step)) { // checksum // appendUndefined(); return; } if (numOfTerms == 0) { return; } int counter = numOfTerms - 1; // number of pluses for (int j = 0; j < numOfAllTerms; j++) { // if (coeffs[offset+j*step]==0) { // continue; // } else if (coeffs[offset+j*step]!=1) { sbToCAS.append(kernel.format(coeffs[offset + j * step], tpl)); sbToCAS.append("*"); // } sbToCAS.append(vars[j]); if (counter > 0) { sbToCAS.append(" + "); counter--; } } } private void appendUndefined(StringBuilder sb1, StringTemplate tpl, double num1) { // sb1.append("\\text{"); // sb1.append(app.getPlain("Undefined")); // sb1.append("}"); // eg SurdText[1.23456789012345] returns 1.23456789012345 DrawEquation.appendNumber(sb1, tpl, kernel.format(num1, StringTemplate.maxPrecision)); } /** * Goal: modifies a StringBuilder object sb to be a radical up to quartic * roots The precision is adapted, according to setting * * @param sb * @param num1 * @param tpl */ protected void PSLQappendQuartic(StringBuilder sb, double num1, StringTemplate tpl) { double[] numPowers = new double[5]; double temp = 1.0; for (int i = 4; i >= 0; i--) { numPowers[i] = temp; temp *= num1; } getKernel(); int[] coeffs = PSLQ(numPowers, Kernel.STANDARD_PRECISION, 10); if (coeffs[0] == 0 && coeffs[1] == 0) { if (coeffs[2] == 0 && coeffs[3] == 0 && coeffs[4] == 0) { appendUndefined(sb, tpl, num1); } else if (coeffs[2] == 0) { // coeffs[1]: denominator; coeffs[2]: numerator int denom = coeffs[3]; int numer = -coeffs[4]; Fractionappend(sb, numer, denom, tpl); } else { // coeffs, if found, shows the equation // coeffs[2]+coeffs[1]x+coeffs[0]x^2=0" // We want x=\frac{a +/- b1\sqrt{b2}}{c} // where c=coeffs[0], a=-coeffs[1], b=coeffs[1]^2 - // 4*coeffs[0]*coeffs[2] int a = -coeffs[3]; int b2 = coeffs[3] * coeffs[3] - 4 * coeffs[2] * coeffs[4]; int b1 = 1; int c = 2 * coeffs[2]; if (b2 <= 0) { // should not happen! appendUndefined(sb, tpl, num1); return; } // free the squares of b2 while (b2 % 4 == 0) { b2 = b2 / 4; b1 = b1 * 2; } for (int s = 3; s <= Math.sqrt(b2); s += 2) while (b2 % (s * s) == 0) { b2 = b2 / (s * s); b1 = b1 * s; } if (c < 0) { a = -a; c = -c; } boolean positive; if (num1 > (a + 0.0) / c) { positive = true; if (b2 == 1) { a += b1; b1 = 0; b2 = 0; } } else { positive = false; if (b2 == 1) { a -= b1; b1 = 0; b2 = 0; } } int gcd = MathUtils.gcd(MathUtils.gcd(a, b1), c); if (gcd != 1) { a = a / gcd; b1 = b1 / gcd; c = c / gcd; } ExpressionNode en; if (Kernel.isZero(b1)) { // eg SurdText[0.33] // eg SurdText[0.235] en = new ExpressionNode(kernel, a); } else { en = (new ExpressionNode(kernel, b2)).sqrt().multiplyR(b1); if (positive) { en = en.plusR(a); } else { en = en.subtractR(a); } } en = en.divide(c); sb.append(en.toString(tpl)); } } else if (coeffs[0] == 0) { sb.append("Root of a cubic equation: "); sb.append(kernel.format(coeffs[1], tpl)); sb.append("x^3 + "); sb.append(kernel.format(coeffs[2], tpl)); sb.append("x^2 + "); sb.append(kernel.format(coeffs[3], tpl)); sb.append("x + "); sb.append(kernel.format(coeffs[4], tpl)); // TODO: what to do in MathML? App.debug(sb.toString()); } else { sb.append("Root of a quartic equation: "); sb.append(kernel.format(coeffs[0], tpl)); sb.append("x^4 + "); sb.append(kernel.format(coeffs[1], tpl)); sb.append("x^3 + "); sb.append(kernel.format(coeffs[2], tpl)); sb.append("x^2 + "); sb.append(kernel.format(coeffs[3], tpl)); sb.append("x + "); sb.append(kernel.format(coeffs[4], tpl)); // TODO: what to do in MathML? App.debug(sb.toString()); } } /** * Quadratic Case. modifies a StringBuilder object sb to be the * quadratic-radical expression of num, within certain precision. * * @param sb * @param num1 * @param tpl */ protected void PSLQappendQuadratic(StringBuilder sb, double num1, StringTemplate tpl) { if (Kernel.isZero(num1)) { DrawEquation.appendNumber(sb, tpl, "0"); return; } double[] numPowers = { num1 * num1, num1, 1.0 }; int[] coeffs = PSLQ(numPowers, 1E-10, 10); if (coeffs == null) { appendUndefined(sb, tpl, num1); return; } // debug0 = coeffs[0]; // debug1 = coeffs[1]; // debug2 = coeffs[2]; // App.debug(coeffs[0]+" "+coeffs[1]+" "+coeffs[2]); if ((coeffs[0] == 0 && coeffs[1] == 0 && coeffs[2] == 0) // try to minimize possibility of wrong answer // and maximize usefulness // numbers determined by commented-out code in compute() method || Math.abs(coeffs[0]) > 570 || Math.abs(coeffs[1]) > 729 || Math.abs(coeffs[2]) > 465) { // App.debug(coeffs[0]+" "+coeffs[1]+" "+coeffs[2]); appendUndefined(sb, tpl, num1); } else if (coeffs[0] == 0) { // coeffs[1]: denominator; coeffs[2]: numerator int denom = coeffs[1]; int numer = -coeffs[2]; Fractionappend(sb, numer, denom, tpl); } else { // coeffs, if found, shows the equation // coeffs[2]+coeffs[1]x+coeffs[0]x^2=0" // We want x=\frac{a +/- b1\sqrt{b2}}{c} // where c=coeffs[0], a=-coeffs[1], b=coeffs[1]^2 - // 4*coeffs[0]*coeffs[2] int a = -coeffs[1]; int b2 = coeffs[1] * coeffs[1] - 4 * coeffs[0] * coeffs[2]; int b1 = 1; int c = 2 * coeffs[0]; if (b2 <= 0) { // should not happen! appendUndefined(sb, tpl, num1); return; } // free the squares of b2 while (b2 % 4 == 0) { b2 = b2 / 4; b1 = b1 * 2; } for (int s = 3; s <= Math.sqrt(b2); s += 2) while (b2 % (s * s) == 0) { b2 = b2 / (s * s); b1 = b1 * s; } if (c < 0) { a = -a; c = -c; } boolean positive; if (num1 > (a + 0.0) / c) { positive = true; if (b2 == 1) { a += b1; b1 = 0; b2 = 0; } } else { positive = false; if (b2 == 1) { a -= b1; b1 = 0; b2 = 0; } } int gcd = MathUtils.gcd(MathUtils.gcd(a, b1), c); if (gcd != 1) { a = a / gcd; b1 = b1 / gcd; c = c / gcd; } ExpressionNode en; if (Kernel.isZero(b1)) { // eg SurdText[0.33] // eg SurdText[0.235] en = new ExpressionNode(kernel, a); } else { en = (new ExpressionNode(kernel, b2)).sqrt().multiplyR(b1); if (positive) { en = en.plusR(a); } else { en = en.subtractR(a); } } en = en.divide(c); sb.append(en.toString(tpl)); } } /** * @param n * @param x * @param AccuracyFactor * @param bound * @return */ int[][] mPSLQ(int n, double[] x, double AccuracyFactor, int bound) { MyDecimalMatrix r2; int rCols = 0; // tracks the number of solutions globally // int[] orthoIndices = new int[n]; // int[][] B1 = new int[n][n]; // int[][] M = new int[n][n]; // int[][] B2 = new int[n][n]; // double[] xB2 = new double[n]; MyDecimalMatrix B_comp; MyDecimalMatrix result2; // now n>=2. int p = n; // length of current x /* * for (int i=0; i<n; i++) { B2[i][i] = 1; } */ // First cycle of the loop has something different to do, so it is // explicitly written here. int oldp = p; int q; IntRelationFinder irf = new IntRelationFinder(oldp, x, fullScale, lessScale, AccuracyFactor, bound); if (irf.result.size() == 0) { return null; } IntRelationFinder.IntRelation m = irf.result.get(0); // the most // significant // resulting // relations // r2 stores all possible results. Numbers are initialized here because // we need the correct field. r2 = new MyDecimalMatrix(m.getBMatrix().getScale(), n, n); result2 = m.getBSolMatrix(); if (result2 != null) q = result2.getColumnDimension(); else q = 0; // store the results to r2 for (int j = 0; j < q; j++) { for (int i = 0; i < n; i++) { r2.setEntry(i, rCols, result2.getEntry(i, j)); } if (rCols == n - 1) Log.warn("There should not be that many solutions."); rCols++; } p = 0; for (int j = 0; j < oldp; j++) { if (m.orthoIndices[j] == 0) { x[p++] = new Double(m.xB1.getEntry(0, j).toString()); } } B_comp = m.getBRestMatrix(); // second and more cycles. If p<oldp means at least one solution has // been found in the last cycle. // also the dimension of x should be at least 2. while (oldp >= 2 && p < oldp && rCols < n - 1) { oldp = p; p = 0; irf = new IntRelationFinder(oldp, x, fullScale, lessScale, AccuracyFactor, bound); if (irf.result.size() == 0) { break; } m = irf.result.get(0); if (m.getBSolMatrix() == null) { break; } result2 = B_comp.multiply(m.getBSolMatrix()); q = result2.getColumnDimension(); // store the resulting columns B_comp.result2 to r2 for (int j = 0; j < q; j++) { // we don't accept all zeros as a relation boolean allZero = true; for (int i = 0; i < n; i++) allZero = allZero && result2.getEntry(i, j).intValue() == 0; if (allZero) break; // we don't accept any entry's absolute value being larger than // bound boolean tooLargeEntry = false; for (int i = 0; i < n; i++) tooLargeEntry = tooLargeEntry || result2.getEntry(i, j).abs().intValue() > bound; if (tooLargeEntry) break; for (int i = 0; i < n; i++) { r2.setEntry(i, rCols, result2.getEntry(i, j)); } if (rCols == n - 1) { Log.warn("There should not be that many solutions."); } rCols++; } // x<-the non-zero elements of new xB p = 0; for (int j = 0; j < oldp; j++) { if (m.orthoIndices[j] == 0) { x[p++] = new Double(m.xB1.getEntry(0, j).toString()); } } // B_comp <- B_comp . the new B_rest (nxq,qxq') if (m.getBRestMatrix() != null) B_comp = B_comp.multiply(m.getBRestMatrix()); } /* * result = new int[n][rCols]; for (int i=0; i<n; i++){ for (int j=0; * j<rCols; j++) { result[i][j]=r[i][j]; } } */ int[][] result = new int[n][rCols]; for (int i = 0; i < n; i++) { for (int j = 0; j < rCols; j++) { result[i][j] = r2.getEntry(i, j).intValue(); } } return result; } private static int[] PSLQ(double[] x, double AccuracyFactor, int bound) { return PSLQ(x.length, x, AccuracyFactor, bound, null, null, null); } /* * Algorithm PSLQ from Ferguson and Bailey (1992) */ private static int[] PSLQ(int n, double[] x_input, double AccuracyFactor, int bound, int[][] B_mutable, double[] xB_mutable, int[] orthoIndices_mutable) { double[] x = new double[n]; for (int i = 0; i < n; i++) { // need a copy of the input x[i] = x_input[i]; } // returning single solution int[] coeffs = new int[n]; // mutable outputs int[] orthoIndices; if (orthoIndices_mutable == null) orthoIndices = new int[n]; else orthoIndices = orthoIndices_mutable; double[] xB; if (xB_mutable == null) xB = new double[n]; else xB = xB_mutable; int[][] B; if (B_mutable == null) B = new int[n][n]; else B = B_mutable; // other working variables double normX; double[] ss; double[][] H, P, newH; int[][] D, E, A, newAorB; double[][][] G; int[][][] R; double gamma, deltaSq; for (int i = 0; i < n; i++) { coeffs[i] = 0; orthoIndices[i] = 0; } if (n <= 1) return coeffs; for (int i = 0; i < n; i++) { if (Double.isNaN(x[i])) return coeffs; } // PSLQ Algorithm (Ferguson et al, 1999) // // Initialize: // Given input array x=(x[0]..x[n-1]), // define partial sum of squares ss[j] = sum of x[k]x[k] from k=j to n-1 // define Hx as n x (n-1) matrix, where H[i][j] = ss[i+1]/ss[i] if // 1<=i=j<=n-1, =-x[i]x[j]/(s[j]s[j+1]) if 1<=j<i<=n, =0 otherwise // define Px as I_n - (xt.x) // [Debug] check the following properties -- (1)Hxt.Hx=I_(n-1), (2) // |Hx|^2=|Px|^2=n-1, (3)x.Hx=0, (4)Pxt=Px, (5)Px=Hx.Hxt // [Important Note] Px.mt = mt for any m such that m.xt = 0 // // // // // set H = Hx, A=B=I_n // perform Hermite reduction on H, producing D // Fix a constant gamma > 2/sqrt(3) // replace x by xD^-1, H by DH, A by DA, B by BD^-1 // // 4-step iteration. Step 1 "Exchange", Step 2 "Corner", Step 3 // "Reduction", Step 4 "Termination" // // Number of iteration: if a relation m exists within norm M, then we // need no more than (n(n-1)/2) log(gamma^(n-1)M)/log(tau) iterations. // Quality of the result: the result found by PSLQ has norm no more than // gamma^(n-2)*M // Multiple relations: when one of the coordinate of xB is zero, the // corresponding column of B will be a relation. Then we may use the // n-1 remaining coordinates to find another relation. // x=(x1..xn), xB=(y1..yn) where yj=0, y=(y1..~yj..yn), yC=(z1..zn-1), // zk=0. // m=B[,j] is a relation for x, m2=C[,k] is a relation for (xB[,1],. // ~xB[,j],..xB[,n]) // x.B[,1] C[1,k] // xBCD // normalize x normX = 0; for (int i = 0; i < n; i++) { normX += x[i] * x[i]; } normX = Math.sqrt(normX); for (int i = 0; i < n; i++) { x[i] = x[i] / normX; } // partial sums of squares ss = new double[n]; ss[n - 1] = x[n - 1] * x[n - 1]; for (int i = n - 2; i >= 0; i--) { ss[i] = ss[i + 1] + x[i] * x[i]; } for (int i = 0; i < n; i++) { ss[i] = Math.sqrt(ss[i]); } // pre-calculate ss[j]*ss[j+1] double[] Pss = new double[n - 1]; for (int i = 0; i < n - 1; i++) { Pss[i] = ss[i] * ss[i + 1]; } // initialize Matrix H (lower trapezoidal H = new double[n][n - 1]; for (int i = 0; i < n; i++) { for (int j = 0; j < i; j++) { H[i][j] = -x[i] * x[j] / Pss[j]; } if (i < n - 1) H[i][i] = ss[i + 1] / ss[i]; for (int j = i + 1; j < n - 1; j++) { H[i][j] = 0; } } // test property of H: the n-1 columns are orthogonal /* * for (int i =0 ; i<n-1; i++) { for (int j=0; j<n-1; j++) { double sum * = 0; for (int k=0; k<n; k++) { sum += H[k][i]*H[k][j]; } * System.out.println(sum); } } */ // matrix P = In - x.x P = new double[n][n]; for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) P[i][j] = -x[i] * x[j]; for (int i = 0; i < n; i++) P[i][i] += 1; // debug: |P|^2=|H|^2 = n-1 // AbstractApplication.debug("Frobenius Norm Squares: \n" // + "|P|^2 = " + frobNormSq(P,n,n) // + "|H|^2 = " + frobNormSq(H,n,n-1) // ); // initialize matrices R R = new int[n - 1][n][n]; for (int j = 0; j < n - 1; j++) { for (int i = 0; i < n; i++) for (int k = 0; k < n; k++) R[j][i][k] = 0; for (int i = 0; i < n; i++) R[j][i][i] = 1; R[j][j][j] = 0; R[j][j][j + 1] = 1; R[j][j + 1][j] = 1; R[j][j + 1][j + 1] = 0; } gamma = 1.5; deltaSq = 3.0 / 4 - (1.0 / gamma) / gamma; // initialize A, B = I_n A = new int[n][n]; for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) A[i][j] = 0; for (int i = 0; i < n; i++) A[i][i] = 1; // B = new int[n][n]; for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) B[i][j] = 0; for (int i = 0; i < n; i++) B[i][i] = 1; // iteration int itCount = 0; double itBound = 2.0 * gamma / deltaSq * n * n * (n + 1) * Math.log(Math.sqrt(bound * bound * n) * n * n) / Math.log(2); // AbstractApplication.debug("itBound = " + itBound); while (itCount < itBound) { // 0. test if we have found a relation in a column of B // xB = new double[n]; boolean solutionFound = false; boolean firstSolutionRecorded = false; for (int i = 0; i < n; i++) { xB[i] = 0; for (int k = 0; k < n; k++) xB[i] += x[k] * B[k][i]; if (Kernel.isEqual(xB[i], 0, AccuracyFactor / normX)) { solutionFound = true; orthoIndices[i] = 1; if (!firstSolutionRecorded) { for (int k = 0; k < n; k++) coeffs[k] = B[k][i]; firstSolutionRecorded = true; } } } if (solutionFound) return coeffs; // 0.5. calculate D, E // matrix D D = new int[n][n]; double[][] D0 = new double[n][n]; // testing for (int i = 0; i < n; i++) { // define backwards. the 0's and 1's should be defined first. for (int j = n - 1; j >= i + 1; j--) { D[i][j] = 0; D0[i][j] = 0; } D[i][i] = 1; D0[i][i] = 1; for (int j = i - 1; j >= 0; j--) { double sum = 0; double sum0 = 0; for (int k = j + 1; k <= i; k++) { sum += D[i][k] * H[k][j]; sum0 += D0[i][k] * H[k][j]; } D[i][j] = (int) Math.floor(-1.0 / H[j][j] * sum + 0.5); D0[i][j] = -1.0 / H[j][j] * sum0; } } // matrix E = D^{-1} E = new int[n][n]; for (int i = 0; i < n; i++) { // define backwards. the 0's and 1's should be defined first. for (int j = n - 1; j >= i + 1; j--) { E[i][j] = 0; } E[i][i] = 1; for (int j = i - 1; j >= 0; j--) { int sum = 0; for (int k = j + 1; k <= i; k++) sum += E[i][k] * D[k][j]; E[i][j] = -sum; } } // 1. replace H by DH newH = new double[n][n - 1]; double[][] newH0 = new double[n][n - 1]; for (int i = 0; i < n; i++) { for (int j = 0; j < n - 1; j++) { newH[i][j] = 0; newH0[i][j] = 0; for (int k = 0; k < n; k++) { newH[i][j] += D[i][k] * H[k][j]; newH0[i][j] += D0[i][k] * H[k][j]; } } } for (int i = 0; i < n; i++) for (int j = 0; j < n - 1; j++) H[i][j] = newH[i][j]; // 2. find j to maximize gamma^j |h_jj| double gammaPow = 1; double temp; double max = 0; int index = 0; for (int j = 0; j < n - 1; j++) { gammaPow *= gamma; temp = gammaPow * Math.abs(H[j][j]); if (max < temp) { max = temp; index = j; } } // 2.5 calculate matrices G[0], G[1],... G[n-2] G = new double[n - 1][n - 1][n - 1]; for (int i = 0; i < n - 1; i++) for (int k = 0; k < n - 1; k++) G[n - 2][i][k] = 0; for (int i = 0; i < n - 1; i++) G[n - 2][i][i] = 1; for (int j = 0; j < n - 2; j++) { double b = H[j + 1][j]; double c = H[j + 1][j + 1]; double d = Math.sqrt(b * b + c * c); for (int i = 0; i < n - 2; i++) for (int k = 0; k < n - 2; k++) G[j][i][k] = 0; for (int i = 0; i < j; i++) G[j][i][i] = 1; for (int i = j + 2; i < n - 1; i++) G[j][i][i] = 1; G[j][j][j] = b / d; G[j][j][j + 1] = -c / d; G[j][j + 1][j] = -G[j][j][j + 1]; // =c/d G[j][j + 1][j + 1] = G[j][j][j]; // = b/d } // 3. replace H by R_jHG_j, A by R_jDA, B by BER_j newH = new double[n][n - 1]; for (int i = 0; i < n; i++) { for (int j = 0; j < n - 1; j++) { newH[i][j] = 0; for (int k = 0; k < n; k++) for (int l = 0; l < n - 1; l++) newH[i][j] += R[index][i][k] * H[k][l] * G[index][l][j]; } } for (int i = 0; i < n; i++) for (int j = 0; j < n - 1; j++) H[i][j] = newH[i][j]; newAorB = new int[n][n]; for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { newAorB[i][j] = 0; for (int k = 0; k < n; k++) for (int l = 0; l < n; l++) newAorB[i][j] += R[index][i][k] * D[k][l] * A[l][j]; } } for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) A[i][j] = newAorB[i][j]; for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { newAorB[i][j] = 0; for (int k = 0; k < n; k++) for (int l = 0; l < n; l++) newAorB[i][j] += B[i][k] * E[k][l] * R[index][l][j]; } } for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) B[i][j] = newAorB[i][j]; itCount++; } return null; } @Override public boolean isLaTeXTextCommand() { return true; } /* * * public class FullDec extends MyDecimal{ * * * protected FullDec() { super(fullScale); // TODO Auto-generated * constructor stub } * * protected FullDec(double x) { super(full,x); * * // TODO Auto-generated constructor stub } * * protected FullDec(MyDecimal md) { super(md); } * * } * * private class DoubleDec extends MyDecimal{ * * protected DoubleDec() { super(decLess); // TODO Auto-generated * constructor stub } * * protected DoubleDec(double x) { super(decLess,x); // TODO Auto-generated * constructor stub } * * protected DoubleDec(FullDec x) { super(decLess, x); } * * protected DoubleDec(MyDecimal md) { super(md); } * * } */ private class IntRelationFinder { // constants (defined later) private MyDecimal ZERO; MyDecimal ZERO_LESS; private MyDecimal ONE_LESS; // parameters private double tau, rho, gamma; double err; // error bound for x private double M; // norm bound; // input private int n; private double[] x; private MyDecimal[] x_full; private MyDecimal[] x_double; // working variables private int fullScale1; private int lessScale1; private MyDecimal xNorm; private MyDecimalMatrix H_full; private MyDecimalMatrix H; MyDecimalMatrix I; private MyDecimalMatrix A; private MyDecimalMatrix B; private MyDecimalMatrix D; // private Array2DRowFieldMatrix<Dfp> B_comp; //B_comp: *= new B_rest // (B_rest is in IntRelation class) private MyDecimalMatrix xB; private MyDecimal b, l, d; private int r; // results ArrayList<IntRelation> result; // PSLQ with initialization for exact calculation IntRelationFinder(int n, double[] x, int fullScale_input, int lessScale_input, double err, double bound) { this.fullScale1 = fullScale_input; this.lessScale1 = lessScale_input; this.n = n; this.err = err; this.M = bound; result = new ArrayList<IntRelation>(); int digitsNeeded = (int) Math.ceil(-Math.log10(err)); // int digitsAllocated = lessScale; lessScale1 = digitsNeeded; fullScale1 = digitsNeeded * n; /* * boolean needSizeChange = false; while (digitsAllocated <= * digitsNeeded) { needSizeChange = true; digitsAllocated *=2; } if * (needSizeChange) { lessScale = digitsAllocated; } * * needSizeChange = false; digitsNeeded = n*digitsNeeded; * digitsAllocated = fullScale; while (digitsAllocated <= * digitsNeeded) { needSizeChange = true; digitsAllocated *=2; } if * (needSizeChange) { fullScale = digitsAllocated; } */ ZERO = new MyDecimal(fullScale1, BigDecimal.ZERO); ZERO_LESS = new MyDecimal(lessScale1, BigDecimal.ZERO); ONE_LESS = new MyDecimal(lessScale1, BigDecimal.ONE); if (n < 1) { result.clear(); return; } if (n == 1) { if (x == null || x[0] >= err) { result.clear(); return; } B = new MyDecimalMatrix(lessScale1, 1, 1); B.setEntry(0, 0, ONE_LESS); xB = new MyDecimalMatrix(lessScale1, 1, 1); xB.setEntry(0, 0, ZERO_LESS); IntRelation m = new IntRelation(n, B, xB, 1); result.add(m); return; } this.x = new double[n]; this.x_full = new MyDecimal[n]; // normalized, full precision this.x_double = new MyDecimal[n]; // normalized, double precision // int[] orthoIndices = new int[n]; for (int i = 0; i < n; i++) { this.x[i] = x[i]; this.x_full[i] = new MyDecimal(fullScale1, x[i]); } rho = 2; tau = 1.5; // arbitrarily chosen between 1+ and 2- gamma = 1 / Math.sqrt(1 / tau / tau - 1 / rho / rho); initialize_full(); b = new MyDecimal(lessScale1); l = new MyDecimal(lessScale1); d = new MyDecimal(lessScale1); boolean loopTillExhausted = true; int iterCount = 0; double iterBound = n * (n + 1) / 2.0 * ((n - 1) * Math.log(gamma) + 0.5 * Math.log(n) + Math.log(M)) / Math.log(tau); while (iterCount < iterBound) { // "Exchange": find r to maximize gamma^r |h_rr|, exchange rows // r and r+1. double gammaPow = 1.0; double temp; double max = 0; for (int index = 0; index < n - 1; index++) { gammaPow *= gamma; temp = gammaPow * H.getEntry(index, index).abs().doubleValue(); if (max < temp) { max = temp; r = index; } } if (r < n - 2) { // for r=n-2 we don't need to define these. // Also l will be undefined H.getEntry(r, r); b = H.getEntry(r + 1, r); l = H.getEntry(r + 1, r + 1); d = b.multiply(b).add(l.multiply(l)).sqrt(); } MyDecimal temp2 = new MyDecimal(xB.getEntry(0, r)); xB.setEntry(0, r, xB.getEntry(0, r + 1)); xB.setEntry(0, r + 1, temp2); MyDecimal[] temp3 = H.getRow(r); H.setRow(r, H.getRow(r + 1)); H.setRow(r + 1, temp3); temp3 = A.getRow(r); A.setRow(r, A.getRow(r + 1)); A.setRow(r + 1, temp3); temp3 = B.getColumn(r); B.setColumn(r, B.getColumn(r + 1)); B.setColumn(r + 1, temp3); // Corner MyDecimal[] temp4; if (r < n - 2) { temp3 = H.getColumn(r); temp4 = H.getColumn(r + 1); for (int i = r; i < n; i++) { H.setEntry(i, r, b.multiply(temp3[i]).add(l.multiply(temp4[i])).divide(d)); H.setEntry(i, r + 1, l.negate().multiply(temp3[i]).add(b.multiply(temp4[i])).divide(d)); } } // pretermination, just for double-check. Not mentioned in the // article. boolean relationExhausted = false; for (int j = 0; j < n - 1; j++) { if (H.getEntry(j, j).abs().doubleValue() < Math.pow(10, -Math.min(H.getScale(), 5))) { relationExhausted = true; Log.warn("relation pre-Exhausted at iteration " + iterCount + "with r = " + j + "where n-1 = " + (n - 1)); } } // reduction if (!relationExhausted) hermiteReduction(); // termination boolean relationFound = false; relationExhausted = false; for (int j = 0; j < n - 1; j++) { if (H.getEntry(j, j).abs().doubleValue() < Math.pow(10, -Math.min(H.getScale(), 5))) { relationExhausted = true; Log.warn("relation Exhausted at iteration " + iterCount + "with r = " + j + "where n-1 = " + (n - 1)); } } for (int j = 0; j < n; j++) { // we look at the j-th column of B double sumBCol = 0; double maxxB = 0; for (int i = 0; i < n; i++) { sumBCol += B.getEntry(i, j).abs().doubleValue(); maxxB = Math.max(maxxB, xB.getEntry(0, i).abs().doubleValue()); } if (xB.getEntry(0, j).signum() == 0 || xB.getEntry(0, j).abs().doubleValue() < sumBCol * err) { relationFound = true; IntRelation m = new IntRelation(n, B, xB, err / maxxB); result.add(m); break; } } // System.out.println(""); if (relationFound && !loopTillExhausted || relationExhausted) break; iterCount++; } } void initialize_full() { xNorm = new MyDecimal(lessScale1); // normalize x for (int i = 0; i < n; i++) { xNorm = xNorm.add(x_full[i].multiply(x_full[i])); } xNorm = xNorm.sqrt(); // System.out.println(xNorm.toFullString()); for (int i = 0; i < n; i++) { x_full[i] = x_full[i].divide(xNorm); x_double[i] = new MyDecimal(lessScale1, x_full[i]); } // partial sums of squares MyDecimal[] ss = new MyDecimal[n]; for (int i = 0; i < n; i++) { ss[i] = new MyDecimal(fullScale1); } ss[n - 1] = x_full[n - 1].multiply(x_full[n - 1]); for (int i = n - 2; i >= 0; i--) { ss[i] = ss[i + 1].add(x_full[i].multiply(x_full[i])); } for (int i = 0; i < n; i++) { ss[i] = ss[i].sqrt(); } // pre-calculate ss[j]*ss[j+1] MyDecimal[] Pss = new MyDecimal[n - 1]; for (int i = 0; i < n - 1; i++) { Pss[i] = new MyDecimal(fullScale1, ss[i].multiply(ss[i + 1])); } // initialize Matrix H (lower trapezoidal H_full = new MyDecimalMatrix(fullScale1, n, n - 1); for (int i = 0; i < n; i++) { for (int j = 0; j < i; j++) { H_full.setEntry(i, j, x_full[i].multiply(x_full[j]).divide(Pss[j]).negate()); } if (i < n - 1) H_full.setEntry(i, i, ss[i + 1].divide(ss[i])); for (int j = i + 1; j < n - 1; j++) { H_full.setEntry(i, j, ZERO); } } // test property of H: the n-1 columns are orthogonal // System.out.println(H.transpose().multiply(H).toString()); // matrix P = In - x.x /* * P = new double[n][n]; for (int i=0; i<n; i++) for (int j=0; j<n; * j++) P[i][j] = -x[i]*x[j]; for (int i=0; i<n; i++) P[i][i]+=1; */ /* * P = new Array2DRowFieldMatrix<Dfp>(decFull, n,n); * * for (int i=0; i<n; i++) { for (int j=0; j<n; j++) { P.setEntry(i, * j, x1[i].multiply(x1[j]).negate()); } } * * * for (int i=0; i<n; i++) { P.setEntry(i, i, P.getEntry(i, * i).add(ONE)); } */ /* * //P = H.multiply( ((Array2DRowFieldMatrix<Dfp>) H.transpose())); * * System.out.println(P.toString()); * * //debug: |P|^2=|H|^2 = n-1 * AbstractApplication.debug("Frobenius Norm Squares: \n" + * "|P|^2 = " + frobNormSq(P,n,n) + "|H|^2 = " + frobNormSq(H,n,n-1) * ); */ H = new MyDecimalMatrix(lessScale1, n, n - 1); for (int i = 0; i < n; i++) { for (int j = 0; j < n - 1; j++) { H.setEntry(i, j, new MyDecimal(lessScale1, H_full.getEntry(i, j))); } } // initialize matrices R /* * R = new int[n-1][n][n]; for (int j=0; j<n-1; j++) { for (int i=0; * i<n; i++) for (int k=0; k<n; k++) R[j][i][k]=0; for (int i=0; * i<n; i++) R[j][i][i]=1; R[j][j][j]=0; R[j][j][j+1]=1; * R[j][j+1][j]=1; R[j][j+1][j+1]=0; } * * gamma = 1.5; deltaSq = 3.0/4 - (1.0/gamma)/gamma; * * //initialize A, B = I_n A = new int[n][n]; for (int i=0; i<n; * i++) for (int j=0; j<n; j++) A[i][j]=0; for (int i=0; i<n; i++) * A[i][i]=1; //B = new int[n][n]; for (int i=0; i<n; i++) for (int * j=0; j<n; j++) B[i][j]=0; for (int i=0; i<n; i++) B[i][i]=1; */ I = new MyDecimalMatrix(lessScale1, n, n); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { if (i == j) I.setEntry(i, i, ONE_LESS); else I.setEntry(i, j, ZERO_LESS); } } A = I.copy(); B = I.copy(); xB = new MyDecimalMatrix(lessScale1, 1, n); for (int i = 0; i < n; i++) { xB.setEntry(0, i, x_double[i]); } // System.out.println(A.toString()); } private void hermiteReduction() { D = I.copy(); for (int i = 1; i < n; i++) { for (int j = i - 1; j >= 0; j--) { // MyDecimal q = new MyDecimal(H.getEntry(i, // j).divide(H.getEntry(j, j)).divide(ONE, 0, // BigDecimal.ROUND_DOWN)); MyDecimal q = new MyDecimal(lessScale1, Math.rint(H.getEntry(i, j).doubleValue() / H.getEntry(j, j).doubleValue())); for (int k = 0; k <= j; k++) { H.setEntry(i, k, H.getEntry(i, k).subtract(q.multiply(H.getEntry(j, k)))); } for (int k = 0; k < n; k++) { D.setEntry(i, k, D.getEntry(i, k).subtract(q.multiply(D.getEntry(j, k)))); A.setEntry(i, k, A.getEntry(i, k).subtract(q.multiply(A.getEntry(j, k)))); B.setEntry(k, j, B.getEntry(k, j).add(q.multiply(B.getEntry(k, i)))); } xB.setEntry(0, j, xB.getEntry(0, j).add(q.multiply(xB.getEntry(0, i)))); } } } private class IntRelation implements Comparable<IntRelation> { private int size; double sig; // private double norm2; //norm of the vector m squared // private int normInf; // these are all copied from IntRelationFinder private int nilDim; // dimension of the nil-subspace corresponding // to vector m, that is, the # cols of B_sol private MyDecimalMatrix B1, B_sol, B_rest; MyDecimalMatrix xB1; int[] orthoIndices; public IntRelation(int n, MyDecimalMatrix B, MyDecimalMatrix xB, double sig) { if (n == 0) return; this.size = n; this.B1 = B.copy(); this.xB1 = xB.copy(); this.sig = sig; orthoIndices = new int[n]; nilDim = 0; for (int j = 0; j < n; j++) { // we look at the j-th column of B double sumBCol = 0; double maxxB = 0; for (int i = 0; i < n; i++) { sumBCol += B.getEntry(i, j).abs().doubleValue(); maxxB = Math.max(maxxB, xB.getEntry(0, i).abs().doubleValue()); } if (xB.getEntry(0, j).equals(ZERO_LESS) || Math.abs(xB.getEntry(0, j).doubleValue()) < sumBCol * err) { orthoIndices[j] = 1; nilDim++; } else { orthoIndices[j] = 0; } } // B_sol: solution columns of B; B_rest: non solution columns if (nilDim > 0) { B_sol = new MyDecimalMatrix(B.getScale(), size, nilDim); } else { B_sol = null; } if (nilDim < size) { B_rest = new MyDecimalMatrix(B.getScale(), size, size - nilDim); } else { B_rest = null; } int is, ir; is = ir = 0; for (int j = 0; j < n; j++) { if (orthoIndices[j] == 1) { B_sol.setColumn(is++, B.getColumn(j)); } else { B_rest.setColumn(ir++, B.getColumn(j)); } } } public MyDecimalMatrix getBMatrix() { return B1.copy(); } public MyDecimalMatrix getBSolMatrix() { if (B_sol != null) { return B_sol.copy(); } return null; } public MyDecimalMatrix getBRestMatrix() { if (B_rest != null) { return B_rest.copy(); } return null; } public int compareTo(IntRelation m2) { if (this.size != m2.size) { return -100 * (this.size - m2.size); // should throw an // exception } if (Kernel.isGreater(this.sig, m2.sig, 10E-7)) { return 1; } else if (Kernel.isGreater(m2.sig, this.sig, 10E-7)) { return -1; } return 0; } } } /** * @author tam * */ public enum AlgebraicFittingType { /** * Given x, we search for the form x = A/B where A,B are integers. */ RATIONAL_NUMBER, /** * Given x and constants Ck, we search for x = (sum of AkCk)/B where Ak * and B are integers. */ LINEAR_COMBINATION, /** * Given x and constants Ck, search for x = (sum of AkCk)/(sum of BkCk) * where Ak and Bk are integers. */ RATIONAL_COMBINATION, /** * Given x and constants Ek, search for x = product of Ek^(pk/qk) where * pk,qk are integers. */ POWER_PRODUCT, /** * Given x and constants Ck, search for x = (A+B sqrt(C))/D, where * A,B,C,D are linear combinations of Ck */ QUADRATIC_RADICAL, /** * Given x and invertible function f, find x = f(y) where y is fit in * the type of RATIONAL_NUMBER */ FUNCTION_OF_RATIONAL_NUMBER, /** * Given x, constants Ck, and invertible function f, find x = f(y) where * y is fit in the type of LINEAR_COMINATION */ FUNCTION_OF_LINEAR_COMBINATION, /** * Given x, constants Ck, and invertible function f, find x = f(y) where * y is fit in the type of POWER_PRODUCT */ FUNCTION_OF_POWER_PRODUCT, /** * Given x, constants Ck, and invertible function f, find x = f(y) where * y is fit in the type of QUADRATIC_RADICAL */ FUNCTION_OF_QUADRATIC_RADICAL } private class AlgebraicFit { // input private double num1; // parameters private int numOfConsts; private int numOfRadicals; private double[] constValues; private String[] constStrings; private int coeffBound; // largest acceptable absolute value of // coefficients private double err; private AlgebraicFittingType aft; private StringTemplate tpl; // working variables private double[] numList; private int[][] coeffs; // relations generated from mPSLQ private int s; // number of candidate relations private int numOfPenalties; private int[][] penalties; private int numOfConstsUsed; // does not include 1 private int maxCoeff; private int sumCoeffs; private boolean isOneUsed; private int bestIndex; private int[] bestRelation; // output public StringBuilder formalSolution; /** * @param constStrings * the strings of the constants * @param constValues * the double value of them * @param aft * @param tpl */ public AlgebraicFit(String[] constStrings, double[] constValues, AlgebraicFittingType aft, StringTemplate tpl) { this.numOfConsts = constValues == null ? 0 : constValues.length; this.numOfRadicals = this.numOfConsts; // this.constValues = constValues.clone(); //not available in GWT if (constValues != null) { this.constValues = new double[constValues.length]; for (int i = 0; i < constValues.length; i++) { this.constValues[i] = constValues[i]; } } // this.constStrings = constStrings.clone(); //not available in GWT if (constStrings != null) { this.constStrings = new String[constStrings.length]; for (int i = 0; i < constStrings.length; i++) { this.constStrings[i] = constStrings[i]; } } this.aft = aft; this.tpl = tpl; getKernel(); getKernel(); err = Math.min(Kernel.MAX_PRECISION, Kernel.STANDARD_PRECISION); coeffBound = 100; formalSolution = new StringBuilder(); formalSolution.setLength(0); // numList = new double[(numOfConstants+1)*(deg+1)]; } public void compute(double number) { switch (aft) { case RATIONAL_NUMBER: computeRationalNumber(number); return; case LINEAR_COMBINATION: computeConstant(number); return; case RATIONAL_COMBINATION: // TODO return; case POWER_PRODUCT: // TODO return; case QUADRATIC_RADICAL: computeQuadratic(number); return; case FUNCTION_OF_RATIONAL_NUMBER: // TODO return; case FUNCTION_OF_LINEAR_COMBINATION: // TODO return; case FUNCTION_OF_POWER_PRODUCT: // TODO return; case FUNCTION_OF_QUADRATIC_RADICAL: // TODO return; } } private void computeQuadratic(double number) { num1 = number; numList = new double[(1 + numOfConsts) * 3]; // (numOfConsts+1)(degree+1) double temp; for (int j = 0; j < numOfConsts; j++) { temp = constValues[j]; for (int i = 0; i < 3; i++) { numList[j * 3 + i] = temp; temp *= num1; } } temp = 1.0; // Note: it turns out to be better to put the 1 term at the end // instead of in the front for (int i = 0; i < 3; i++) { numList[numOfConsts * 3 + i] = temp; temp *= num1; } coeffs = mPSLQ((numOfConsts + 1) * 3, numList, err, coeffBound); if (coeffs == null || coeffs[0].length == 0) { return; } s = coeffs[0].length; // penalties calculation int[][] termCount = new int[numOfConsts + 1][s]; // count number of // terms within // one big // coefficient int[][] termMax = new int[numOfConsts + 1][s]; // max abs value of // coefficient // within one big // coefficient double[][] w = new double[4][s]; // w[0]+w[1]x+w[2]x^2=0, // w[3]=w[1]^2-4w[0]*w[2] numOfPenalties = 7; penalties = new int[numOfPenalties][s]; for (int j = 0; j < s; j++) { // for the j-th solution, check its // characteristics and make // penalties // first penalties[0][j]: is the largest coefficient in a big // coefficient greater than 100? Yes: +1, No: 0 // second penalties[1][j]: number of constants used // third penalties[2][j]: maximum number of terms in each "big" // coefficient (i.e. linear combination of constants) // fourth penalties[3][j]: form -- "A" - 0, "A/B" - 1, // "sqrt(A)/B" - 2, (B+sqrt(A)) - 3, (B+sqrt(A))/C - 4 // fifth penalties[4][j]: coefficient -- its absolute value when // less than 100. 100 or more -- 100. // sixth penalties[5][j]: is multiple of "1" used? Yes: 1, No:0 // last penalties[6][j]: -j (the larger index the better) for (int i = 0; i < numOfPenalties; i++) { penalties[i][j] = 0; } for (int a = 0; a < numOfConsts + 1; a++) { termCount[a][j] = 0; termMax[a][j] = 0; for (int b = 0; b < 3; b++) { if (coeffs[a * 3 + b][j] != 0) { termCount[a][j]++; } termMax[a][j] = Math.max(termMax[a][j], Math.abs(coeffs[a * 3 + b][j])); } } int[] wj = new int[(numOfConsts + 1) * 3]; for (int i = 0; i < wj.length; i++) { wj[i] = coeffs[i][j]; } for (int b = 0; b < 3; b++) { w[b][j] = evaluateCombination(numOfConsts, constValues, wj, b, 3) + coeffs[numOfConsts * 3 + b][j]; } w[3][j] = w[1][j] * w[1][j] - 4 * w[0][j] * w[2][j]; if (Kernel.isZero(w[2][j])) { // w[0][j]+w[1][j]x=0 if (Kernel.isZero(w[1][j])) penalties[3][j] = 10000; // bad case else if (Kernel.isEqual(Math.abs(w[1][j]), 1.0)) // naturally // an // integer penalties[3][j] = 0; else penalties[3][j] = 1; } else { if (Kernel.isEqual(Math.abs(w[2][j]), 0.5)) { // 0 or 2 or 4 if (Kernel.isZero(w[3][j])) penalties[3][j] = 0; else if (Kernel.isZero(w[1][j])) penalties[3][j] = 2; else penalties[3][j] = 4; } else { // 1 or 3 or 5 if (Kernel.isZero(w[3][j])) penalties[3][j] = 1; else if (Kernel.isZero(w[1][j])) penalties[3][j] = 3; else penalties[3][j] = 5; } } for (int a = 0; a < numOfConsts + 1; a++) { penalties[0][j] += termMax[a][j] > coeffBound ? 1 : 0; penalties[1][j] += termCount[a][j] > 0 ? 1 : 0; penalties[2][j] = Math.max(penalties[2][j], termCount[a][j]); // penalties[3][j]: form -- integer - 0, fraction - 1, // sqrt(w3) - 2 // sqrt(w3)/2w2" - 3, (-w1+sqrt(w3)) - 4, (-w1+sqrt(w3))/2w2 // - 5 if (termMax[a][j] >= 100) { penalties[4][j] += 100; } else { penalties[4][j] += termMax[a][j]; } } boolean isOneUsed1 = false; for (int b = 0; b < 3; b++) { isOneUsed1 = isOneUsed1 || coeffs[numOfConsts * 3 + b][j] != 0; } penalties[5][j] = isOneUsed1 ? 1 : 0; penalties[6][j] = -j; } bestIndex = leastPenaltyIndex(); bestRelation = new int[(1 + numOfConsts) * 3]; for (int a = 0; a < (1 + numOfConsts) * 3; a++) { bestRelation[a] = coeffs[a][bestIndex]; } // clear the solutions that do not fulfill strict requirements if (penalties[0][bestIndex] >= 1) { bestIndex = -1; formalSolution.setLength(0); return; } // Suppose A+Bx+Cx^2 = 0, where A,B,C are linear combinations of 1 // and values in constValueboolean isAZero = true; boolean isAZero = true; boolean isARational = true; boolean isBZero = true; boolean isBRational = true; boolean isCZero = true; boolean isCRational = true; if (bestRelation[numOfConsts * 3] != 0) { isAZero = false; } if (bestRelation[numOfConsts * 3 + 1] != 0) { isBZero = false; } if (bestRelation[numOfConsts * 3 + 2] != 0) { isCZero = false; } for (int j = 0; j < numOfConsts; j++) { if (bestRelation[j * 3] != 0) { isAZero = false; isARational = false; } if (bestRelation[j * 3 + 1] != 0) { isBZero = false; isBRational = false; } if (bestRelation[j * 3 + 2] != 0) { isCZero = false; isCRational = false; } } /* * if (isARational && isBRational && isCRational) { //TODO: optimize * this PSLQappendQuadratic(sb, num, tpl); //return; } */ if (isAZero && isBZero && isCZero) { formalSolution.setLength(0); } else if (isCZero) { if (isBZero) { formalSolution.setLength(0); } else { StringBuilder AString = new StringBuilder(); StringBuilder BString = new StringBuilder(); AString.append(kernel.format(bestRelation[numOfConsts * 3], tpl)); if (!isARational) { AString.append("+"); // appendCombination(AString,(bestRelation[numOfConsts*3]==0)? // numOfTermsInA : numOfTermsInA-1, constStrings, // bestRelation, 0, 3, tpl); appendCombination(AString, numOfConsts, constStrings, bestRelation, 0, 3, tpl); } BString.append(kernel.format(bestRelation[numOfConsts * 3 + 1], tpl)); if (!isBRational) { BString.append("+"); appendCombination(BString, numOfConsts, constStrings, bestRelation, 1, 3, tpl); } formalSolution.append("-("); formalSolution.append(AString.toString()); formalSolution.append(")/("); formalSolution.append(BString.toString()); formalSolution.append(")"); } } else { double Avalue = bestRelation[numOfConsts * 3] + evaluateCombination(numOfConsts, constValues, bestRelation, 0, 3); double Bvalue = bestRelation[numOfConsts * 3 + 1] + evaluateCombination(numOfConsts, constValues, bestRelation, 1, 3); double Cvalue = bestRelation[numOfConsts * 3 + 2] + evaluateCombination(numOfConsts, constValues, bestRelation, 2, 3); double discr = Bvalue * Bvalue - 4 * Avalue * Cvalue; StringBuilder AString = new StringBuilder(); StringBuilder BString = new StringBuilder(); StringBuilder CString = new StringBuilder(); AString.append(kernel.format(bestRelation[numOfConsts * 3], tpl)); if (!isARational) { AString.append("+"); appendCombination(AString, numOfConsts, constStrings, bestRelation, 0, 3, tpl); } BString.append(kernel.format(bestRelation[numOfConsts * 3 + 1], tpl)); if (!isBRational) { BString.append("+"); appendCombination(BString, numOfConsts, constStrings, bestRelation, 1, 3, tpl); } CString.append(kernel.format(bestRelation[numOfConsts * 3 + 2], tpl)); if (!isCRational) { CString.append("+"); appendCombination(CString, numOfConsts, constStrings, bestRelation, 2, 3, tpl); } formalSolution.append("("); formalSolution.append("-("); formalSolution.append(BString.toString()); formalSolution.append(")"); if (!Kernel.isZero(discr)) { if (num1 * 2 * Cvalue + Bvalue >= 0) { formalSolution.append("+"); } else formalSolution.append("-"); formalSolution.append("sqrt("); formalSolution.append("("); formalSolution.append(BString.toString()); formalSolution.append(")^2"); formalSolution.append("-4*("); formalSolution.append(CString.toString()); formalSolution.append(")*("); formalSolution.append(AString.toString()); formalSolution.append(")"); formalSolution.append(")"); } formalSolution.append(")/("); formalSolution.append("2*("); formalSolution.append(CString.toString()); formalSolution.append(")"); formalSolution.append(")"); } } /** * Assume that number = A/B, then we need to find relation (B,-A) to the * vector (number,1) * * @param number */ private void computeRationalNumber(double number) { numOfConsts = 0; numList = new double[] { number, 1 }; coeffs = mPSLQ(2, numList, err, coeffBound); if (coeffs == null) return; // get number of solutions s = coeffs[0].length; // compute penalties, using leastPenaltyIndex() to find the // bestIndex numOfPenalties = 3; penalties = new int[numOfPenalties][s]; for (int j = 0; j < s; j++) { maxCoeff = Math.max(Math.abs(coeffs[0][j]), Math.abs(coeffs[1][j])); penalties[0][j] = maxCoeff > coeffBound ? 1 : 0; penalties[1][j] = Math.abs(coeffs[0][j]); // magnitude of // denominator penalties[2][j] = Math.abs(coeffs[1][j]); } bestIndex = leastPenaltyIndex(); // retrieve the bestRelation bestRelation = new int[numOfConsts + 2]; for (int i = 0; i < numOfConsts + 2; i++) { bestRelation[i] = coeffs[i][bestIndex]; } // clear the solutions that do not fulfill strict requirements if (penalties[0][bestIndex] == 1) { bestIndex = -1; formalSolution.setLength(0); return; } // construct CAS String of the formal solution formalSolution.append(kernel.format(-bestRelation[1], tpl)); formalSolution.append("/("); formalSolution.append(kernel.format(bestRelation[0], tpl)); formalSolution.append(")"); } // constant test public void computeConstant(double number) { numList = new double[numOfConsts + 2]; // {the constants} U {1} U // {num} for (int j = 0; j < numOfConsts; j++) { numList[j] = constValues[j]; } numList[numOfConsts] = 1.0; numList[numOfConsts + 1] = number; coeffs = mPSLQ(numOfConsts + 2, numList, err, coeffBound); if (coeffs == null) { return; } s = coeffs[0].length; // penalties calculation int numOfRadicalsUsed = 0; int numOfOthersUsed = 0; for (int j = 0; j < s; j++) { for (int i = 0; i < numOfRadicals; i++) if (coeffs[i][j] != 0) numOfRadicalsUsed++; for (int i = numOfRadicals; i < numOfConsts; i++) if (coeffs[i][j] != 0) numOfOthersUsed++; numOfConstsUsed = numOfRadicalsUsed + numOfOthersUsed; isOneUsed = coeffs[numOfConsts][j] == 1; maxCoeff = 0; sumCoeffs = 0; for (int i = 0; i < numOfConsts + 1; i++) { sumCoeffs += Math.abs(coeffs[i][j]); maxCoeff = Math.max(maxCoeff, Math.abs(coeffs[i][j])); } } numOfPenalties = 7; penalties = new int[numOfPenalties][s]; // candidates = new boolean[s]; for (int j = 0; j < s; j++) { // for the j-th solution, check its // characteristics and make // penalties // penalties[0][j]: coefficient of num can't be zero. Is zero: // 1, No: 0 // penalties[1][j]: is the largest coefficient in a big // coefficient greater than acceptable bound? Yes: +1, No: 0 // penalties[2][j]: is non-algebraic? Yes: 1, no: 0 // penalties[3][j]: number of constants used // penalties[4][j]: coefficient -- its absolute value when no // greater than bound. bound+1 or more -- bound+1. // penalties[5][j]: is multiple of "1" used? Yes: 1, No:0 // penalties[6][j]: -j (the larger index the better) penalties[0][j] = (coeffs[numOfConsts + 1][j] == 0) ? 1 : 0; penalties[1][j] = (maxCoeff > coeffBound) ? 1 : 0; penalties[2][j] = numOfOthersUsed > 0 ? 1 : 0; penalties[3][j] = numOfConstsUsed; penalties[4][j] = Math.min(sumCoeffs, coeffBound + 1); penalties[5][j] = isOneUsed ? 1 : 0; penalties[6][j] = -j; } bestIndex = leastPenaltyIndex(); bestRelation = new int[numOfConsts + 2]; for (int i = 0; i < numOfConsts + 2; i++) { bestRelation[i] = coeffs[i][bestIndex]; } if (penalties[0][bestIndex] == 1 || penalties[1][bestIndex] == 1) { bestIndex = -1; formalSolution.setLength(0); return; } // construct a formal solution in CAS format formalSolution.append("("); appendCombination(formalSolution, numOfConsts, constStrings, bestRelation, 0, 1, tpl); formalSolution.append("+"); formalSolution.append(kernel.format(bestRelation[numOfConsts], tpl)); formalSolution.append(")/("); formalSolution.append(kernel.format(-bestRelation[numOfConsts + 1], tpl)); formalSolution.append(")"); return; } private int leastPenaltyIndex() { boolean[] candidates = new boolean[s]; // find the best index int bestIndex1 = -1; for (int j = 0; j < s; j++) { candidates[j] = true; } for (int k = 0; k < numOfPenalties; k++) { int minPenalty = Integer.MAX_VALUE; for (int j = 0; j < s; j++) { if (candidates[j]) { if (penalties[k][j] < minPenalty) { minPenalty = penalties[k][j]; } } } for (int j = 0; j < s; j++) { if (candidates[j]) { if (penalties[k][j] > minPenalty) { candidates[j] = false; } else { bestIndex1 = j; } } } } return bestIndex1; } /* * * void computeLinear(double num) { * * } * * void testZero(double num) { * * } */ public void setCoeffBound(int b) { coeffBound = b; } /* * public int getCoeffBound() { return coeffBound; } * * public void setConsts(int n, String[] listOfNames, double[] * listOfValues) { * * if (listOfNames.length < n || listOfValues.length != * listOfNames.length) { App.debug("error: size does not match"); * return; } numOfConsts = n; constStrings = listOfNames.clone(); * constValues = listOfValues.clone(); } */ /** * By default, it is just an identity function. User can change the * call() method according to the underlying function and the type T of * the argument. Requirements: the function should be an invertible one, * and the type of the values should also be T. * * @author lightest * * @param <T> */ /* * public class FunctionForFit<T> implements Callable { * * private T arg; public FunctionForFit(T arg) { this.arg = arg; } * * public T call() throws Exception { return arg; } * * } */ } // TODO Consider locusequability }