@Test
public void testJacobiEvaluationAt1() {
for (int v = 0; v < 10; ++v) {
for (int w = 0; w < 10; ++w) {
for (int i = 0; i < 10; ++i) {
PolynomialFunction jacobi = PolynomialsUtils.createJacobiPolynomial(i, v, w);
double binomial = CombinatoricsUtils.binomialCoefficient(v + i, i);
Assert.assertTrue(Precision.equals(binomial, jacobi.value(1.0), 1));
}
}
}
}
@Override
public boolean equals(Object o) {
if (this == o) {
return true;
}
if (!(o instanceof TesparSymbol)) {
return false;
} else {
TesparSymbol symbol = (TesparSymbol) o;
return duration == symbol.getDuration()
&& shape == symbol.getShape()
&& Precision.equals(amplitude, symbol.getAmplitude(), TimeSeriesPrecision.EPSILON);
}
}
/* 106: */
/* 107: */ protected void solvePhase1(SimplexTableau tableau)
/* 108: */ throws MaxCountExceededException, UnboundedSolutionException,
NoFeasibleSolutionException
/* 109: */ {
/* 110:169 */ if (tableau.getNumArtificialVariables() == 0) {
/* 111:170 */ return;
/* 112: */ }
/* 113:173 */ while (!tableau.isOptimal()) {
/* 114:174 */ doIteration(tableau);
/* 115: */ }
/* 116:178 */ if (!Precision.equals(
tableau.getEntry(0, tableau.getRhsOffset()), 0.0D, this.epsilon)) {
/* 117:179 */ throw new NoFeasibleSolutionException();
/* 118: */ }
/* 119: */ }
/**
* Solves Phase 1 of the Simplex method.
*
* @param tableau Simple tableau for the problem.
* @throws TooManyIterationsException if the allowed number of iterations has been exhausted.
* @throws UnboundedSolutionException if the model is found not to have a bounded solution.
* @throws NoFeasibleSolutionException if there is no feasible solution?
*/
protected void solvePhase1(final SimplexTableau tableau)
throws TooManyIterationsException, UnboundedSolutionException, NoFeasibleSolutionException {
// make sure we're in Phase 1
if (tableau.getNumArtificialVariables() == 0) {
return;
}
while (!tableau.isOptimal()) {
doIteration(tableau);
}
// if W is not zero then we have no feasible solution
if (!Precision.equals(tableau.getEntry(0, tableau.getRhsOffset()), 0d, epsilon)) {
throw new NoFeasibleSolutionException();
}
}
/* 43: */
/* 44: */ private Integer getPivotRow(SimplexTableau tableau, int col) /* 45: */ {
/* 46: 90 */ List<Integer> minRatioPositions = new ArrayList();
/* 47: 91 */ double minRatio = 1.7976931348623157E+308D;
/* 48: 92 */ for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getHeight(); i++)
/* 49: */ {
/* 50: 93 */ double rhs = tableau.getEntry(i, tableau.getWidth() - 1);
/* 51: 94 */ double entry = tableau.getEntry(i, col);
/* 52: 96 */ if (Precision.compareTo(entry, 0.0D, this.maxUlps) > 0)
/* 53: */ {
/* 54: 97 */ double ratio = rhs / entry;
/* 55: 98 */ int cmp = Precision.compareTo(ratio, minRatio, this.maxUlps);
/* 56: 99 */ if (cmp == 0)
/* 57: */ {
/* 58:100 */ minRatioPositions.add(Integer.valueOf(i));
/* 59: */ }
/* 60:101 */ else if (cmp < 0)
/* 61: */ {
/* 62:102 */ minRatio = ratio;
/* 63:103 */ minRatioPositions = new ArrayList();
/* 64:104 */ minRatioPositions.add(Integer.valueOf(i));
/* 65: */ }
/* 66: */ }
/* 67: */ }
/* 68:109 */ if (minRatioPositions.size() == 0) {
/* 69:110 */ return null;
/* 70: */ }
/* 71:111 */ if (minRatioPositions.size() > 1) {
/* 72:114 */ for (Integer row : minRatioPositions) {
/* 73:115 */ for (int i = 0; i < tableau.getNumArtificialVariables(); i++)
/* 74: */ {
/* 75:116 */ int column = i + tableau.getArtificialVariableOffset();
/* 76:117 */ double entry = tableau.getEntry(row.intValue(), column);
/* 77:118 */ if ((Precision.equals(entry, 1.0D, this.maxUlps))
&& (row.equals(tableau.getBasicRow(column)))) {
/* 78:120 */ return row;
/* 79: */ }
/* 80: */ }
/* 81: */ }
/* 82: */ }
/* 83:125 */ return (Integer) minRatioPositions.get(0);
/* 84: */ }
/**
* Returns the row with the minimum ratio as given by the minimum ratio test (MRT).
*
* @param tableau Simple tableau for the problem.
* @param col Column to test the ratio of (see {@link #getPivotColumn(SimplexTableau)}).
* @return the row with the minimum ratio.
*/
private Integer getPivotRow(SimplexTableau tableau, final int col) {
// create a list of all the rows that tie for the lowest score in the minimum ratio test
List<Integer> minRatioPositions = new ArrayList<Integer>();
double minRatio = Double.MAX_VALUE;
for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getHeight(); i++) {
final double rhs = tableau.getEntry(i, tableau.getWidth() - 1);
final double entry = tableau.getEntry(i, col);
if (Precision.compareTo(entry, 0d, maxUlps) > 0) {
final double ratio = rhs / entry;
// check if the entry is strictly equal to the current min ratio
// do not use a ulp/epsilon check
final int cmp = Double.compare(ratio, minRatio);
if (cmp == 0) {
minRatioPositions.add(i);
} else if (cmp < 0) {
minRatio = ratio;
minRatioPositions = new ArrayList<Integer>();
minRatioPositions.add(i);
}
}
}
if (minRatioPositions.size() == 0) {
return null;
} else if (minRatioPositions.size() > 1) {
// there's a degeneracy as indicated by a tie in the minimum ratio test
// 1. check if there's an artificial variable that can be forced out of the basis
if (tableau.getNumArtificialVariables() > 0) {
for (Integer row : minRatioPositions) {
for (int i = 0; i < tableau.getNumArtificialVariables(); i++) {
int column = i + tableau.getArtificialVariableOffset();
final double entry = tableau.getEntry(row, column);
if (Precision.equals(entry, 1d, maxUlps) && row.equals(tableau.getBasicRow(column))) {
return row;
}
}
}
}
// 2. apply Bland's rule to prevent cycling:
// take the row for which the corresponding basic variable has the smallest index
//
// see http://www.stanford.edu/class/msande310/blandrule.pdf
// see http://en.wikipedia.org/wiki/Bland%27s_rule (not equivalent to the above paper)
//
// Additional heuristic: if we did not get a solution after half of maxIterations
// revert to the simple case of just returning the top-most row
// This heuristic is based on empirical data gathered while investigating MATH-828.
if (getEvaluations() < getMaxEvaluations() / 2) {
Integer minRow = null;
int minIndex = tableau.getWidth();
final int varStart = tableau.getNumObjectiveFunctions();
final int varEnd = tableau.getWidth() - 1;
for (Integer row : minRatioPositions) {
for (int i = varStart; i < varEnd && !row.equals(minRow); i++) {
final Integer basicRow = tableau.getBasicRow(i);
if (basicRow != null && basicRow.equals(row) && i < minIndex) {
minIndex = i;
minRow = row;
}
}
}
return minRow;
}
}
return minRatioPositions.get(0);
}
/** {@inheritDoc} */
@Override
protected UnivariatePointValuePair doOptimize() {
final boolean isMinim = getGoalType() == GoalType.MINIMIZE;
final double lo = getMin();
final double mid = getStartValue();
final double hi = getMax();
// Optional additional convergence criteria.
final ConvergenceChecker<UnivariatePointValuePair> checker = getConvergenceChecker();
double a;
double b;
if (lo < hi) {
a = lo;
b = hi;
} else {
a = hi;
b = lo;
}
double x = mid;
double v = x;
double w = x;
double d = 0;
double e = 0;
double fx = computeObjectiveValue(x);
if (!isMinim) {
fx = -fx;
}
double fv = fx;
double fw = fx;
UnivariatePointValuePair previous = null;
UnivariatePointValuePair current = new UnivariatePointValuePair(x, isMinim ? fx : -fx);
// Best point encountered so far (which is the initial guess).
UnivariatePointValuePair best = current;
while (true) {
final double m = 0.5 * (a + b);
final double tol1 = relativeThreshold * FastMath.abs(x) + absoluteThreshold;
final double tol2 = 2 * tol1;
// Default stopping criterion.
final boolean stop = FastMath.abs(x - m) <= tol2 - 0.5 * (b - a);
if (!stop) {
double p = 0;
double q = 0;
double r = 0;
double u = 0;
if (FastMath.abs(e) > tol1) { // Fit parabola.
r = (x - w) * (fx - fv);
q = (x - v) * (fx - fw);
p = (x - v) * q - (x - w) * r;
q = 2 * (q - r);
if (q > 0) {
p = -p;
} else {
q = -q;
}
r = e;
e = d;
if (p > q * (a - x) && p < q * (b - x) && FastMath.abs(p) < FastMath.abs(0.5 * q * r)) {
// Parabolic interpolation step.
d = p / q;
u = x + d;
// f must not be evaluated too close to a or b.
if (u - a < tol2 || b - u < tol2) {
if (x <= m) {
d = tol1;
} else {
d = -tol1;
}
}
} else {
// Golden section step.
if (x < m) {
e = b - x;
} else {
e = a - x;
}
d = GOLDEN_SECTION * e;
}
} else {
// Golden section step.
if (x < m) {
e = b - x;
} else {
e = a - x;
}
d = GOLDEN_SECTION * e;
}
// Update by at least "tol1".
if (FastMath.abs(d) < tol1) {
if (d >= 0) {
u = x + tol1;
} else {
u = x - tol1;
}
} else {
u = x + d;
}
double fu = computeObjectiveValue(u);
if (!isMinim) {
fu = -fu;
}
// User-defined convergence checker.
previous = current;
current = new UnivariatePointValuePair(u, isMinim ? fu : -fu);
best = best(best, best(previous, current, isMinim), isMinim);
if (checker != null && checker.converged(getIterations(), previous, current)) {
return best;
}
// Update a, b, v, w and x.
if (fu <= fx) {
if (u < x) {
b = x;
} else {
a = x;
}
v = w;
fv = fw;
w = x;
fw = fx;
x = u;
fx = fu;
} else {
if (u < x) {
a = u;
} else {
b = u;
}
if (fu <= fw || Precision.equals(w, x)) {
v = w;
fv = fw;
w = u;
fw = fu;
} else if (fu <= fv || Precision.equals(v, x) || Precision.equals(v, w)) {
v = u;
fv = fu;
}
}
} else { // Default termination (Brent's criterion).
return best(best, best(previous, current, isMinim), isMinim);
}
incrementIterationCount();
}
}
/** {@inheritDoc} */
@Override
protected double doSolve() {
// prepare arrays with the first points
final double[] x = new double[maximalOrder + 1];
final double[] y = new double[maximalOrder + 1];
x[0] = getMin();
x[1] = getStartValue();
x[2] = getMax();
verifySequence(x[0], x[1], x[2]);
// evaluate initial guess
y[1] = computeObjectiveValue(x[1]);
if (Precision.equals(y[1], 0.0, 1)) {
// return the initial guess if it is a perfect root.
return x[1];
}
// evaluate first endpoint
y[0] = computeObjectiveValue(x[0]);
if (Precision.equals(y[0], 0.0, 1)) {
// return the first endpoint if it is a perfect root.
return x[0];
}
int nbPoints;
int signChangeIndex;
if (y[0] * y[1] < 0) {
// reduce interval if it brackets the root
nbPoints = 2;
signChangeIndex = 1;
} else {
// evaluate second endpoint
y[2] = computeObjectiveValue(x[2]);
if (Precision.equals(y[2], 0.0, 1)) {
// return the second endpoint if it is a perfect root.
return x[2];
}
if (y[1] * y[2] < 0) {
// use all computed point as a start sampling array for solving
nbPoints = 3;
signChangeIndex = 2;
} else {
throw new NoBracketingException(x[0], x[2], y[0], y[2]);
}
}
// prepare a work array for inverse polynomial interpolation
final double[] tmpX = new double[x.length];
// current tightest bracketing of the root
double xA = x[signChangeIndex - 1];
double yA = y[signChangeIndex - 1];
double absYA = FastMath.abs(yA);
int agingA = 0;
double xB = x[signChangeIndex];
double yB = y[signChangeIndex];
double absYB = FastMath.abs(yB);
int agingB = 0;
// search loop
while (true) {
// check convergence of bracketing interval
final double xTol =
getAbsoluteAccuracy()
+ getRelativeAccuracy() * FastMath.max(FastMath.abs(xA), FastMath.abs(xB));
if (((xB - xA) <= xTol) || (FastMath.max(absYA, absYB) < getFunctionValueAccuracy())) {
switch (allowed) {
case ANY_SIDE:
return absYA < absYB ? xA : xB;
case LEFT_SIDE:
return xA;
case RIGHT_SIDE:
return xB;
case BELOW_SIDE:
return (yA <= 0) ? xA : xB;
case ABOVE_SIDE:
return (yA < 0) ? xB : xA;
default:
// this should never happen
throw new MathInternalError(null);
}
}
// target for the next evaluation point
double targetY;
if (agingA >= MAXIMAL_AGING) {
// we keep updating the high bracket, try to compensate this
final int p = agingA - MAXIMAL_AGING;
final double weightA = (1 << p) - 1;
final double weightB = p + 1;
targetY = (weightA * yA - weightB * REDUCTION_FACTOR * yB) / (weightA + weightB);
} else if (agingB >= MAXIMAL_AGING) {
// we keep updating the low bracket, try to compensate this
final int p = agingB - MAXIMAL_AGING;
final double weightA = p + 1;
final double weightB = (1 << p) - 1;
targetY = (weightB * yB - weightA * REDUCTION_FACTOR * yA) / (weightA + weightB);
} else {
// bracketing is balanced, try to find the root itself
targetY = 0;
}
// make a few attempts to guess a root,
double nextX;
int start = 0;
int end = nbPoints;
do {
// guess a value for current target, using inverse polynomial interpolation
System.arraycopy(x, start, tmpX, start, end - start);
nextX = guessX(targetY, tmpX, y, start, end);
if (!((nextX > xA) && (nextX < xB))) {
// the guessed root is not strictly inside of the tightest bracketing interval
// the guessed root is either not strictly inside the interval or it
// is a NaN (which occurs when some sampling points share the same y)
// we try again with a lower interpolation order
if (signChangeIndex - start >= end - signChangeIndex) {
// we have more points before the sign change, drop the lowest point
++start;
} else {
// we have more points after sign change, drop the highest point
--end;
}
// we need to do one more attempt
nextX = Double.NaN;
}
} while (Double.isNaN(nextX) && (end - start > 1));
if (Double.isNaN(nextX)) {
// fall back to bisection
nextX = xA + 0.5 * (xB - xA);
start = signChangeIndex - 1;
end = signChangeIndex;
}
// evaluate the function at the guessed root
final double nextY = computeObjectiveValue(nextX);
if (Precision.equals(nextY, 0.0, 1)) {
// we have found an exact root, since it is not an approximation
// we don't need to bother about the allowed solutions setting
return nextX;
}
if ((nbPoints > 2) && (end - start != nbPoints)) {
// we have been forced to ignore some points to keep bracketing,
// they are probably too far from the root, drop them from now on
nbPoints = end - start;
System.arraycopy(x, start, x, 0, nbPoints);
System.arraycopy(y, start, y, 0, nbPoints);
signChangeIndex -= start;
} else if (nbPoints == x.length) {
// we have to drop one point in order to insert the new one
nbPoints--;
// keep the tightest bracketing interval as centered as possible
if (signChangeIndex >= (x.length + 1) / 2) {
// we drop the lowest point, we have to shift the arrays and the index
System.arraycopy(x, 1, x, 0, nbPoints);
System.arraycopy(y, 1, y, 0, nbPoints);
--signChangeIndex;
}
}
// insert the last computed point
// (by construction, we know it lies inside the tightest bracketing interval)
System.arraycopy(x, signChangeIndex, x, signChangeIndex + 1, nbPoints - signChangeIndex);
x[signChangeIndex] = nextX;
System.arraycopy(y, signChangeIndex, y, signChangeIndex + 1, nbPoints - signChangeIndex);
y[signChangeIndex] = nextY;
++nbPoints;
// update the bracketing interval
if (nextY * yA <= 0) {
// the sign change occurs before the inserted point
xB = nextX;
yB = nextY;
absYB = FastMath.abs(yB);
++agingA;
agingB = 0;
} else {
// the sign change occurs after the inserted point
xA = nextX;
yA = nextY;
absYA = FastMath.abs(yA);
agingA = 0;
++agingB;
// update the sign change index
signChangeIndex++;
}
}
}