@Test(expectedExceptions = IllegalArgumentException.class)
public void linearFuncDiffTwiceMultiTest() {
double[] xValues = new double[] {1, 2, 3, 4};
DoubleMatrix2D coefsMatrix =
new DoubleMatrix2D(
new double[][] {{1., -3.}, {0., 5.}, {1., 0.}, {0., 5.}, {1., 3.}, {0., 5.}});
double[] xKeys = new double[] {-2, 1, 2, 2.5};
final int dim = 2;
final int nCoefs = 2;
PiecewisePolynomialResult pp =
new PiecewisePolynomialResult(new DoubleMatrix1D(xValues), coefsMatrix, nCoefs, dim);
PiecewisePolynomialFunction1D function = new PiecewisePolynomialFunction1D();
function.differentiateTwice(pp, xKeys);
}
/** Sample function is f(x) = (x-1)^4 */
@Test
public void GeneralIntegrateDifferentiateTest() {
final double[] knots = new double[] {1., 2., 3., 4};
final double[][] coefMat =
new double[][] {{1., 0., 0., 0., 0.}, {1., 4., 6., 4., 1.}, {1., 8., 24., 32., 16.}};
final double[] xKeys = new double[] {-2, 1, 2.5, 4.};
final double[] initials = new double[] {1., 2.5, 23. / 7., 7.};
final int nKeys = xKeys.length;
final int nInit = initials.length;
final double[][] integrateExp = new double[nInit][nKeys];
for (int i = 0; i < nInit; ++i) {
for (int j = 0; j < nKeys; ++j) {
integrateExp[i][j] = Math.pow(xKeys[j] - 1., 5.) / 5. - Math.pow(initials[i] - 1., 5.) / 5.;
}
}
final double[] differentiateExp = new double[] {-108., 0., 27. / 2., 108.};
final double[] differentiateTwiceExp = new double[nKeys];
for (int i = 0; i < nKeys; ++i) {
differentiateTwiceExp[i] = 12. * (xKeys[i] - 1.) * (xKeys[i] - 1.);
}
PiecewisePolynomialFunction1D function = new PiecewisePolynomialFunction1D();
PiecewisePolynomialResult result =
new PiecewisePolynomialResult(new DoubleMatrix1D(knots), new DoubleMatrix2D(coefMat), 5, 1);
final double[] differentiate = function.differentiate(result, xKeys).getData()[0];
final double[] differentiateTwice = function.differentiateTwice(result, xKeys).getData()[0];
final double[][] integrate = new double[nInit][nKeys];
for (int i = 0; i < nInit; ++i) {
for (int j = 0; j < nKeys; ++j) {
integrate[i][j] = function.integrate(result, initials[i], xKeys).getData()[j];
}
}
for (int i = 0; i < nKeys; ++i) {
final double ref = differentiateExp[i] == 0. ? 1. : Math.abs(differentiateExp[i]);
assertEquals(differentiate[i], differentiateExp[i], ref * EPS);
}
for (int i = 0; i < nKeys; ++i) {
final double ref = differentiateTwiceExp[i] == 0. ? 1. : Math.abs(differentiateTwiceExp[i]);
assertEquals(differentiateTwice[i], differentiateTwiceExp[i], ref * EPS);
}
for (int j = 0; j < nInit; ++j) {
for (int i = 0; i < nKeys; ++i) {
final double ref = integrateExp[j][i] == 0. ? 1. : Math.abs(integrateExp[j][i]);
assertEquals(integrate[j][i], integrateExp[j][i], ref * EPS);
}
}
{
final double ref = differentiateExp[0] == 0. ? 1. : Math.abs(differentiateExp[0]);
assertEquals(
function.differentiate(result, xKeys[0]).getData()[0], differentiateExp[0], ref * EPS);
}
{
final double ref = differentiateExp[3] == 0. ? 1. : Math.abs(differentiateExp[3]);
assertEquals(
function.differentiate(result, xKeys[3]).getData()[0], differentiateExp[3], ref * EPS);
}
{
final double ref = differentiateTwiceExp[0] == 0. ? 1. : Math.abs(differentiateTwiceExp[0]);
assertEquals(
function.differentiateTwice(result, xKeys[0]).getData()[0],
differentiateTwiceExp[0],
ref * EPS);
}
{
final double ref = differentiateTwiceExp[3] == 0. ? 1. : Math.abs(differentiateTwiceExp[3]);
assertEquals(
function.differentiateTwice(result, xKeys[3]).getData()[0],
differentiateTwiceExp[3],
ref * EPS);
}
{
final double ref = integrateExp[0][0] == 0. ? 1. : Math.abs(integrateExp[0][0]);
assertEquals(
function.integrate(result, initials[0], xKeys[0]), integrateExp[0][0], ref * EPS);
}
{
final double ref = integrateExp[0][3] == 0. ? 1. : Math.abs(integrateExp[0][3]);
assertEquals(
function.integrate(result, initials[0], xKeys[3]), integrateExp[0][3], ref * EPS);
}
{
final double ref = integrateExp[3][0] == 0. ? 1. : Math.abs(integrateExp[3][0]);
assertEquals(
function.integrate(result, initials[3], xKeys[0]), integrateExp[3][0], ref * EPS);
}
{
final double ref = integrateExp[1][0] == 0. ? 1. : Math.abs(integrateExp[1][0]);
assertEquals(
function.integrate(result, initials[1], xKeys[0]), integrateExp[1][0], ref * EPS);
}
}
@Override
public PiecewisePolynomialResult interpolate(final double[] xValues, final double[] yValues) {
ArgumentChecker.notNull(xValues, "xValues");
ArgumentChecker.notNull(yValues, "yValues");
ArgumentChecker.isTrue(
xValues.length == yValues.length | xValues.length + 2 == yValues.length,
"(xValues length = yValues length) or (xValues length + 2 = yValues length)");
ArgumentChecker.isTrue(xValues.length > 2, "Data points should be more than 2");
final int nDataPts = xValues.length;
final int yValuesLen = yValues.length;
for (int i = 0; i < nDataPts; ++i) {
ArgumentChecker.isFalse(Double.isNaN(xValues[i]), "xValues containing NaN");
ArgumentChecker.isFalse(Double.isInfinite(xValues[i]), "xValues containing Infinity");
}
for (int i = 0; i < yValuesLen; ++i) {
ArgumentChecker.isFalse(Double.isNaN(yValues[i]), "yValues containing NaN");
ArgumentChecker.isFalse(Double.isInfinite(yValues[i]), "yValues containing Infinity");
}
for (int i = 0; i < nDataPts - 1; ++i) {
for (int j = i + 1; j < nDataPts; ++j) {
ArgumentChecker.isFalse(xValues[i] == xValues[j], "xValues should be distinct");
}
}
double[] xValuesSrt = Arrays.copyOf(xValues, nDataPts);
double[] yValuesSrt = new double[nDataPts];
if (nDataPts == yValuesLen) {
yValuesSrt = Arrays.copyOf(yValues, nDataPts);
} else {
yValuesSrt = Arrays.copyOfRange(yValues, 1, nDataPts + 1);
}
ParallelArrayBinarySort.parallelBinarySort(xValuesSrt, yValuesSrt);
final double[] intervals = _solver.intervalsCalculator(xValuesSrt);
final double[] slopes = _solver.slopesCalculator(yValuesSrt, intervals);
final PiecewisePolynomialResult result = _method.interpolate(xValues, yValues);
ArgumentChecker.isTrue(result.getOrder() >= 3, "Primary interpolant should be degree >= 2");
final double[] initialFirst = _function.differentiate(result, xValuesSrt).getData()[0];
final double[] initialSecond = _function.differentiateTwice(result, xValuesSrt).getData()[0];
double[] first = firstDerivativeCalculator(yValuesSrt, intervals, slopes, initialFirst);
boolean modFirst = false;
int k;
double[] aValues = aValuesCalculator(slopes, first);
double[] bValues = bValuesCalculator(slopes, first);
double[][] intervalsA = getIntervalsA(intervals, slopes, first, bValues);
double[][] intervalsB = getIntervalsB(intervals, slopes, first, aValues);
while (modFirst == false) {
k = 0;
for (int i = 0; i < nDataPts - 2; ++i) {
if (first[i + 1] > 0.) {
if (intervalsA[i + 1][1] + Math.abs(intervalsA[i + 1][1]) * ERROR
< intervalsB[i][0] - Math.abs(intervalsB[i][0]) * ERROR
| intervalsA[i + 1][0] - Math.abs(intervalsA[i + 1][0]) * ERROR
> intervalsB[i][1] + Math.abs(intervalsB[i][1]) * ERROR) {
++k;
first[i + 1] = firstDerivativesRecalculator(intervals, slopes, aValues, bValues, i + 1);
}
}
}
if (k == 0) {
modFirst = true;
}
aValues = aValuesCalculator(slopes, first);
bValues = bValuesCalculator(slopes, first);
intervalsA = getIntervalsA(intervals, slopes, first, bValues);
intervalsB = getIntervalsB(intervals, slopes, first, aValues);
}
final double[] second = secondDerivativeCalculator(initialSecond, intervalsA, intervalsB);
final double[][] coefs = _solver.solve(yValuesSrt, intervals, slopes, first, second);
for (int i = 0; i < nDataPts - 1; ++i) {
for (int j = 0; j < 6; ++j) {
ArgumentChecker.isFalse(Double.isNaN(coefs[i][j]), "Too large input");
ArgumentChecker.isFalse(Double.isInfinite(coefs[i][j]), "Too large input");
}
}
return new PiecewisePolynomialResult(
new DoubleMatrix1D(xValuesSrt), new DoubleMatrix2D(coefs), 6, 1);
}
@Test
public void quadraticAllTest() {
final double[] knots = new double[] {1., 3.};
final DoubleMatrix2D coefsMatrix = new DoubleMatrix2D(new double[][] {{-1., 2., 1.}});
final double[] xKeys = new double[] {-2, 1, 2.5, 4.};
final double[] initials = new double[] {-0.5, 1., 2.5, 5.};
final int nKeys = xKeys.length;
final int nInit = initials.length;
final double[] valuesExp = new double[] {-14., 1., 7. / 4., -2.};
final double[][] integrateExp = new double[nInit][nKeys];
for (int i = 0; i < nInit; ++i) {
for (int j = 0; j < nKeys; ++j) {
integrateExp[i][j] =
-1.
/ 3.
* (xKeys[j] - initials[i])
* (xKeys[j] * xKeys[j]
+ initials[i] * initials[i]
- 6. * xKeys[j]
- 6. * initials[i]
+ 6.
+ initials[i] * xKeys[j]);
}
}
final double[] differentiateExp = new double[nKeys];
for (int j = 0; j < nKeys; ++j) {
differentiateExp[j] = -2. * (xKeys[j] - 1) + 2.;
}
final double[] differentiateTwiceExp = new double[nKeys];
for (int j = 0; j < nKeys; ++j) {
differentiateTwiceExp[j] = -2.;
}
PiecewisePolynomialResult result =
new PiecewisePolynomialResult(new DoubleMatrix1D(knots), coefsMatrix, 3, 1);
PiecewisePolynomialFunction1D function = new PiecewisePolynomialFunction1D();
final double[] values = function.evaluate(result, xKeys).getData()[0];
final double[] differentiate = function.differentiate(result, xKeys).getData()[0];
final double[] differentiateTwice = function.differentiateTwice(result, xKeys).getData()[0];
final double[][] integrate = new double[nInit][nKeys];
for (int i = 0; i < nInit; ++i) {
for (int j = 0; j < nKeys; ++j) {
integrate[i][j] = function.integrate(result, initials[i], xKeys).getData()[j];
}
}
for (int i = 0; i < nKeys; ++i) {
final double ref = valuesExp[i] == 0. ? 1. : Math.abs(valuesExp[i]);
assertEquals(values[i], valuesExp[i], ref * EPS);
}
for (int i = 0; i < nKeys; ++i) {
final double ref = differentiateExp[i] == 0. ? 1. : Math.abs(differentiateExp[i]);
assertEquals(differentiate[i], differentiateExp[i], ref * EPS);
}
for (int i = 0; i < nKeys; ++i) {
final double ref = differentiateTwiceExp[i] == 0. ? 1. : Math.abs(differentiateTwiceExp[i]);
assertEquals(differentiateTwice[i], differentiateTwiceExp[i], ref * EPS);
}
{
final double ref = differentiateTwiceExp[1] == 0. ? 1. : Math.abs(differentiateTwiceExp[1]);
assertEquals(differentiateTwice[1], differentiateTwiceExp[1], ref * EPS);
}
for (int j = 0; j < nInit; ++j) {
for (int i = 0; i < nKeys; ++i) {
final double ref = integrateExp[j][i] == 0. ? 1. : Math.abs(integrateExp[j][i]);
assertEquals(integrate[j][i], integrateExp[j][i], ref * EPS);
}
}
}
@Override
public PiecewisePolynomialResultsWithSensitivity interpolateWithSensitivity(
final double[] xValues, final double[] yValues) {
ArgumentChecker.notNull(xValues, "xValues");
ArgumentChecker.notNull(yValues, "yValues");
ArgumentChecker.isTrue(
xValues.length == yValues.length | xValues.length + 2 == yValues.length,
"(xValues length = yValues length) or (xValues length + 2 = yValues length)");
ArgumentChecker.isTrue(xValues.length > 2, "Data points should be more than 2");
final int nDataPts = xValues.length;
final int yValuesLen = yValues.length;
for (int i = 0; i < nDataPts; ++i) {
ArgumentChecker.isFalse(Double.isNaN(xValues[i]), "xValues containing NaN");
ArgumentChecker.isFalse(Double.isInfinite(xValues[i]), "xValues containing Infinity");
}
for (int i = 0; i < yValuesLen; ++i) {
ArgumentChecker.isFalse(Double.isNaN(yValues[i]), "yValues containing NaN");
ArgumentChecker.isFalse(Double.isInfinite(yValues[i]), "yValues containing Infinity");
}
for (int i = 0; i < nDataPts - 1; ++i) {
for (int j = i + 1; j < nDataPts; ++j) {
ArgumentChecker.isFalse(xValues[i] == xValues[j], "xValues should be distinct");
}
}
double[] yValuesSrt = new double[nDataPts];
if (nDataPts == yValuesLen) {
yValuesSrt = Arrays.copyOf(yValues, nDataPts);
} else {
yValuesSrt = Arrays.copyOfRange(yValues, 1, nDataPts + 1);
}
final double[] intervals = _solver.intervalsCalculator(xValues);
final double[] slopes = _solver.slopesCalculator(yValuesSrt, intervals);
double[][] slopesSensitivity = _solver.slopeSensitivityCalculator(intervals);
final DoubleMatrix1D[] firstWithSensitivity = new DoubleMatrix1D[nDataPts + 1];
final DoubleMatrix1D[] secondWithSensitivity = new DoubleMatrix1D[nDataPts + 1];
final PiecewisePolynomialResult result = _method.interpolate(xValues, yValues);
ArgumentChecker.isTrue(result.getOrder() >= 3, "Primary interpolant should be degree >= 2");
final double[] initialFirst = _function.differentiate(result, xValues).getData()[0];
final double[] initialSecond = _function.differentiateTwice(result, xValues).getData()[0];
double[] first = firstDerivativeCalculator(yValuesSrt, intervals, slopes, initialFirst);
boolean modFirst = false;
int k;
double[] aValues = aValuesCalculator(slopes, first);
double[] bValues = bValuesCalculator(slopes, first);
double[][] intervalsA = getIntervalsA(intervals, slopes, first, bValues);
double[][] intervalsB = getIntervalsB(intervals, slopes, first, aValues);
while (modFirst == false) {
k = 0;
for (int i = 0; i < nDataPts - 2; ++i) {
if (first[i + 1] > 0.) {
if (intervalsA[i + 1][1] + Math.abs(intervalsA[i + 1][1]) * ERROR
< intervalsB[i][0] - Math.abs(intervalsB[i][0]) * ERROR
| intervalsA[i + 1][0] - Math.abs(intervalsA[i + 1][0]) * ERROR
> intervalsB[i][1] + Math.abs(intervalsB[i][1]) * ERROR) {
++k;
first[i + 1] = firstDerivativesRecalculator(intervals, slopes, aValues, bValues, i + 1);
}
}
}
if (k == 0) {
modFirst = true;
}
aValues = aValuesCalculator(slopes, first);
bValues = bValuesCalculator(slopes, first);
intervalsA = getIntervalsA(intervals, slopes, first, bValues);
intervalsB = getIntervalsB(intervals, slopes, first, aValues);
}
final double[] second = secondDerivativeCalculator(initialSecond, intervalsA, intervalsB);
firstWithSensitivity[0] = new DoubleMatrix1D(first);
secondWithSensitivity[0] = new DoubleMatrix1D(second);
/*
* Centered finite difference method is used for computing node sensitivity
*/
int nExtra = (nDataPts == yValuesLen) ? 0 : 1;
final double[] yValuesUp = Arrays.copyOf(yValues, nDataPts + 2 * nExtra);
final double[] yValuesDw = Arrays.copyOf(yValues, nDataPts + 2 * nExtra);
final double[][] tmpFirst = new double[nDataPts][nDataPts];
final double[][] tmpSecond = new double[nDataPts][nDataPts];
for (int l = nExtra; l < nDataPts + nExtra; ++l) {
final double den = Math.abs(yValues[l]) < SMALL ? EPS : yValues[l] * EPS;
yValuesUp[l] = Math.abs(yValues[l]) < SMALL ? EPS : yValues[l] * (1. + EPS);
yValuesDw[l] = Math.abs(yValues[l]) < SMALL ? -EPS : yValues[l] * (1. - EPS);
final double[] yValuesSrtUp = Arrays.copyOfRange(yValuesUp, nExtra, nDataPts + nExtra);
final double[] yValuesSrtDw = Arrays.copyOfRange(yValuesDw, nExtra, nDataPts + nExtra);
final DoubleMatrix1D[] yValuesUpDw =
new DoubleMatrix1D[] {new DoubleMatrix1D(yValuesUp), new DoubleMatrix1D(yValuesDw)};
final DoubleMatrix1D[] yValuesSrtUpDw =
new DoubleMatrix1D[] {new DoubleMatrix1D(yValuesSrtUp), new DoubleMatrix1D(yValuesSrtDw)};
final DoubleMatrix1D[] firstSecondUpDw = new DoubleMatrix1D[4];
for (int ii = 0; ii < 2; ++ii) {
final double[] slopesUpDw =
_solver.slopesCalculator(yValuesSrtUpDw[ii].getData(), intervals);
final PiecewisePolynomialResult resultUpDw =
_method.interpolate(xValues, yValuesUpDw[ii].getData());
final double[] initialFirstUpDw = _function.differentiate(resultUpDw, xValues).getData()[0];
final double[] initialSecondUpDw =
_function.differentiateTwice(resultUpDw, xValues).getData()[0];
double[] firstUpDw =
firstDerivativeCalculator(
yValuesSrtUpDw[ii].getData(), intervals, slopesUpDw, initialFirstUpDw);
boolean modFirstUpDw = false;
double[] aValuesUpDw = aValuesCalculator(slopesUpDw, firstUpDw);
double[] bValuesUpDw = bValuesCalculator(slopesUpDw, firstUpDw);
double[][] intervalsAUpDw = getIntervalsA(intervals, slopesUpDw, firstUpDw, bValuesUpDw);
double[][] intervalsBUpDw = getIntervalsB(intervals, slopesUpDw, firstUpDw, aValuesUpDw);
while (modFirstUpDw == false) {
k = 0;
for (int i = 0; i < nDataPts - 2; ++i) {
if (firstUpDw[i + 1] > 0.) {
if (intervalsAUpDw[i + 1][1] + Math.abs(intervalsAUpDw[i + 1][1]) * ERROR
< intervalsBUpDw[i][0] - Math.abs(intervalsBUpDw[i][0]) * ERROR
| intervalsAUpDw[i + 1][0] - Math.abs(intervalsAUpDw[i + 1][0]) * ERROR
> intervalsBUpDw[i][1] + Math.abs(intervalsBUpDw[i][1]) * ERROR) {
++k;
firstUpDw[i + 1] =
firstDerivativesRecalculator(
intervals, slopesUpDw, aValuesUpDw, bValuesUpDw, i + 1);
}
}
}
if (k == 0) {
modFirstUpDw = true;
}
aValuesUpDw = aValuesCalculator(slopesUpDw, firstUpDw);
bValuesUpDw = bValuesCalculator(slopesUpDw, firstUpDw);
intervalsAUpDw = getIntervalsA(intervals, slopesUpDw, firstUpDw, bValuesUpDw);
intervalsBUpDw = getIntervalsB(intervals, slopesUpDw, firstUpDw, aValuesUpDw);
}
final double[] secondUpDw =
secondDerivativeCalculator(initialSecondUpDw, intervalsAUpDw, intervalsBUpDw);
firstSecondUpDw[ii] = new DoubleMatrix1D(firstUpDw);
firstSecondUpDw[2 + ii] = new DoubleMatrix1D(secondUpDw);
}
for (int j = 0; j < nDataPts; ++j) {
tmpFirst[j][l - nExtra] =
0.5 * (firstSecondUpDw[0].getData()[j] - firstSecondUpDw[1].getData()[j]) / den;
tmpSecond[j][l - nExtra] =
0.5 * (firstSecondUpDw[2].getData()[j] - firstSecondUpDw[3].getData()[j]) / den;
}
yValuesUp[l] = yValues[l];
yValuesDw[l] = yValues[l];
}
for (int i = 0; i < nDataPts; ++i) {
firstWithSensitivity[i + 1] = new DoubleMatrix1D(tmpFirst[i]);
secondWithSensitivity[i + 1] = new DoubleMatrix1D(tmpSecond[i]);
}
final DoubleMatrix2D[] resMatrix =
_solver.solveWithSensitivity(
yValuesSrt,
intervals,
slopes,
slopesSensitivity,
firstWithSensitivity,
secondWithSensitivity);
for (int l = 0; l < nDataPts; ++l) {
DoubleMatrix2D m = resMatrix[l];
final int rows = m.getNumberOfRows();
final int cols = m.getNumberOfColumns();
for (int i = 0; i < rows; ++i) {
for (int j = 0; j < cols; ++j) {
ArgumentChecker.isTrue(
Doubles.isFinite(m.getEntry(i, j)), "Matrix contains a NaN or infinite");
}
}
}
final DoubleMatrix2D coefMatrix = resMatrix[0];
final DoubleMatrix2D[] coefSenseMatrix = new DoubleMatrix2D[nDataPts - 1];
System.arraycopy(resMatrix, 1, coefSenseMatrix, 0, nDataPts - 1);
final int nCoefs = coefMatrix.getNumberOfColumns();
return new PiecewisePolynomialResultsWithSensitivity(
new DoubleMatrix1D(xValues), coefMatrix, nCoefs, 1, coefSenseMatrix);
}