Completed support fo asin, acos and atan in DSCompiler.
git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1371805 13f79535-47bb-0310-9956-ffa450edef68
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Luc Maisonobe
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@ -1097,10 +1097,39 @@ public class DSCompiler {
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final double x = operand[operandOffset];
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function[0] = FastMath.acos(x);
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if (order > 0) {
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function[1] = -1.0 / FastMath.sqrt(1 - x * x);
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for (int i = 2; i <= order; ++i) {
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// TODO compute higher order derivatives
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function[i] = Double.NaN;
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// the nth order derivative of acos has the form:
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// dn(acos(x)/dxn = P_n(x) / [1 - x^2]^((2n-1)/2)
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// where P_n(x) is a degree n-1 polynomial with same parity as n-1
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// P_1(x) = -1, P_2(x) = -x, P_3(x) = -2x^2 - 1 ...
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// the general recurrence relation for P_n is:
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// P_n(x) = (1-x^2) P_(n-1)'(x) + (2n-3) x P_(n-1)(x)
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// as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
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final double[] p = new double[order];
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p[0] = -1;
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final double x2 = x * x;
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final double f = 1.0 / (1 - x2);
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double coeff = FastMath.sqrt(f);
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function[1] = coeff * p[0];
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for (int n = 2; n <= order; ++n) {
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// update and evaluate polynomial P_n(x)
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double v = 0;
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p[n - 1] = (n - 1) * p[n - 2];
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for (int k = n - 1; k >= 0; k -= 2) {
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v = v * x2 + p[k];
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if (k > 2) {
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p[k - 2] = (k - 1) * p[k - 1] + (2 * n - k) * p[k - 3];
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} else if (k == 2) {
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p[0] = p[1];
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}
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}
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if ((n & 0x1) == 0) {
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v *= x;
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}
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coeff *= f;
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function[n] = coeff * v;
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}
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}
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@ -1125,10 +1154,39 @@ public class DSCompiler {
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final double x = operand[operandOffset];
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function[0] = FastMath.asin(x);
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if (order > 0) {
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function[1] = 1.0 / FastMath.sqrt(1 - x * x);
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for (int i = 2; i <= order; ++i) {
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// TODO compute higher order derivatives
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function[i] = Double.NaN;
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// the nth order derivative of asin has the form:
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// dn(asin(x)/dxn = P_n(x) / [1 - x^2]^((2n-1)/2)
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// where P_n(x) is a degree n-1 polynomial with same parity as n-1
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// P_1(x) = 1, P_2(x) = x, P_3(x) = 2x^2 + 1 ...
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// the general recurrence relation for P_n is:
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// P_n(x) = (1-x^2) P_(n-1)'(x) + (2n-3) x P_(n-1)(x)
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// as per polynomial parity, we can store coefficients of both P_(n-1) and P_n in the same array
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final double[] p = new double[order];
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p[0] = 1;
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final double x2 = x * x;
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final double f = 1.0 / (1 - x2);
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double coeff = FastMath.sqrt(f);
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function[1] = coeff * p[0];
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for (int n = 2; n <= order; ++n) {
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// update and evaluate polynomial P_n(x)
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double v = 0;
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p[n - 1] = (n - 1) * p[n - 2];
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for (int k = n - 1; k >= 0; k -= 2) {
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v = v * x2 + p[k];
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if (k > 2) {
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p[k - 2] = (k - 1) * p[k - 1] + (2 * n - k) * p[k - 3];
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} else if (k == 2) {
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p[0] = p[1];
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}
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}
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if ((n & 0x1) == 0) {
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v *= x;
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}
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coeff *= f;
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function[n] = coeff * v;
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}
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}
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@ -1153,10 +1211,39 @@ public class DSCompiler {
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final double x = operand[operandOffset];
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function[0] = FastMath.atan(x);
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if (order > 0) {
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function[1] = 1.0 / (1 + x * x);
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for (int i = 2; i <= order; ++i) {
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// TODO compute higher order derivatives
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function[i] = Double.NaN;
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// the nth order derivative of atan has the form:
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// dn(atan(x)/dxn = Q_n(x) / (1 + x^2)^n
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// where Q_n(x) is a degree n-1 polynomial with same parity as n-1
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// Q_1(x) = 1, Q_2(x) = -2x, Q_3(x) = 6x^2 - 2 ...
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// the general recurrence relation for Q_n is:
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// Q_n(x) = (1+x^2) Q_(n-1)'(x) - 2(n-1) x Q_(n-1)(x)
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// as per polynomial parity, we can store coefficients of both Q_(n-1) and Q_n in the same array
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final double[] q = new double[order];
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q[0] = 1;
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final double x2 = x * x;
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final double f = 1.0 / (1 + x2);
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double coeff = f;
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function[1] = coeff * q[0];
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for (int n = 2; n <= order; ++n) {
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// update and evaluate polynomial Q_n(x)
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double v = 0;
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q[n - 1] = -n * q[n - 2];
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for (int k = n - 1; k >= 0; k -= 2) {
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v = v * x2 + q[k];
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if (k > 2) {
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q[k - 2] = (k - 1) * q[k - 1] + (k - 1 - 2 * n) * q[k - 3];
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} else if (k == 2) {
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q[0] = q[1];
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}
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}
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if ((n & 0x1) == 0) {
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v *= x;
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}
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coeff *= f;
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function[n] = coeff * v;
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}
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}
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@ -459,6 +459,51 @@ public class DerivativeStructureTest {
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}
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}
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@Test
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public void testSinAsin() {
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double[] epsilon = new double[] { 3.0e-16, 5.0e-16, 3.0e-15, 2.0e-14, 4.0e-13 };
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for (int maxOrder = 0; maxOrder < 5; ++maxOrder) {
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for (double x = 0.1; x < 1.2; x += 0.001) {
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DerivativeStructure dsX = new DerivativeStructure(1, maxOrder, 0, x);
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DerivativeStructure rebuiltX = dsX.sin().asin();
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DerivativeStructure zero = rebuiltX.subtract(dsX);
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for (int n = 0; n <= maxOrder; ++n) {
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Assert.assertEquals(0.0, zero.getPartialDerivative(n), epsilon[n]);
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}
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}
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}
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}
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@Test
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public void testCosAcos() {
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double[] epsilon = new double[] { 6.0e-16, 6.0e-15, 2.0e-13, 4.0e-12, 2.0e-10 };
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for (int maxOrder = 0; maxOrder < 5; ++maxOrder) {
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for (double x = 0.1; x < 1.2; x += 0.001) {
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DerivativeStructure dsX = new DerivativeStructure(1, maxOrder, 0, x);
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DerivativeStructure rebuiltX = dsX.cos().acos();
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DerivativeStructure zero = rebuiltX.subtract(dsX);
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for (int n = 0; n <= maxOrder; ++n) {
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Assert.assertEquals(0.0, zero.getPartialDerivative(n), epsilon[n]);
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}
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}
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}
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}
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@Test
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public void testTanAtan() {
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double[] epsilon = new double[] { 6.0e-17, 2.0e-16, 2.0e-15, 4.0e-14, 2.0e-12 };
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for (int maxOrder = 0; maxOrder < 5; ++maxOrder) {
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for (double x = 0.1; x < 1.2; x += 0.001) {
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DerivativeStructure dsX = new DerivativeStructure(1, maxOrder, 0, x);
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DerivativeStructure rebuiltX = dsX.tan().atan();
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DerivativeStructure zero = rebuiltX.subtract(dsX);
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for (int n = 0; n <= maxOrder; ++n) {
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Assert.assertEquals(0.0, zero.getPartialDerivative(n), epsilon[n]);
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}
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}
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}
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}
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@Test
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public void testTangentDefinition() {
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double[] epsilon = new double[] { 5.0e-16, 2.0e-15, 3.0e-14, 5.0e-13, 2.0e-11 };
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