org.apache.commons.math4.analysis.differentiation

## Class DSCompiler

• public class DSCompiler
extends Object
Class holding "compiled" computation rules for derivative structures.

This class implements the computation rules described in Dan Kalman's paper Doubly Recursive Multivariate Automatic Differentiation, Mathematics Magazine, vol. 75, no. 3, June 2002. However, in order to avoid performances bottlenecks, the recursive rules are "compiled" once in an unfold form. This class does this recursion unrolling and stores the computation rules as simple loops with pre-computed indirection arrays.

This class maps all derivative computation into single dimension arrays that hold the value and partial derivatives. The class does not hold these arrays, which remains under the responsibility of the caller. For each combination of number of free parameters and derivation order, only one compiler is necessary, and this compiler will be used to perform computations on all arrays provided to it, which can represent hundreds or thousands of different parameters kept together with all their partial derivatives.

The arrays on which compilers operate contain only the partial derivatives together with the 0th derivative, i.e. the value. The partial derivatives are stored in a compiler-specific order, which can be retrieved using methods getPartialDerivativeIndex and getPartialDerivativeOrders(int). The value is guaranteed to be stored as the first element (i.e. the getPartialDerivativeIndex method returns 0 when called with 0 for all derivation orders and getPartialDerivativeOrders returns an array filled with 0 when called with 0 as the index).

Note that the ordering changes with number of parameters and derivation order. For example given 2 parameters x and y, df/dy is stored at index 2 when derivation order is set to 1 (in this case the array has three elements: f, df/dx and df/dy). If derivation order is set to 2, then df/dy will be stored at index 3 (in this case the array has six elements: f, df/dx, df/dxdx, df/dy, df/dxdy and df/dydy).

Given this structure, users can perform some simple operations like adding, subtracting or multiplying constants and negating the elements by themselves, knowing if they want to mutate their array or create a new array. These simple operations are not provided by the compiler. The compiler provides only the more complex operations between several arrays.

This class is mainly used as the engine for scalar variable DerivativeStructure. It can also be used directly to hold several variables in arrays for more complex data structures. User can for example store a vector of n variables depending on three x, y and z free parameters in one array as follows:

   // parameter 0 is x, parameter 1 is y, parameter 2 is z
int parameters = 3;
DSCompiler compiler = DSCompiler.getCompiler(parameters, order);
int size = compiler.getSize();

// pack all elements in a single array
double[] array = new double[n * size];
for (int i = 0; i < n; ++i) {

// we know value is guaranteed to be the first element
array[i * size] = v[i];

// we don't know where first derivatives are stored, so we ask the compiler
array[i * size + compiler.getPartialDerivativeIndex(1, 0, 0) = dvOnDx[i][0];
array[i * size + compiler.getPartialDerivativeIndex(0, 1, 0) = dvOnDy[i][0];
array[i * size + compiler.getPartialDerivativeIndex(0, 0, 1) = dvOnDz[i][0];

// we let all higher order derivatives set to 0

}


Then in another function, user can perform some operations on all elements stored in the single array, such as a simple product of all variables:

   // compute the product of all elements
double[] product = new double[size];
prod[0] = 1.0;
for (int i = 0; i < n; ++i) {
double[] tmp = product.clone();
compiler.multiply(tmp, 0, array, i * size, product, 0);
}

// value
double p = product[0];

// first derivatives
double dPdX = product[compiler.getPartialDerivativeIndex(1, 0, 0)];
double dPdY = product[compiler.getPartialDerivativeIndex(0, 1, 0)];
double dPdZ = product[compiler.getPartialDerivativeIndex(0, 0, 1)];

// cross derivatives (assuming order was at least 2)
double dPdXdX = product[compiler.getPartialDerivativeIndex(2, 0, 0)];
double dPdXdY = product[compiler.getPartialDerivativeIndex(1, 1, 0)];
double dPdXdZ = product[compiler.getPartialDerivativeIndex(1, 0, 1)];
double dPdYdY = product[compiler.getPartialDerivativeIndex(0, 2, 0)];
double dPdYdZ = product[compiler.getPartialDerivativeIndex(0, 1, 1)];
double dPdZdZ = product[compiler.getPartialDerivativeIndex(0, 0, 2)];

Since:
3.1
DerivativeStructure
• ### Method Summary

All Methods
Modifier and Type Method and Description
void acos(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute arc cosine of a derivative structure.
void acosh(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute inverse hyperbolic cosine of a derivative structure.
void add(double[] lhs, int lhsOffset, double[] rhs, int rhsOffset, double[] result, int resultOffset)
Perform addition of two derivative structures.
void asin(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute arc sine of a derivative structure.
void asinh(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute inverse hyperbolic sine of a derivative structure.
void atan(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute arc tangent of a derivative structure.
void atan2(double[] y, int yOffset, double[] x, int xOffset, double[] result, int resultOffset)
Compute two arguments arc tangent of a derivative structure.
void atanh(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute inverse hyperbolic tangent of a derivative structure.
void checkCompatibility(DSCompiler compiler)
Check rules set compatibility.
void compose(double[] operand, int operandOffset, double[] f, double[] result, int resultOffset)
Compute composition of a derivative structure by a function.
void cos(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute cosine of a derivative structure.
void cosh(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute hyperbolic cosine of a derivative structure.
void divide(double[] lhs, int lhsOffset, double[] rhs, int rhsOffset, double[] result, int resultOffset)
Perform division of two derivative structures.
void exp(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute exponential of a derivative structure.
void expm1(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute exp(x) - 1 of a derivative structure.
static DSCompiler getCompiler(int parameters, int order)
Get the compiler for number of free parameters and order.
int getFreeParameters()
Get the number of free parameters.
int getOrder()
Get the derivation order.
int getPartialDerivativeIndex(int... orders)
Get the index of a partial derivative in the array.
int[] getPartialDerivativeOrders(int index)
Get the derivation orders for a specific index in the array.
int getSize()
Get the array size required for holding partial derivatives data.
void linearCombination(double a1, double[] c1, int offset1, double a2, double[] c2, int offset2, double[] result, int resultOffset)
Compute linear combination.
void linearCombination(double a1, double[] c1, int offset1, double a2, double[] c2, int offset2, double a3, double[] c3, int offset3, double[] result, int resultOffset)
Compute linear combination.
void linearCombination(double a1, double[] c1, int offset1, double a2, double[] c2, int offset2, double a3, double[] c3, int offset3, double a4, double[] c4, int offset4, double[] result, int resultOffset)
Compute linear combination.
void log(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute natural logarithm of a derivative structure.
void log10(double[] operand, int operandOffset, double[] result, int resultOffset)
Computes base 10 logarithm of a derivative structure.
void log1p(double[] operand, int operandOffset, double[] result, int resultOffset)
Computes shifted logarithm of a derivative structure.
void multiply(double[] lhs, int lhsOffset, double[] rhs, int rhsOffset, double[] result, int resultOffset)
Perform multiplication of two derivative structures.
void pow(double[] x, int xOffset, double[] y, int yOffset, double[] result, int resultOffset)
Compute power of a derivative structure.
void pow(double[] operand, int operandOffset, double p, double[] result, int resultOffset)
Compute power of a derivative structure.
void pow(double[] operand, int operandOffset, int n, double[] result, int resultOffset)
Compute integer power of a derivative structure.
void pow(double a, double[] operand, int operandOffset, double[] result, int resultOffset)
Compute power of a double to a derivative structure.
void remainder(double[] lhs, int lhsOffset, double[] rhs, int rhsOffset, double[] result, int resultOffset)
Perform remainder of two derivative structures.
void rootN(double[] operand, int operandOffset, int n, double[] result, int resultOffset)
Compute nth root of a derivative structure.
void sin(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute sine of a derivative structure.
void sinh(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute hyperbolic sine of a derivative structure.
void subtract(double[] lhs, int lhsOffset, double[] rhs, int rhsOffset, double[] result, int resultOffset)
Perform subtraction of two derivative structures.
void tan(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute tangent of a derivative structure.
void tanh(double[] operand, int operandOffset, double[] result, int resultOffset)
Compute hyperbolic tangent of a derivative structure.
double taylor(double[] ds, int dsOffset, double... delta)
Evaluate Taylor expansion of a derivative structure.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Method Detail

• #### getCompiler

public static DSCompiler getCompiler(int parameters,
int order)
throws NumberIsTooLargeException
Get the compiler for number of free parameters and order.
Parameters:
parameters - number of free parameters
order - derivation order
Returns:
cached rules set
Throws:
NumberIsTooLargeException - if order is too large
• #### getPartialDerivativeIndex

public int getPartialDerivativeIndex(int... orders)
throws DimensionMismatchException,
NumberIsTooLargeException
Get the index of a partial derivative in the array.

If all orders are set to 0, then the 0th order derivative is returned, which is the value of the function.

The indices of derivatives are between 0 and getSize() - 1. Their specific order is fixed for a given compiler, but otherwise not publicly specified. There are however some simple cases which have guaranteed indices:

• the index of 0th order derivative is always 0
• if there is only 1 free parameter, then the derivatives are sorted in increasing derivation order (i.e. f at index 0, df/dp at index 1, d2f/dp2 at index 2 ... dkf/dpk at index k),
• if the derivation order is 1, then the derivatives are sorted in increasing free parameter order (i.e. f at index 0, df/dx1 at index 1, df/dx2 at index 2 ... df/dxk at index k),
• all other cases are not publicly specified

This method is the inverse of method getPartialDerivativeOrders(int)

Parameters:
orders - derivation orders with respect to each parameter
Returns:
index of the partial derivative
Throws:
DimensionMismatchException - if the numbers of parameters does not match the instance
NumberIsTooLargeException - if sum of derivation orders is larger than the instance limits
getPartialDerivativeOrders(int)
• #### getFreeParameters

public int getFreeParameters()
Get the number of free parameters.
Returns:
number of free parameters
• #### getOrder

public int getOrder()
Get the derivation order.
Returns:
derivation order
• #### getSize

public int getSize()
Get the array size required for holding partial derivatives data.

This number includes the single 0 order derivative element, which is guaranteed to be stored in the first element of the array.

Returns:
array size required for holding partial derivatives data
• #### linearCombination

public void linearCombination(double a1,
double[] c1,
int offset1,
double a2,
double[] c2,
int offset2,
double[] result,
int resultOffset)
Compute linear combination. The derivative structure built will be a1 * ds1 + a2 * ds2
Parameters:
a1 - first scale factor
c1 - first base (unscaled) component
offset1 - offset of first operand in its array
a2 - second scale factor
c2 - second base (unscaled) component
offset2 - offset of second operand in its array
result - array where result must be stored (it may be one of the input arrays)
resultOffset - offset of the result in its array
• #### linearCombination

public void linearCombination(double a1,
double[] c1,
int offset1,
double a2,
double[] c2,
int offset2,
double a3,
double[] c3,
int offset3,
double[] result,
int resultOffset)
Compute linear combination. The derivative structure built will be a1 * ds1 + a2 * ds2 + a3 * ds3 + a4 * ds4
Parameters:
a1 - first scale factor
c1 - first base (unscaled) component
offset1 - offset of first operand in its array
a2 - second scale factor
c2 - second base (unscaled) component
offset2 - offset of second operand in its array
a3 - third scale factor
c3 - third base (unscaled) component
offset3 - offset of third operand in its array
result - array where result must be stored (it may be one of the input arrays)
resultOffset - offset of the result in its array
• #### linearCombination

public void linearCombination(double a1,
double[] c1,
int offset1,
double a2,
double[] c2,
int offset2,
double a3,
double[] c3,
int offset3,
double a4,
double[] c4,
int offset4,
double[] result,
int resultOffset)
Compute linear combination. The derivative structure built will be a1 * ds1 + a2 * ds2 + a3 * ds3 + a4 * ds4
Parameters:
a1 - first scale factor
c1 - first base (unscaled) component
offset1 - offset of first operand in its array
a2 - second scale factor
c2 - second base (unscaled) component
offset2 - offset of second operand in its array
a3 - third scale factor
c3 - third base (unscaled) component
offset3 - offset of third operand in its array
a4 - fourth scale factor
c4 - fourth base (unscaled) component
offset4 - offset of fourth operand in its array
result - array where result must be stored (it may be one of the input arrays)
resultOffset - offset of the result in its array

public void add(double[] lhs,
int lhsOffset,
double[] rhs,
int rhsOffset,
double[] result,
int resultOffset)
Perform addition of two derivative structures.
Parameters:
lhs - array holding left hand side of addition
lhsOffset - offset of the left hand side in its array
rhs - array right hand side of addition
rhsOffset - offset of the right hand side in its array
result - array where result must be stored (it may be one of the input arrays)
resultOffset - offset of the result in its array
• #### subtract

public void subtract(double[] lhs,
int lhsOffset,
double[] rhs,
int rhsOffset,
double[] result,
int resultOffset)
Perform subtraction of two derivative structures.
Parameters:
lhs - array holding left hand side of subtraction
lhsOffset - offset of the left hand side in its array
rhs - array right hand side of subtraction
rhsOffset - offset of the right hand side in its array
result - array where result must be stored (it may be one of the input arrays)
resultOffset - offset of the result in its array
• #### multiply

public void multiply(double[] lhs,
int lhsOffset,
double[] rhs,
int rhsOffset,
double[] result,
int resultOffset)
Perform multiplication of two derivative structures.
Parameters:
lhs - array holding left hand side of multiplication
lhsOffset - offset of the left hand side in its array
rhs - array right hand side of multiplication
rhsOffset - offset of the right hand side in its array
result - array where result must be stored (for multiplication the result array cannot be one of the input arrays)
resultOffset - offset of the result in its array
• #### divide

public void divide(double[] lhs,
int lhsOffset,
double[] rhs,
int rhsOffset,
double[] result,
int resultOffset)
Perform division of two derivative structures.
Parameters:
lhs - array holding left hand side of division
lhsOffset - offset of the left hand side in its array
rhs - array right hand side of division
rhsOffset - offset of the right hand side in its array
result - array where result must be stored (for division the result array cannot be one of the input arrays)
resultOffset - offset of the result in its array
• #### remainder

public void remainder(double[] lhs,
int lhsOffset,
double[] rhs,
int rhsOffset,
double[] result,
int resultOffset)
Perform remainder of two derivative structures.
Parameters:
lhs - array holding left hand side of remainder
lhsOffset - offset of the left hand side in its array
rhs - array right hand side of remainder
rhsOffset - offset of the right hand side in its array
result - array where result must be stored (it may be one of the input arrays)
resultOffset - offset of the result in its array
• #### pow

public void pow(double a,
double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute power of a double to a derivative structure.
Parameters:
a - number to exponentiate
operand - array holding the power
operandOffset - offset of the power in its array
result - array where result must be stored (for power the result array cannot be the input array)
resultOffset - offset of the result in its array
Since:
3.3
• #### pow

public void pow(double[] operand,
int operandOffset,
double p,
double[] result,
int resultOffset)
Compute power of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
p - power to apply
result - array where result must be stored (for power the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### pow

public void pow(double[] operand,
int operandOffset,
int n,
double[] result,
int resultOffset)
Compute integer power of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
n - power to apply
result - array where result must be stored (for power the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### pow

public void pow(double[] x,
int xOffset,
double[] y,
int yOffset,
double[] result,
int resultOffset)
Compute power of a derivative structure.
Parameters:
x - array holding the base
xOffset - offset of the base in its array
y - array holding the exponent
yOffset - offset of the exponent in its array
result - array where result must be stored (for power the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### rootN

public void rootN(double[] operand,
int operandOffset,
int n,
double[] result,
int resultOffset)
Compute nth root of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
n - order of the root
result - array where result must be stored (for nth root the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### exp

public void exp(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute exponential of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for exponential the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### expm1

public void expm1(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute exp(x) - 1 of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for exponential the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### log

public void log(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute natural logarithm of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for logarithm the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### log1p

public void log1p(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Computes shifted logarithm of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for shifted logarithm the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### log10

public void log10(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Computes base 10 logarithm of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for base 10 logarithm the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### cos

public void cos(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute cosine of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for cosine the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### sin

public void sin(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute sine of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for sine the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### tan

public void tan(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute tangent of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for tangent the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### acos

public void acos(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute arc cosine of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for arc cosine the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### asin

public void asin(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute arc sine of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for arc sine the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### atan

public void atan(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute arc tangent of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for arc tangent the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### atan2

public void atan2(double[] y,
int yOffset,
double[] x,
int xOffset,
double[] result,
int resultOffset)
Compute two arguments arc tangent of a derivative structure.
Parameters:
y - array holding the first operand
yOffset - offset of the first operand in its array
x - array holding the second operand
xOffset - offset of the second operand in its array
result - array where result must be stored (for two arguments arc tangent the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### cosh

public void cosh(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute hyperbolic cosine of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for hyperbolic cosine the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### sinh

public void sinh(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute hyperbolic sine of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for hyperbolic sine the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### tanh

public void tanh(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute hyperbolic tangent of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for hyperbolic tangent the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### acosh

public void acosh(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute inverse hyperbolic cosine of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for inverse hyperbolic cosine the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### asinh

public void asinh(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute inverse hyperbolic sine of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for inverse hyperbolic sine the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### atanh

public void atanh(double[] operand,
int operandOffset,
double[] result,
int resultOffset)
Compute inverse hyperbolic tangent of a derivative structure.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
result - array where result must be stored (for inverse hyperbolic tangent the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### compose

public void compose(double[] operand,
int operandOffset,
double[] f,
double[] result,
int resultOffset)
Compute composition of a derivative structure by a function.
Parameters:
operand - array holding the operand
operandOffset - offset of the operand in its array
f - array of value and derivatives of the function at the current point (i.e. at operand[operandOffset]).
result - array where result must be stored (for composition the result array cannot be the input array)
resultOffset - offset of the result in its array
• #### taylor

public double taylor(double[] ds,
int dsOffset,
double... delta)
throws MathArithmeticException
Evaluate Taylor expansion of a derivative structure.
Parameters:
ds - array holding the derivative structure
dsOffset - offset of the derivative structure in its array
delta - parameters offsets (Δx, Δy, ...)
Returns:
value of the Taylor expansion at x + Δx, y + Δy, ...
Throws:
MathArithmeticException - if factorials becomes too large
• #### checkCompatibility

public void checkCompatibility(DSCompiler compiler)
throws DimensionMismatchException
Check rules set compatibility.
Parameters:
compiler - other compiler to check against instance
Throws:
DimensionMismatchException - if number of free parameters or orders are inconsistent