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* (the "License"); you may not use this file except in compliance with
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* Unless required by applicable law or agreed to in writing, software
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package org.hipparchus.ode.nonstiff;

import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;

import org.hipparchus.fraction.BigFraction;
import org.hipparchus.linear.Array2DRowFieldMatrix;
import org.hipparchus.linear.Array2DRowRealMatrix;
import org.hipparchus.linear.ArrayFieldVector;
import org.hipparchus.linear.FieldDecompositionSolver;
import org.hipparchus.linear.FieldLUDecomposition;
import org.hipparchus.linear.FieldMatrix;
import org.hipparchus.linear.MatrixUtils;
import org.hipparchus.linear.QRDecomposition;
import org.hipparchus.linear.RealMatrix;

/** Transformer to Nordsieck vectors for Adams integrators.
* classical representation with several previous first derivatives and Nordsieck
* representation with higher order scaled derivatives.</p>
*
* <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
* <pre>
* s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
* s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
* s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
* ...
* s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative
* </pre></p>
*
* <p>With the previous definition, the classical representation of multistep methods
* uses first derivatives only, i.e. it handles y<sub>n</sub>, s<sub>1</sub>(n) and
* q<sub>n</sub> where q<sub>n</sub> is defined as:
* <pre>
*   q<sub>n</sub> = [ s<sub>1</sub>(n-1) s<sub>1</sub>(n-2) ... s<sub>1</sub>(n-(k-1)) ]<sup>T</sup>
* </pre>
* (we omit the k index in the notation for clarity).</p>
*
* <p>Another possible representation uses the Nordsieck vector with
* higher degrees scaled derivatives all taken at the same step, i.e it handles y<sub>n</sub>,
* s<sub>1</sub>(n) and r<sub>n</sub>) where r<sub>n</sub> is defined as:
* <pre>
* r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
* </pre>
* (here again we omit the k index in the notation for clarity)
* </p>
*
* <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be
* computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact
* for degree k polynomials.
* <pre>
* s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + &sum;<sub>j&gt;0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</sub>(n)
* </pre>
* The previous formula can be used with several values for i to compute the transform between
* classical representation and Nordsieck vector at step end. The transform between r<sub>n</sub>
* and q<sub>n</sub> resulting from the Taylor series formulas above is:
* <pre>
* q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
* </pre>
* where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)&times;(k-1) matrix built
* with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being
* the column number starting from 1:
* <pre>
*        [  -2   3   -4    5  ... ]
*        [  -4  12  -32   80  ... ]
*   P =  [  -6  27 -108  405  ... ]
*        [  -8  48 -256 1280  ... ]
*        [          ...           ]
* </pre></p>
*
* <p>Changing -i into +i in the formula above can be used to compute a similar transform between
* classical representation and Nordsieck vector at step start. The resulting matrix is simply
* the absolute value of matrix P.</p>
*
* at step n+1 is computed from the Nordsieck vector at step n as follows:
* <ul>
*   <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
*   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
*   <li>r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li>
* </ul>
* where A is a rows shifting matrix (the lower left part is an identity matrix):
* <pre>
*        [ 0 0   ...  0 0 | 0 ]
*        [ ---------------+---]
*        [ 1 0   ...  0 0 | 0 ]
*    A = [ 0 1   ...  0 0 | 0 ]
*        [       ...      | 0 ]
*        [ 0 0   ...  1 0 | 0 ]
*        [ 0 0   ...  0 1 | 0 ]
* </pre></p>
*
* at step n+1 is computed from the Nordsieck vector at step n as follows:
* <ul>
*   <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li>
*   <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li>
*   <li>R<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li>
* </ul>
* From this predicted vector, the corrected vector is computed as follows:
* <ul>
*   <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... &plusmn;1 ] r<sub>n+1</sub></li>
*   <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
*   <li>r<sub>n+1</sub> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li>
* </ul>
* where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
* predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
* represent the corrected states.</p>
*
* <p>We observe that both methods use similar update formulas. In both cases a P<sup>-1</sup>u
* vector and a P<sup>-1</sup> A P matrix are used that do not depend on the state,
* they only depend on k. This class handles these transformations.</p>
*
*/

/** Cache for already computed coefficients. */
private static final Map<Integer, AdamsNordsieckTransformer> CACHE = new HashMap<>();

/** Update matrix for the higher order derivatives h<sup>2</sup>/2 y'', h<sup>3</sup>/6 y''' ... */
private final Array2DRowRealMatrix update;

/** Update coefficients of the higher order derivatives wrt y'. */
private final double[] c1;

/** Simple constructor.
* @param n number of steps of the multistep method
* (excluding the one being computed)
*/

final int rows = n - 1;

// compute exact coefficients
FieldMatrix<BigFraction> bigP = buildP(rows);
FieldDecompositionSolver<BigFraction> pSolver =
new FieldLUDecomposition<BigFraction>(bigP).getSolver();

BigFraction[] u = new BigFraction[rows];
Arrays.fill(u, BigFraction.ONE);
BigFraction[] bigC1 = pSolver.solve(new ArrayFieldVector<BigFraction>(u, false)).toArray();

// update coefficients are computed by combining transform from
// Nordsieck to multistep, then shifting rows to represent step advance
// then applying inverse transform
BigFraction[][] shiftedP = bigP.getData();
for (int i = shiftedP.length - 1; i > 0; --i) {
// shift rows
shiftedP[i] = shiftedP[i - 1];
}
shiftedP = new BigFraction[rows];
Arrays.fill(shiftedP, BigFraction.ZERO);
FieldMatrix<BigFraction> bigMSupdate =
pSolver.solve(new Array2DRowFieldMatrix<BigFraction>(shiftedP, false));

// convert coefficients to double
update         = MatrixUtils.bigFractionMatrixToRealMatrix(bigMSupdate);
c1             = new double[rows];
for (int i = 0; i < rows; ++i) {
c1[i] = bigC1[i].doubleValue();
}

}

/** Get the Nordsieck transformer for a given number of steps.
* @param nSteps number of steps of the multistep method
* (excluding the one being computed)
* @return Nordsieck transformer for the specified number of steps
*/
public static AdamsNordsieckTransformer getInstance(final int nSteps) { // NOPMD - PMD false positive
synchronized(CACHE) {
if (t == null) {
CACHE.put(nSteps, t);
}
return t;
}
}

/** Build the P matrix.
* <p>The P matrix general terms are shifted (j+1) (-i)<sup>j</sup> terms
* with i being the row number starting from 1 and j being the column
* number starting from 1:
* <pre>
*        [  -2   3   -4    5  ... ]
*        [  -4  12  -32   80  ... ]
*   P =  [  -6  27 -108  405  ... ]
*        [  -8  48 -256 1280  ... ]
*        [          ...           ]
* </pre></p>
* @param rows number of rows of the matrix
* @return P matrix
*/
private FieldMatrix<BigFraction> buildP(final int rows) {

final BigFraction[][] pData = new BigFraction[rows][rows];

for (int i = 1; i <= pData.length; ++i) {
// build the P matrix elements from Taylor series formulas
final BigFraction[] pI = pData[i - 1];
final int factor = -i;
int aj = factor;
for (int j = 1; j <= pI.length; ++j) {
pI[j - 1] = new BigFraction(aj * (j + 1));
aj *= factor;
}
}

return new Array2DRowFieldMatrix<BigFraction>(pData, false);

}

/** Initialize the high order scaled derivatives at step start.
* @param h step size to use for scaling
* @param t first steps times
* @param y first steps states
* @param yDot first steps derivatives
* @return Nordieck vector at start of first step (h<sup>2</sup>/2 y''<sub>n</sub>,
* h<sup>3</sup>/6 y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>)
*/

public Array2DRowRealMatrix initializeHighOrderDerivatives(final double h, final double[] t,
final double[][] y,
final double[][] yDot) {

// using Taylor series with di = ti - t0, we get:
//  y(ti)  - y(t0)  - di y'(t0) =   di^2 / h^2 s2 + ... +   di^k     / h^k sk + O(h^k)
//  y'(ti) - y'(t0)             = 2 di   / h^2 s2 + ... + k di^(k-1) / h^k sk + O(h^(k-1))
// we write these relations for i = 1 to i= 1+n/2 as a set of n + 2 linear
// equations depending on the Nordsieck vector [s2 ... sk rk], so s2 to sk correspond
// to the appropriately truncated Taylor expansion, and rk is the Taylor remainder.
// The goal is to have s2 to sk as accurate as possible considering the fact the sum is
// truncated and we don't want the error terms to be included in s2 ... sk, so we need
// to solve also for the remainder
final double[][] a     = new double[c1.length + 1][c1.length + 1];
final double[][] b     = new double[c1.length + 1][y.length];
final double[]   y0    = y;
final double[]   yDot0 = yDot;
for (int i = 1; i < y.length; ++i) {

final double di    = t[i] - t;
final double ratio = di / h;
double dikM1Ohk    =  1 / h;

// linear coefficients of equations
// y(ti) - y(t0) - di y'(t0) and y'(ti) - y'(t0)
final double[] aI    = a[2 * i - 2];
final double[] aDotI = (2 * i - 1) < a.length ? a[2 * i - 1] : null;
for (int j = 0; j < aI.length; ++j) {
dikM1Ohk *= ratio;
aI[j]     = di      * dikM1Ohk;
aDotI[j]  = (j + 2) * dikM1Ohk;
}
}

// expected value of the previous equations
final double[] yI    = y[i];
final double[] yDotI = yDot[i];
final double[] bI    = b[2 * i - 2];
final double[] bDotI = (2 * i - 1) < b.length ? b[2 * i - 1] : null;
for (int j = 0; j < yI.length; ++j) {
bI[j]    = yI[j] - y0[j] - di * yDot0[j];
if (bDotI != null) {
bDotI[j] = yDotI[j] - yDot0[j];
}
}

}

// solve the linear system to get the best estimate of the Nordsieck vector [s2 ... sk],
// with the additional terms s(k+1) and c grabbing the parts after the truncated Taylor expansion
final QRDecomposition decomposition = new QRDecomposition(new Array2DRowRealMatrix(a, false));
final RealMatrix x = decomposition.getSolver().solve(new Array2DRowRealMatrix(b, false));

// extract just the Nordsieck vector [s2 ... sk]
final Array2DRowRealMatrix truncatedX = new Array2DRowRealMatrix(x.getRowDimension() - 1, x.getColumnDimension());
for (int i = 0; i < truncatedX.getRowDimension(); ++i) {
for (int j = 0; j < truncatedX.getColumnDimension(); ++j) {
truncatedX.setEntry(i, j, x.getEntry(i, j));
}
}
return truncatedX;

}

/** Update the high order scaled derivatives for Adams integrators (phase 1).
* <p>The complete update of high order derivatives has a form similar to:
* <pre>
* r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub>
* </pre>
* this method computes the P<sup>-1</sup> A P r<sub>n</sub> part.</p>
* @param highOrder high order scaled derivatives
* (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k))
* @return updated high order derivatives
* @see #updateHighOrderDerivativesPhase2(double[], double[], Array2DRowRealMatrix)
*/
public Array2DRowRealMatrix updateHighOrderDerivativesPhase1(final Array2DRowRealMatrix highOrder) {
return update.multiply(highOrder);
}

/** Update the high order scaled derivatives Adams integrators (phase 2).
* <p>The complete update of high order derivatives has a form similar to:
* <pre>
* r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub>
* </pre>
* this method computes the (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u part.</p>
* <p>Phase 1 of the update must already have been performed.</p>
* @param start first order scaled derivatives at step start
* @param end first order scaled derivatives at step end
* @param highOrder high order scaled derivatives, will be modified
* (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k))
* @see #updateHighOrderDerivativesPhase1(Array2DRowRealMatrix)
*/
public void updateHighOrderDerivativesPhase2(final double[] start,
final double[] end,
final Array2DRowRealMatrix highOrder) {
final double[][] data = highOrder.getDataRef();
for (int i = 0; i < data.length; ++i) {
final double[] dataI = data[i];
final double c1I = c1[i];
for (int j = 0; j < dataI.length; ++j) {
dataI[j] += c1I * (start[j] - end[j]);
}
}
}

}
```