commons-math-legacy/src/main/java/org/apache/commons/math4/legacy/ode/nonstiff/AdamsBashforthFieldIntegrator.java [238:282]:
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    @Override
    public FieldODEStateAndDerivative<T> integrate(final FieldExpandableODE<T> equations,
                                                   final FieldODEState<T> initialState,
                                                   final T finalTime)
        throws NumberIsTooSmallException, DimensionMismatchException,
               MaxCountExceededException, NoBracketingException {

        sanityChecks(initialState, finalTime);
        final T   t0 = initialState.getTime();
        final T[] y  = equations.getMapper().mapState(initialState);
        setStepStart(initIntegration(equations, t0, y, finalTime));
        final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0;

        // compute the initial Nordsieck vector using the configured starter integrator
        start(equations, getStepStart(), finalTime);

        // reuse the step that was chosen by the starter integrator
        FieldODEStateAndDerivative<T> stepStart = getStepStart();
        FieldODEStateAndDerivative<T> stepEnd   =
                        AdamsFieldStepInterpolator.taylor(stepStart,
                                                          stepStart.getTime().add(getStepSize()),
                                                          getStepSize(), scaled, nordsieck);

        // main integration loop
        setIsLastStep(false);
        do {

            T[] predictedY = null;
            final T[] predictedScaled = MathArrays.buildArray(getField(), y.length);
            Array2DRowFieldMatrix<T> predictedNordsieck = null;
            T error = getField().getZero().add(10);
            while (error.subtract(1.0).getReal() >= 0.0) {

                // predict a first estimate of the state at step end
                predictedY = stepEnd.getState();

                // evaluate the derivative
                final T[] yDot = computeDerivatives(stepEnd.getTime(), predictedY);

                // predict Nordsieck vector at step end
                for (int j = 0; j < predictedScaled.length; ++j) {
                    predictedScaled[j] = getStepSize().multiply(yDot[j]);
                }
                predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck);
                updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck);
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commons-math-legacy/src/main/java/org/apache/commons/math4/legacy/ode/nonstiff/AdamsMoultonFieldIntegrator.java [213:257]:
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    @Override
    public FieldODEStateAndDerivative<T> integrate(final FieldExpandableODE<T> equations,
                                                   final FieldODEState<T> initialState,
                                                   final T finalTime)
        throws NumberIsTooSmallException, DimensionMismatchException,
               MaxCountExceededException, NoBracketingException {

        sanityChecks(initialState, finalTime);
        final T   t0 = initialState.getTime();
        final T[] y  = equations.getMapper().mapState(initialState);
        setStepStart(initIntegration(equations, t0, y, finalTime));
        final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0;

        // compute the initial Nordsieck vector using the configured starter integrator
        start(equations, getStepStart(), finalTime);

        // reuse the step that was chosen by the starter integrator
        FieldODEStateAndDerivative<T> stepStart = getStepStart();
        FieldODEStateAndDerivative<T> stepEnd   =
                        AdamsFieldStepInterpolator.taylor(stepStart,
                                                          stepStart.getTime().add(getStepSize()),
                                                          getStepSize(), scaled, nordsieck);

        // main integration loop
        setIsLastStep(false);
        do {

            T[] predictedY = null;
            final T[] predictedScaled = MathArrays.buildArray(getField(), y.length);
            Array2DRowFieldMatrix<T> predictedNordsieck = null;
            T error = getField().getZero().add(10);
            while (error.subtract(1.0).getReal() >= 0.0) {

                // predict a first estimate of the state at step end (P in the PECE sequence)
                predictedY = stepEnd.getState();

                // evaluate a first estimate of the derivative (first E in the PECE sequence)
                final T[] yDot = computeDerivatives(stepEnd.getTime(), predictedY);

                // update Nordsieck vector
                for (int j = 0; j < predictedScaled.length; ++j) {
                    predictedScaled[j] = getStepSize().multiply(yDot[j]);
                }
                predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck);
                updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck);
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