/**
   *
   *
   * <h2>Remove Emittance Growth Through an RF Gap</h2>
   *
   * <p>Method to modify the covariance matrix when simulating emittance growth through RF
   * accelerating gaps. (The method only considers the case of propagation through an <code>
   * IdealRfGap</code> element). If the <code>IElement</code> argument is any other type of element,
   * nothing is done. The argument <code>matTau</code> is the covariance matrix after the usual
   * propagation through the <code>elem</code> element.
   *
   * <p>Note that this method is essentially the complement of the method {@link
   * EnvelopeTracker#addEmittanceGrowth(EnvelopeProbe, IElement, PhaseMatrix)}. Whereas <code>
   * addEmittanceGrowth()</code> augments the momentum elements of <b>&sigma;</b>, this method
   * reduces them by the same amount. Specifically, let <i>x</i> be either transverse phase space
   * variable. The emittance growth effect is achieved by first multiplying the element &lt;x'|x&gt;
   * of the RF gap transfer matrix <b>&Phi;</b> by the factor
   * <i>F<sub>t</sub></i>(&Delta;<i>&phi;</i>) returned by method {@link
   * EnvelopeTrackerBase#compTransFourierTransform(double)} (see {@link
   * EnvelopeTrackerBase#modTransferMatrixForEmitGrowth(double, PhaseMatrix)}). Currently this
   * action is done in {@link #compTransferMatrix(double, EnvelopeProbe, IElement)}. Once the
   * covariance matrix <b>&tau;</b> is back-propagated by the modified transfer matrix <b>&Phi;</b>,
   * the moment &lt;<i>x'</i><sup>2</sup>&gt; is reduced by the result of this function.
   *
   * <p>The discussion below is taken directly from {@link
   * EnvelopeTracker#addEmittanceGrowth(EnvelopeProbe, IElement, PhaseMatrix)}. It is applicable
   * here if the emittance is reduced by &Delta;&lt;<i>x'<sub>f</sub></i><sup>2</sup>&gt; rather
   * than increased by it.
   *
   * <p>The before gap and after gap transverse RMS divergence angles, <i>x'<sub>i</sub></i> and
   * <i>x'<sub>f</sub></i>, respectively, are related by the following formula: <br>
   * <br>
   * &nbsp; &lt;<i>x'<sub>f</sub></i><sup>2</sup>&gt; =
   * &Delta;&lt;<i>x'<sub>f</sub></i><sup>2</sup>&gt; + &lt;<i>x'<sub>i</sub></i><sup>2</sup>&gt;
   * <br>
   * <br>
   * where &Delta;&lt;<i>x'<sub>f</sub></i><sup>2</sup>&gt; is the emittance growth factor given by
   * <br>
   * <br>
   * &nbsp; &Delta;&lt;<i>x'<sub>f</sub></i><sup>2</sup>&gt; &equiv;
   * <i>k<sub>t</sub></i><sup>2</sup>
   * <i>G<sub>t</sub></i>(<i>&phi;<sub>s</sub></i>,&Delta;<i>&phi;</i>)
   * &lt;<i>x<sub>i</sub></i></i><sup>2</sup>&gt;. <br>
   * <br>
   * where <i>G<sub>t</sub></i>(<i>&phi;<sub>s</sub></i>,&Delta;<i>&phi;</i>) is the transverse
   * 3-dimensional emittance growth function, and <i>x<sub>i</sub></i> represents the before-gap
   * position for <em>either</em> transverse phase plane. The action of this method is described by
   * the original equation.
   *
   * <p>The resulting action on the before gap and after gap transverse RMS emittances,
   * <i>&epsilon;<sub>t,i</sub></i> and <i>&epsilon;<sub>t,f</sub></i>, respectively, is now
   * described by the following formula: <br>
   * <br>
   * &nbsp; <i>&epsilon;<sub>t,f</sub></i><sup>2</sup> =
   * <i>&eta;</i><sup>2</sup><i>&epsilon;<sub>t,i</sub></i><sup>2</sup> +
   * &Delta;<i>&epsilon;<sub>t,f</sub></i><sup>2</sup> <br>
   * <br>
   * where <i>&eta;</i> is the momentum compaction due to acceleration <br>
   * <br>
   * <i>&eta;</i> &equiv;
   * <i>&beta;<sub>i</sub>&gamma;<sub>i</sub></i>/<i>&beta;<sub>f</sub>&gamma;<sub>f</sub></i> <br>
   * <br>
   * and &Delta;<i>&epsilon;<sub>t,f</sub></i> is the emittance increase term <br>
   * <br>
   * &nbsp; &Delta;<i>&epsilon;<sub>t,f</sub></i><sup>2</sup> &equiv;
   * &Delta;&lt;<i>x'<sub>f</sub></i><sup>2</sup>&gt;
   * &lt;<i>x<sub>f</sub></i></i><sup>2</sup>&gt;<sup>2</sup>. <br>
   * <br>
   * There are analogous formulas for the before and after gap longitudinal plane emittances
   * <i>&epsilon;<sub>z,i</sub></i> and <i>&epsilon;<sub>z,f</sub></i>, respectively, with
   * <i>G<sub>t</sub></i>(<i>&phi;<sub>s</sub></i>,&Delta;<i>&phi;</i>) replaced by
   * <i>G<sub>z</sub></i>(<i>&phi;<sub>s</sub></i>,&Delta;<i>&phi;</i>) and
   * <i>x</i><sub>(<i>f,i</i>)</sub> replaced by <i>z</i><sub>(<i>f,i</i>)</sub>.
   *
   * <p><strong>NOTES</strong>: CKA <br>
   * &middot; Since we are modeling the RF gap as a thin lens, only the momentum (divergance angle)
   * is modified, &lt;<i>x</i><sup>2</sup>&gt;, &lt;<i>y</i><sup>2</sup>&gt;, and
   * &lt;<i>z</i><sup>2</sup>&gt; remain unaffected. Thus, &lt;<i>x<sub>f</sub></i><sup>2</sup>&gt;
   * = &lt;<i>x<sub>i</sub></i><sup>2</sup>&gt; and &lt;<i>z<sub>f</sub></i><sup>2</sup>&gt; =
   * &lt;<i>z<sub>i</sub></i><sup>2</sup>&gt; and may be computed as such in the above. <br>
   * &middot; The &lt;<i>x'</i><sup>2</sup>&gt; element is modified by the formula <br>
   * <br>
   * &nbsp; &lt;<i>x'</i><sup>2</sup>&gt; = &lt;<i>x'</i><sup>2</sup>&gt; +
   * <i>c<sub>eg</sub></i>&lt;<i>x</i><sup>2</sup>&gt; <br>
   * <br>
   * where <i>c<sub>eg</sub></i> is the emittance growth coefficent. There are similar equations for
   * the other phase planes. The emittance growth coefficents are computed in the base class <code>
   * EnvelopeTrackerBase</code> by the methods <code>emitGrowthCoefTrans(EnvelopeProbe, IdealRfGap)
   * </code> and <code>emitGrowthCoefLong(EnvelopeProbe, IdealRfGap)</code>.
   *
   * <p><strong>NOTES</strong>: (H. SAKO) <br>
   * &middot; Increase emittance using same (nonlinear) procedure on the second moments as in
   * Trace3D.
   *
   * @param iElem <code>IElement</code> element for exceptional processing
   * @param probe <code>IProbe</code> object associated with correlation matrix
   * @param matTau correlation matrix after (normal) propagation thru <code>elem</code>
   * @return covariance matrix of <code>probe</code> after adjusting for emittance growth
   * @throws ModelException unknown/unsupported emittance growth model, or unknown/unsupported phase
   *     plane
   * @see #compTransferMatrix(double, EnvelopeProbe, IElement)
   * @see EnvelopeTrackerBase#compTransFourierTransform(double)
   * @see EnvelopeTrackerBase#compLongFourierTransform(double)
   * @see EnvelopeTracker#addEmittanceGrowth(EnvelopeProbe, IElement, PhaseMatrix)
   * @author Hiroyuki Sako
   * @author Christopher K. Allen
   */
  private PhaseMatrix removeEmittanceGrowth(EnvelopeProbe probe, IElement iElem, PhaseMatrix matTau)
      throws ModelException {

    // Check for RF Gap
    if (!(iElem instanceof IdealRfGap)) return matTau;

    if (!this.getEmittanceGrowth()) return matTau;

    // Get the synchronous phase and compute the phase spread
    IdealRfGap elemRfGap = (IdealRfGap) iElem;

    double W = probe.getKineticEnergy();
    double dW = elemRfGap.energyGain(probe);
    probe.setKineticEnergy(W - dW);

    double phi_s = elemRfGap.getPhase();
    double dphi = this.effPhaseSpread(probe, elemRfGap);

    // Compute the divergence angle increment coefficients
    //  (emittance growth coefficients)
    double dxp_2; // transverse divergence angle augmentation factor
    double dzp_2; // longitudinal divergence angle augmentation factor

    //        if (this.getEmitGrowthModel() == EmitGrowthModel.TRACE3D) {
    //
    //            dxp_2 = this.emitGrowthCoefTrans(probe, elemRfGap);
    //            dzp_2 = this.emitGrowthCoefLong(probe, elemRfGap);
    //
    //        } else {

    double Gt = this.compEmitGrowthFunction(PhasePlane.TRANSVERSE, phi_s, dphi);
    double kt = elemRfGap.compTransFocusing(probe);
    dxp_2 = kt * kt * Gt;

    double Gz = this.compEmitGrowthFunction(PhasePlane.LONGITUDINAL, phi_s, dphi);
    double kz = elemRfGap.compLongFocusing(probe);
    //            double  gf    = elemRfGap.gammaFinal(probe);
    //            double  gf_2  = gf*gf;
    //            dzp_2 = kz*kz*Gz/(gf_2*gf_2);
    //            dzp_2 = gf_2*gf_2*kz*kz*Gz;
    dzp_2 = kz * kz * Gz;

    //        }

    probe.setKineticEnergy(W);

    // Compute new correlation matrix
    //      Transverse planes
    double x_2 = matTau.getElem(0, 0);
    double xp_2 = matTau.getElem(1, 1);
    double xp_2eg = xp_2 - dxp_2 * x_2;
    matTau.setElem(1, 1, xp_2eg);

    double y_2 = matTau.getElem(2, 2);
    double yp_2 = matTau.getElem(3, 3);
    double yp_2eg = yp_2 - dxp_2 * y_2;
    matTau.setElem(3, 3, yp_2eg);

    //      Longitudinal plane
    double z_2 = matTau.getElem(4, 4);
    double zp_2 = matTau.getElem(5, 5);
    double zp_2eg = zp_2 - dzp_2 * z_2;
    matTau.setElem(5, 5, zp_2eg);

    return matTau;
  }
  /**
   *
   *
   * <h2>Compute Transfer Matrix Including Space Charge</h2>
   *
   * <p>Computes the back-propagating transfer matrix over the incremental distance <code>dblLen
   * </code> for the beamline modeling element <code>ifcElem</code>, and for the given <code>probe
   * </code>. We include space charge and emittance growth effects if specified.
   *
   * <p><strong>NOTE</strong>: (CKA) <br>
   * &middot; If space charge is included, the space charge matrix is computed for length <code>
   * dblLen</code>, but at a half-step location behind the current probe position. This method is
   * the same technique used by Trace3D. The space charge matrix is then pre- and post- multiplied
   * by the element transfer matrix for a half-step before and after the mid-step position,
   * respectively. <br>
   * &middot; I do not know if this (leap-frog) technique buys us much more accuracy then full
   * stepping.
   *
   * @param dblLen incremental path length
   * @param probe beam probe under simulation
   * @param ifcElem beamline element propagating probe
   * @return transfer matrix for given element
   * @throws ModelException bubbles up from IElement#transferMap()
   * @see EnvelopeTracker#compScheffMatrix(double, EnvelopeProbe, PhaseMatrix)
   * @see EnvelopeTracker#transferEmitGrowth(EnvelopeProbe, IElement, PhaseMatrix)
   * @see EnvelopeTracker#modTransferMatrixForDisplError(double, double, double, PhaseMatrix)
   */
  private PhaseMatrix compTransferMatrix(double dblLen, EnvelopeProbe probe, IElement ifcElem)
      throws ModelException {

    // Returned value
    PhaseMatrix matPhi; // transfer matrix including all effects

    // Check for exceptional circumstance and modify transfer matrix accordingly
    if (ifcElem instanceof IdealRfGap) {
      IdealRfGap elemRfGap = (IdealRfGap) ifcElem;
      double dW = elemRfGap.energyGain(probe, dblLen);
      double W = probe.getKineticEnergy();
      probe.setKineticEnergy(W - dW);
      PhaseMatrix matPhiI = elemRfGap.transferMap(probe, dblLen).getFirstOrder();

      if (this.getEmittanceGrowth()) {
        double dphi = this.effPhaseSpread(probe, elemRfGap);

        matPhiI = super.modTransferMatrixForEmitGrowth(dphi, matPhiI);
      }
      matPhi = matPhiI.inverse();
      probe.setKineticEnergy(W);

      return matPhi;
    }

    if (dblLen == 0.0) {
      matPhi = ifcElem.transferMap(probe, dblLen).getFirstOrder();

      return matPhi;
    }

    // Check for easy case of no space charge
    if (!this.getUseSpacecharge()) {
      matPhi = ifcElem.transferMap(probe, dblLen).getFirstOrder();

      // we must treat space charge
    } else {

      // Store the current probe state (for rollback)
      EnvelopeProbeState state0 = probe.cloneCurrentProbeState();
      // ProbeState  state0 = probe.createProbeState();

      // Get half-step back-propagation matrix at current probe location
      //  NOTE: invert by computing for negative propagation length
      PhaseMap mapElem0 = ifcElem.transferMap(probe, -dblLen / 2.0);
      PhaseMatrix matPhi0 = mapElem0.getFirstOrder();

      // Get the RMS envelopes at probe location
      CovarianceMatrix covTau0 = probe.getCovariance(); // covariance matrix at entrance

      // Move probe back a half step for position-dependent transfer maps
      double pos = probe.getPosition() - dblLen / 2.0;
      PhaseMatrix matTau1 = covTau0.conjugateTrans(matPhi0);
      CovarianceMatrix covTau1 = new CovarianceMatrix(matTau1);

      probe.setPosition(pos);
      probe.setCovariance(covTau1);

      // space charge transfer matrix
      //  NOTE: invert by computing for negative propagation length
      PhaseMatrix matPhiSc = this.compScheffMatrix(-dblLen, probe, ifcElem);

      // Compute half-step transfer matrix at new probe location
      PhaseMap mapElem1 = ifcElem.transferMap(probe, -dblLen / 2.0);
      PhaseMatrix matPhi1 = mapElem1.getFirstOrder();

      // Restore original probe state
      probe.applyState(state0);

      // Compute the full transfer matrix for the distance dblLen
      matPhi = matPhi1.times(matPhiSc.times(matPhi0));
    }

    if (ifcElem instanceof IdealMagQuad) {
      // sako  put alignment error in sigma matrix
      //  NOTE the use of negative displacements for back-propagation
      IdealMagQuad elemQuad = (IdealMagQuad) ifcElem;

      double delx = -elemQuad.getAlignX();
      double dely = -elemQuad.getAlignY();
      double delz = -elemQuad.getAlignZ();

      matPhi = this.modTransferMatrixForDisplError(delx, dely, delz, matPhi);
    }

    return matPhi;
  }