Ellipsoidal Toolbox (ET): ell_fusionlambda(a, q1, Q1, q2, Q2, n) - File Exchange

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Highlights from
Ellipsoidal Toolbox (ET)

ell_center_ode(t, x, mydata, ... % ELL_CENTER_ODE - ODE for the center of the reach set.
ell_demo1
ell_demo2
ell_demo3
ell_eesm_ode(t, X, l0, mydata... % ELL_EESM_ODE - ODE for the shape matrix of the external ellipsoid.
ell_eesm_ode(t, X, l0, mydata... % ELL_EEDIST_ODE - ODE for the shape matrix of the external ellipsoid
ell_enclose(V) % ELL_ENCLOSE - computes minimum volume ellipsoid that contains given vectors.
ell_fusionlambda(a, q1, Q1, q... % ELL_FUSIONLAMBDA - function whose root in the interval (0, 1) determines
ell_iesm_ode(t, X, l0, mydata... % ELL_IEDIST_ODE - ODE for the shape matrix of the internal ellipsoid
ell_iesm_ode(t, X, xl0, l0, m... % ELL_IESM_ODE - ODE for the shape matrix of the internal ellipsoid.
ell_iesm_ode(t, X, xl0, l0, m... % ELL_IESM_ODE - ODE for the shape matrix of the internal ellipsoid.
ell_inv(A) % ELL_INV - computes matrix inverse treating ill-conditioned matrices properly.
ell_nlfnlc(objf, x0, nlcf, Op... % ELL_NLFNLC - computes minimum of nonlinear function with nonlinear constraints.
ell_ode_solver(fn, t, x0, var... % ELL_ODE_SOLVER - caller for particular ODE solver.
ell_plot(x, varargin)
ell_regularize(Q, delta) % ELL_REGULARIZE - regularization of singular matrix.
ell_simdiag(A, B) % ELL_SIMDIAG - computes the transformation matrix that simultaneously
ell_square_facets(epoints_num... % ELL_SQUARE_FACETS - generates square facets to be used in PATCH function call.
ell_stm_ode(t, x, mydata, n, ... % ELL_STM_ODE - ODE for state transition matrix.
ell_triag_facets(epoints_num,... % ELL_TRIAG_FACETS - generates triangular facets to be used in PATCH function call.
ell_unitball(n) % ELL_UNITBALL - creates unit ball object
ell_valign(v, x) % ELL_VALIGN - given two vectors in R^n, computes orthogonal matrix that rotates
ell_value_extract(X, t, dims) % ELL_VALUE_EXTRACT - extracts matrix value from ppform or vector array.
ellipsoids_init(varargin) % ELLIPSOIDS_INIT - initializes Ellipsoidal Toolbox.
hyperplane2polytope(HA) % HYPERPLANE2POLYTOPE - converts array of hyperplanes into polytope
install(root) % Install Ellipsoidal Toolbox.
polytope2hyperplane(P) % POLYTOPE2HYPERPLANE - converts given polytope object into the array
ellipsoid(varargin) % ELLIPSOID - constructor of the ellipsoid object.
hyperplane(v, c) % HYPERPLANE - creates hyperplane structure (or array of hyperplane structures).
linsys(A, B, U, G, V, C, W, D... % LINSYS - constructor for linear system object.
reach(lsys, X0, L0, T, Option... % REACH - computes reach set approximation of the linear system for the given
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from Ellipsoidal Toolbox (ET) by Alex Kurzhanskiy
Implementation of the ellipsoidal calculus and ellipsoidal methods for reachability analysis.

ell_fusionlambda(a, q1, Q1, q2, Q2, n)

function f = ell_fusionlambda(a, q1, Q1, q2, Q2, n)
%
% ELL_FUSIONLAMBDA - function whose root in the interval (0, 1) determines
%                    the minimal volume ellipsoid overapproximating the
%                    intersection of two ellipsoids.
%
% This function is called from ELLIPSOID/INTERSECTION_EA by FZERO.
%

%
% Author:
% -------
%
%    Alex Kurzhanskiy <akurzhan@eecs.berkeley.edu>
%

  X = a*Q1 + (1 - a)*Q2;
  Y = inv(X);
  Y = 0.5*(Y + Y');
  k = 1 - a*(1 - a)*(q2 - q1)'*Q2*Y*Q1*(q2 - q1);
  q = Y*(a*Q1*q1 + (1 - a)*Q2*q2);
  
  f = k*det(X)*trace(det(X)*Y*(Q1 - Q2)) - n*((det(X))^2)* ...
      (2*q'*Q1*q1 - 2*q'*Q2*q2 + q'*(Q2 - Q1)*q - q1'*Q1*q1 + q2'*Q2*q2);

  return;

Highlights from Ellipsoidal Toolbox (ET)

Highlights from
Ellipsoidal Toolbox (ET)