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approx_Constr_Hess


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 -- Command: [J] = approx_Constr_Hess (@FUN, X)

     This function file can be used to approximate the Hessian of the
     constraints in a constrained Hamiltonian system solved with
     ‘odeRATTLE’.

     If no explicit expression for the Hessian is given to ‘odeRATTLE’,
     it calls this function to approximate it.

     The output argument is a three-dimensional matrix where the last
     dimension refers to each different constraint.

     The first input argument must be a function_handle or an inline
     function and must define the gradient of the constraints.

     The second input argument is just the point at which the Hessian
     will be evaluated.

     See also: odepkg.


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This function file can be used to approximate the Hessian of the
constraints ...



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approx_Constr_grad


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 -- Command: [J] = approx_Constr_grad (@FUN, X)

     This function file can be used to approximate the gradient of the
     constraints in a constrained Hamiltonian system solved with
     ‘odeRATTLE’.

     If no explicit expression for the gradient is given to ‘odeRATTLE’,
     it calls this function to approximate it.

     The output argument is a matrix where the first dimension refers to
     each different constraint.

     The first input argument must be a function_handle or an inline
     function and must define the constraints.

     The second input argument is just the point at which the gradient
     will be evaluated.

     See also: odepkg.


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This function file can be used to approximate the gradient of the
constraints...



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approx_Hamilt_Hess


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 -- Command: [J] = approx_Hamilt_Hess (@FUN, T, X)

     This function file can be used to approximate the Hessian of the
     Hamiltonian in a generic Hamiltonian system solved with ‘odeSE’,
     ‘odeSV’, ‘odeSPVI’, ‘odeRATTLE’.

     If no explicit expression for the Hessian is given to the solver,
     it calls this function to approximate it.

     The output argument is a square matrix.

     The first input argument must be a function_handle or an inline
     function and must define the Hamilton’s equations of motion: q' =
     dH/dp (t,[q;p]) p' = - dH/dq (t,[q;p]).

     The second and third input arguments are just the time and the
     point at which the Hessian will be evaluated.

     See also: odepkg.


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This function file can be used to approximate the Hessian of the
Hamiltonian ...



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golub_welsch


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 -- Command: [NODES] = golub_welsch (M,N)
 -- Command: [NODES, WEIGHTS] = golub_welsch (M,N)
 -- Command: [NODES, WEIGHTS, VALUES] = golub_welsch (M,N)
 -- Command: [NODES, WEIGHTS, VALUES, DER] = golub_welsch (M,N)

     This function can be used to compute Gauss quadrature nodes and
     weights and values of Legendre polynomials (and their derivatives)
     at Gauss quadrature nodes just in one shot, whitout using an
     iterative method.  For all the theory about this topic see [1].

     First output parameter contains Gauss quadrature nodes in (-1,1).

     Second output argument contains computed Gauss quadrature weights.

     Third output parameter contains the values of Legendre polynomials
     at computed Gauss quadrature nodes.

     Fourth output argument contains the values of the derivatives of
     Legendre polynomials at Gauss quadrature nodes.

     First input argument must be a positive integer scalar and
     represents the maximum degree of Legendre polynomials.

     Second input argument must be a positive integer scalar and
     represents the Gauss quadrature order.

     References: [1] G.H. Golub and J.H. Welsch, "Calculation of Gauss
     Quadrature Rules."  Mathematics of computation, Vol.  23, No.  106
     (Apr.  1969), pp.  221-230+s1-s10, American Mathematical Society.

     See also: odepkg.


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This function can be used to compute Gauss quadrature nodes and weights
and v...





