InitDAE for computing consistent initial values¶

DAEinitialization module¶

@author: estevez schwarz/lamour

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DAEinitialization.consistentValues.JacobianDerivativeArray(X, taylorObject, ti, K, dim)¶: JacobianDerivativeArray creates the Jacobian matrix of the derivativeArray as two dimensional ndarray

DAEinitialization.consistentValues.consistentValues(example, TaylorOrder, alpha, x0, ti, augmentation=False, ftol=1e-08, disp=True, maxiter=100, info_level=3)¶

consistentValues computes consistent values of the DAE example in the timepoint ti such that || P(x(ti)-alpha) ||_2 -> min! using the modul minimize with the method SLSQP

param:	example - object of example class TaylorOrder - integer, order of the highest derivative of Taylor coefficients alpha - reference vector as ndarray of size dim = dimension of the DAE x0 - ndarray containing initial guess of length dimTaylorOrder ti - timepoint augmentation* if False: using the TaylorOrder if True: augmentation of number of Taylor coefficients starting with 1 up to TaylorOrder+1 ftol = 1e-8 - (parameter for python module minimize) disp = True - (parameter for python module minimize) maxiter = 100 - (parameter for python module minimize): maximal number of iterations info_level = 0 - integer, higher level implies more information, 0 ... 3 If info_level = 0 no output with the exception of error messages. If info_level > 0 then two log-files are generated: example_name_array_date.log and example_name_consi_date.log array contains Jacobian matrices, projectors, ranks, SVD values etc. consi contains values related to the computation of consistent initial values and the solution in full accuracy (16 digits)
raised:	exceptions: MinimizationNotSuccessful ValueError
return:	solution as ndarray of length dim(TaylorOrder+1) containing values for x(t0), x’(t0),x”(t0)/2!, ..., x ^{(TaylorOrder)}(t0)/(TaylorOrder)! Caution: Only x(t0), x’(t0),x”(t0)/2!, ..., x ^(s)(t0)/s! are consistent for s = TaylorOrder+1-index (see [1]) TaylorEx* Object of class TaylorDAE of the example (to be used in index_Condition)

DAEinitialization.consistentValues.consistentValuesIndex(example, TaylorOrder, alpha, x0, t0, augmentation=False, ftol=1e-08, disp=False, maxiter=100, info_level=0)¶

consistentValuesIndex wraps the moduls consistentValues and index_Condition.

param:

example - object of example class
TaylorOrder - integer, order of the highest derivative of Taylor coefficients
alpha - reference vector as ndarray of size dim = dimension of the DAE
x0 - ndarray containing initial guess of length dim*TaylorOrder
t0 - timepoint
augmentation = False
- if False: using the TaylorOrder
- if True: augmentation of number of Taylor coefficients starting with 1 up to TaylorOrder+1
ftol = 1e-8 - (parameter for python module minimize)
disp = False - (parameter for python module minimize)

If disp = True also results of the iteration of the minimizer are displayed.
maxiter = 100 - (parameter for python module minimize): maximal number of iterations
info_level = 0 - integer, higher level implies more information, 0 ... 3

If info_level = 0 no output with the exception of error messages.

If info_level > 0 then two log-files are generated: example_name_array_date.log and example_name_consi_date.log

array contains Jacobian matrices, projectors, ranks, SVD values etc.

consi contains values related to the computation of consistent initial values and the solution in full accuracy (16 digits)

return:

solution as ndarray of length dim*(TaylorOrder+1) containing values for x(t0), x’(t0),x”(t0)/2!, ..., x ^{(TaylorOrder)}(t0)/(TaylorOrder)!

Caution: Only x(t0), x’(t0),x”(t0)/2!, ..., x ^(s)(t0)/s! are consistent for s = TaylorOrder+1-index (see [1])
index of the DAE in the computed consistent point
cond - condition number of the index computation
computation_robust - If True -> The relation TaylorOrder+1-index-s==0 is fulfilled

DAEinitialization.consistentValues.constraint(taylorObject, ti, K, dim)¶: constraint delivers the dictionary containing the derivative array of order K of the example (in taylorObject) at timpoint ti.

DAEinitialization.consistentValues.derivativeArray(X, taylorObject, ti, K, dim)¶: derivativeArray creates the derivativeArray as ndarray

DAEinitialization.consistentValues.func(X, P, alpha, ex)¶: Objective function ||P(x_0-alpha)||_2, where X = (x_0,x_1,...) :param X - ndarray of Taylor variable :param P - orthogonal projector :param alpha - ‘initial value’ :param ex

DAEinitialization.consistentValues.func_deriv(X, P, alpha, ex)¶: Derivative of objective function

DAEinitialization.consistentValues.index_Condition(Example, TaylorEx, dimOfDAE, TaylorVector, ti, info_level=0)¶

index_Condition calculates the index of the DAE and the matrix condition of the “Jacobian” of the derivative array

param:	Example - object of the example TaylorEx - object of class TaylorDAE of the example dimOfDAE - dimension of the DAE TaylorVector - TaylorVector as ndarray (e.g. solution of consistentValues) ti - timepoint
return:	index - index of the DAE cond - condition number s - level of s-fullness

DAEanalysis module¶

Created on 20.05.2014/ last change 29.1.18

@author: estevez schwarz/lamour

class DAEanalysis.DerivativeArray.TaylorDAE(example, K, tol=None, info_level=2)¶

TaylorDAE object has the attributes

attribute:	example - object of type “example” (see Implementation of an example) K - Number of Taylor coefficients 0,...,K, i.e., we have K+1 coefficients tol - absolute accuracy for rank determination info_level controls the information output 0 - no output 1 ... 4 output increases

CheckOmegaTaylorAndProjectors(Coeff_Matrix, n)¶

CheckOmegaTaylorAndProjectors checks the index of the DAE and computes the projectors T_j.

param:	Coeff_Matrix - Jacobian matrix of the derivative array with additional A at the first block row n - Number of equations of the original DAE
return:	T - Projectors T_j, all in one matrix index - Differentiation index of the DAE

Check_s_full(CoeffMatrix, n)¶: Check_s_full computes the value s, which characterizes the accuracy of the result.

IndexAndCondition(tti, zt)¶

IndexAndCondition computes the index and the condition number of the DAE.

param:	tti - Taylor object of ti zt - Taylor object of z(t)
return:	index - index of the DAE condM - condition number of index determination s - level of s-fullness

Jacobian_forCoefficentMatrix(tti, zt)¶

The method Jacobian_forCoefficentMatrix computes the Jacobian matrices Atilde and Btilde.

param:	tti - Taylor object of timepoint zt - MyUTPM object of z
return:	The output depends of the structure of the DAE. For type_of_DAE = ‘proper’ Atilde= A.D, Btilde= A D’+ B with f( d’(x,t),x,t) and A = df/dx’(zt,ti), D = dd/dx(zt,ti), B = df/dx(zt,ti) type_of_DAE = ‘standard’ Atilde= A, Btilde= B with f( x’,x,t) type_of_DAE = ‘linear’ Atilde= eval_A(t), Btilde= eval_B(t) with f( x’,x,t) = eval_A x’+eval_B x-q
raised:	Exception - unknown type of DAE

Jacobian_matrices_AB(ti, zt)¶

The method Jacobian_matrices_AB computes the Jacobian matrices

param:	ti - Taylor object zt - MyUTPM object
return:	A = df/dx’(zt,ti) B = df/dx(zt,ti) f_Taylor - derivative array of f (see MyUTPM.tracing_A_B)

Jacobian_matrices_ADB(ti, zt)¶

The method Jacobian_matrices_ADB computes the Jacobian matrices

param:	ti - Taylor object zt - MyUTPM object
return:	A = df/dx’(zt,ti) D = dd/dx(zt,ti) B = df/dx(zt,ti)

Jacobian_matrix_u(zt)¶

The method Jacobian_matrix_u computes the Jacobian matrix of u(x)

param:	zt - MyUTPM object
return:	u_x = du/dx(zt)

coefficent_matrix(tti, xt)¶

coefficient matrix of Taylor method

param:	tti - Taylor representation of ti (time point) xt - Taylor coefficients of x(ti)
return:	Coeff_Matrix - see structure below block_dimension - n

structure of Coeff_Matrix for constant standard DAE Ax’+Bx = q

A
B	A				n(K+1) x n(K+1)
	B	A			self.K+1 - number of Taylor coefficients
		...	...		A, B nxn matrices
			B	A

coefficent_matrix_optimization(tti, xt)¶

coefficient matrix of Taylor method

param:	tti - Taylor representation of ti (time point) xt - Taylor coefficients of x(ti)
return:	Coeff_Matrix - block_dimension -

structure of Coeff_Matrix for constant standard DAE Ax’+Bx = q

B	A				n*K x n(K+1)
	B	A			self.K+1 - number of Taylor coefficients
		...	...		A, B nxn matrices
			B	A

f_taylor(x, t)¶

f_taylor computes the Taylor coefficients of f depending of the DAE type

param:	x - Taylor object of x(t) t - Taylor object of t
return:	f - Taylor expansion of f
raised:	Exception - unknown type of DAE

log_files()¶

log_files provides a list which contains the Logger for the different files.

The following numerical submethods are logged:

0 - computation of consistent initial values
1 - Newton method (not active for InitDAE)
2 - linear algebra results of derivative array computations
3 - integration of IVPs (not active for InitDAE)

static write_Values2file(name, message, dim, r, S1, Sr, Sr1, Send, tol, info_level)¶

write_Values2file writes the values of the rank determination to a file

param:	name message dim r - see toolboxLA.svdinfo.CompCondSing S1 Sr Sr1 Send tol info_level - if >2 information printed on terminal

static write_Values2logger(logger, info_level, print_level, message)¶

write_Values2logger stores messages depending of the info_level in the logger.

param:

logger - used logger

info_level - if > 0 creates an output. The greater the more.

print_level - determines the level of the message.

message - string containing the message

Created on 13.01.2016

@author: lamour

exception DAEanalysis.NumericalException.FNotComputable¶: Bases: exceptions.Exception

exception DAEanalysis.NumericalException.IncreasingTaylorCoefficient(K_start, K, K_end)¶: Bases: exceptions.Exception

exception DAEanalysis.NumericalException.LittleNumberOfTaylorCoefficients(K_start)¶: Bases: exceptions.Exception

exception DAEanalysis.NumericalException.MinimalDampingParameter¶: Bases: exceptions.Exception

exception DAEanalysis.NumericalException.MinimizationNotSuccessful¶: Bases: exceptions.Exception

exception DAEanalysis.NumericalException.SingularDAE(string)¶: Bases: exceptions.Exception

toolboxLA module¶

Created on Thu Mar 05 09:07:30 2015

@author: estevez schwarz

toolboxLA.svdinfo.CompCondSing(M, tol=None)¶

CompCondSing computes the ratio between the largest and the smallest nonzero singular value of a matrix M and returns some related information.

param:	M - Matrix M tol - tolerance for the rank decision (optional). The computation of the default tolerance corresponds to linalg.matrix_rank
return:	cond - S[0]/S[rankM-1]: ratio between the largest and the smallest nonzero singular value of M rankM - considered rank of the matrix M S[0] - largest singular value S[rankM-1] - smallest singular value considered as positive S[min(len(S)-1,rankM)] - largest singular value considered as zero S[-1] - smallest singular value tol - used tolerance

toolboxLA.svdinfo.OrthogonalBasisProjectorAlongIm(A, tol=None)¶

OrthogonalBasisProjectorAlongIm computes the orthogonal basis to build the projector along im(A) for a given matrix A

param:	A - Matrix A
return:	Ur.T - resulting orthogonal basis rS - considered rank of the matrix A (and corresponding S from SVD) S[0] - largest singular value of A S[rS-1] - smallest singular value considered as positive S[min(len(S)-1,rS)] - largest singular value considered as zero S[-1] - smallest singular value

toolboxLA.svdinfo.OrthogonalBasisProjectorOntoKer(A, tol=None)¶

OrthogonalBasisProjectorOntoKer computes the orthogonal basis to build the projector onto ker(A) for a given matrix A

param:	A - Matrix A
return:	VTr - resulting orthogonal basis rS - considered rank of the matrix A (and corresponding S from SVD) S[0] - largest singular value of A S[rS-1] - smallest singular value cosidered as positive S[min(len(S)-1,rS)] - largest singular value considered as zero S[-1] - smallest singular value

toolboxLA.svdinfo.OrthogonalProjectorAlongIm(A, tol=None)¶

OrthogonalProjectorAlongIm computes the orthogonal projector along im(A) for a given matrix A

param:	A - Matrix A
return:	W - resulting orthogonal projector rS - considered rank of the matrix A (and corresponding S from SVD) S1 - largest singular value of A Sr - smallest singular value considered as positive Sr1 - largest singular value considered as zero Send - smallest singular value

toolboxLA.svdinfo.OrthogonalProjectorOntoKer(A, tol=None)¶

OrthogonalProjectorOntoKer computes the orthogonal projector onto ker(A) for a given matrix A

param:	A - Matrix A
return:	Q - resulting orthogonal projector rS - considered rank of the matrix A (and corresponding S from SVD) S1 - largest singular value of A Sr - smallest singular value cosidered as positive Sr1 - largest singular value considered as zero Send - smallest singular value

toolboxLA.svdinfo.RankSigma(S, tol=None)¶

RankSigma computes the rank of the diagonal matrix Sigma (S) from the SVD

param:	S - Matrix S tol - tolerance for the rank decision (optional). The computation of the default tolerance corresponds to linalg.matrix_rank
return:	rank - rank(S) tol - used tolerance

toolboxAD module¶

Created on 13.01.2016

@author: lamour

class toolboxAD.MyUTPM.MyUTPM(X)¶

Bases: algopy.utpm.utpm.UTPM

A_B(cgAB, f, t)¶

Calculation of the Jacobians A and B

param:	cgAB - computational graph f - function f t - UTPM object - taylor_init of t
return:	A - Taylor object of Jacobian df/dx’ B - Taylor object of Jacobian df/dx

A_D_B_new(cg, cgAB, d, f, t)¶

Calculation of the Jacobians A, D and B

param:	cg - computational graph of f cgAB - computational graph of A, B d - function d of proper formulation f - function f t - UTPM object - taylor_init of t
return:	D - Taylor object of Jacobian matrix dd/dx A - Taylor object of Jacobian matrix df(z,x,t)/dz B - Taylor object of Jacobian matrix df(z,x,t)/dx

diff()¶: differentiation for UTPM objects by shifting the coefficients

static taylor_init(D, t0)¶: initialization of independent variable t at t0

tracing_A_B(f, t)¶

The function traces the function f to compute the Jacobians A and B

param:	f - function f t - time point
return:	cgAB - computational graph of f Fyxtt.x.data - ndarray of Taylorcoefficients of f

tracing_A_D_B(d, f, t)¶: The function traces the functions d and f to compute the Jacobians D, A and B

InitDAE for computing consistent initial values¶

DAEinitialization module¶

DAEanalysis module¶

toolboxLA module¶

toolboxAD module¶

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