## Tau Toolbox functions

Here are listed all functions found in Tau Toolbox. New changes from previous versions are informed as (New). You can save it as .m file separated or also download full .rar containg all functions.

@ctau

cos | Cosine for ctau object. | |

cosh | Hiperbolic cosine for ctau object. | |

ctau | Class for coefficients of tau method. | |

exp | Exponential for ctau object. | |

log | Logarithm for ctau object. | |

minus | Minus operation for ctau object. | |

mpower | Power operation for ctau object. | |

mtimes | Product operation for ctau object. | |

plot | Plot for ctau object. | |

plus | Plus operation for ctau object. | |

sin | Sine for ctau object. | |

sinh | Hiperbolic sine for ctau object. | |

uminus | - Unitary minus operation for tau object. | |

uplus | Unitary plus operation for ctau object. |

@dtau

diff | Differentiation for dtau object. | |

dtau | Class for dependent tau variable. | |

fred | Fredholm integral for dtau object. | |

int | Integration for dtau object. | |

matrixC | Builds the matrix C, related with the system [C; D]a = b. | |

matrixCsys | Builds the matrix C, related with the system [C; D]a = b. | |

matrixN | N matrix related with differentiation. | |

matrixO | Sparse O matrix related with integration. | |

minus | Minus operation for tau objects. | |

mrdivide | Division operation for tau objects. | |

mtimes | Product operation for tau objects. | |

plus | Plus operation for tau objects. | |

uminus | - Unitary minus operation for tau objects. | |

uplus | Unitary plus operation for tau objects. | |

volt | Volterra integral term. |

@etau

diff | Differenciation for empty tau object in systems. | |

etau | Class to make zeros y in [a b] in terms of a polynomial basis. | |

fred | Fredholm operator for empty tau object in systems. | |

int | Integration for empty tau object in systems. | |

matrixN | Differentiation for empty tau object in systems. | |

matrixO | Integration for empty tau object in systems. | |

minus | Minus operations for empty tau object in systems. | |

mrdivide | Division for empty tau object in systems. | |

mtimes | Multiplication for empty tau object in systems. | |

plus | Sum for empty tau object in systems. | |

uminus | Unitary minus for empty tau object in systems. | |

uplus | Unitary sum for empty tau object in systems. | |

volt | Volterra operator for empty tau object in systems. |

@itau

cos | Cosine for itau object. | |

cosh | Hiperbolic cosine for itau object. | |

coshi | Hiperbolic cosine by orthogonal interpolation for itau object. | |

coshm | Hiperbolic cosine for matrix for itau object. | |

cosht | Hiperbolic cosine by tau approximation for itau object. | |

cosi | Cosine by orthogonal interpolation for itau object. | |

cosm | Cosine for matrix for itau object. | |

cost | Cosine by tau approximation for itau object. | |

exp | Exponential for itau object. | |

expi | Exponential by orthogonal interpolation for itau object. | |

expm | Exponential for matrix for itau object. | |

expt | Exponential by tau approximation for itau object. | |

itau | Class for independent tau variable. | |

linspace | Linear space for itau object. | |

log | Logarithm for itau object. | |

logi | Logarithm by orthogonal interpolation for itau object. | |

logm | Logarithm for matrix for itau object. | |

logt | Logarithm by tau approximation for itau object. | |

matrixM | Sparse M matrix. | |

minus | - Minus operation for tau object. | |

mpower | Power operation for tau object. | |

mtimes | Product operation for tau object. | |

plus | Plus operation for tau object. | |

pol | Polynomial evaluation. | |

pow2orth | Change from power series to orthogonal basis. | |

powMrec | Power of M matrix by recurrence. | |

sin | Sine for itau object. | |

sinh | Hiperbolic sine for itau object. | |

sinhi | Hiperbolic sine by orthogonal interpolation for itau object. | |

sinhm | Hiperbolic sine for matrix for itau object. | |

sinht | Hiperbolic sine by tau approximation for itau object. | |

sini | Sine by orthogonal interpolation for itau object. | |

sinm | Sine for matrix for itau object. | |

sint | Sine by tau approximation for itau object. | |

sqrt | Square root for itau object. | |

sqrti | Square root by orthogonal interpolation for itau object. | |

sqrtm | Square root for matrix for itau object. | |

sqrtt | Square root by tau approximation for itau object. | |

uminus | - Unitary minus operation for tau object. | |

uplus | Unitary plus operation for tau object. |

@rtau

cos | Cosine by tau approximation for rtau object. | |

cosh | Hiperbolic cosine by tau approximation for rtau object. | |

exp | Exponential by tau approximation for rtau object. | |

log | Logarithm by tau approximation for rtau object. | |

minus | Minus operation for rtau object. | |

mpower | Power operation for rtau object. | |

mtimes | Product operation for rtau object. | |

plus | Plus operation for ctau object. | |

rtau | Class to rigth member of tau problem. | |

sin | Sine by tau approximation for rtau object. | |

sinh | Hiperbolic sine by tau approximation for rtau object. | |

uminus | - Unitary minus operation for rtau object. | |

uplus | Unitary plus operation for rtau object. |

BuildOrthogonalBasis

chebyshevTpoints | Chebyshev points in [a b]. | |

chebyshevUpoints | Chebyshev points in [a b]. | |

gegenbauerCpoints | GegenbauerC points in [a b]. | |

legendrePpoints | Legendre points in [a b]. | |

orth2powmatrix | Orthogonal to power matrix. | |

orthobasispoints | Orthogonal basis points in [a b]. | |

plotpoly | Plot orthogonal polynomials. | |

polbesselY | Bessel polynomial basis in unitary circle. | |

polchebyshevT | Chebyshev polynomial basis (T) in [-1 1]. | |

polchebyshevTab | Chebyshev polynomial (T) basis in [a b]. | |

polchebyshevU | Chebyshev polynoamil basis (U) in [-1 1]. | |

polchebyshevUab | Chebyshev polynomial (U) basisin [a b]. | |

polgegenbauerC | Gegenbauer polynomial basis in [-1 1]. | |

polgegenbauerCab | Gegenbauer polynomial basis in [a b]. | |

polhermiteH | Hermite polynomial basis in [-Inf Inf]. | |

pollaguerreL | Laguerre polynomial basis in [0 Inf]. | |

pollegendreP | Legendre polynomial basis in [-1 1]. | |

pollegendrePab | Legendre polynomial basis in [a b]. | |

pow2orthmatrix | Power to orthogonal matrix. | |

shift2ab | Change interval of orthogonality to [a b]. |

Examples

FunOfCoef4nonlinear

cosho | Orthogonal hiperbolic cosine. | |

coso | Orthogonal cosine. | |

expo | Orthogonal exponential. | |

logo | Orthogonal logarithm. | |

sinho | Orthogonal hiperbolic sine. | |

sino | Orthogonal sine. | |

sqrto | Orthogonal square root. |

FunctionOfMatrix

coshm | Matrix hiperbolic cosine. | |

cosht | Cos by tau approximation. | |

cosm | Matrix cosine. | |

cost | Cos by tau approximation. | |

expt | Exp by tau approximation. | |

logt | Log by tau approximation. | |

sinhm | Matrix hyperbolic sine. | |

sinht | Sinh by tau approximation. | |

sinm | Matrix sine. | |

sint | Sin by tau approximation. |

OperatorsEvaluation

taudiff | Tau derivative evaluation. | |

taufred | Tau Fredholm evaluation. | |

tauint | Tau integration evaluation. | |

tauvolt | Tau Volterra evaluation. | |

verapproximation | Verify approximation. |

OrthogonalEvaluation

orthoval | Orthogonal evaluation. | |

orthovalM | Orthogonal evaluation for matrices. | |

orthovalMj | j-th orthogonal evaluation for M matrices. | |

orthovald | Orthogonal (and its derivatives) evaluation. | |

orthovalv | Orthogonal evaluation. | |

orthovalvj | j-th orthogonal evaluation. |

OrthogonalInterpolation

interporth | Orthogonal polynomial interpolation. | |

interporth2 | Orthogonal polynomial interpolation for 2 dimension. | |

interporth2_error | Error in the orthogonal interpolation of F(x,t). | |

interporth2inc | Incremental orthogonal polynomial interpolation for 2D. | |

interporthinc | Incremental orthogonal polynomial interpolation. |

ProductOfPolynomials

chebypolypow | Power of Chebyshev polynomials. | |

chebypolyprod | Product of Chebyshev polynomials. | |

legpolypow | Power of Legendre polynoamials. | |

legpolyprod | Product of Legendre polynomials. | |

orthopolyprod | Product of orthogonal polynomials. |

TauGui

createmfile_advanced_ode | Creates an m-file for ode in advanced mode. | |

createmfile_advanced_ode_pw | Creates an m-file for ode (piecewise) | |

createmfile_advanced_sys | Creates an m-file for systems of odes | |

createmfile_advanced_sys_pw | Creates an m-file for systems of odes | |

createmfile_beginner | Creates an m-file for the problem in beginner mode. | |

findsolver | Find the appropriate solver from taugui. | |

getalldata | Get all data. | |

informations | Informations for taugui. | |

taugui | TAUGUI MATLAB code for taugui.fig |

TauSolvers

schur_iter | Schur iterative. | |

schur_luinc | LU incemental. | |

schur_luinc2 | LU incemental. | |

schur_nflu | Schur with non fixed LU. | |

schurc | Schur complements. | |

schursolver | Iterative Lanczos' tau method based on Schur complements. | |

schursolverpw | Iterative Lanczos' tau method based on Schur complements. | |

tauode | Lanczos' tau method for linear ODEs. | |

tauodenewton | Tau method for linearized problems by newton iterations. | |

tauodepw | Piecewise Lanczos' tau method for linear ODEs. | |

tauodepwpvf | Piecewise Lanczos' tau method for linear systems of BVP. | |

tausolver | Lanczos' tau method for linear differential problems. | |

tausys | Lanczos' tau method for system of linear ODEs. | |

tausysnewton | Tau method for linearized problems by newton iterations. | |

tausyspw | Piecewise Lanczos' tau method for system of linear ODEs. |

Tests

TEST_Mrec_vs_Mformal | This .m is an m-script. | |

TEST_Mrec_vs_Mformal_problem | This .m is an m-script. | |

TEST_coef_fun_by_Schurcomplements | This .m is an m-script. | |

TEST_coeflinearization_vs_convolution | This .m is an m-script. | |

TEST_differents_kind_of_orthoval | This .m is an m-script. | |

TEST_invVAV_vs_recA | This .m is an m-script. | |

TEST_invVAV_vs_recA_problem | This .m is an m-script. | |

TEST_invVAV_vs_recA_problem2 | This .m is an m-script. | |

TEST_invVAV_vs_recA_spy | This .m is an m-script. | |

TEST_invV_vc_W | This .m is an m-script. | |

TEST_invV_vc_W_problem | This .m is an m-script. | |

TEST_polyval_vs_orthoval | This .m is an m-script. | |

TEST_polyval_vs_orthoval_kernel | This .m is an m-script. |

Tools4Tau

Cauchy_error | norm-2 of the error between two approximations with | |

MbyLU | M preconditioner by LU factorization. | |

altura | Find the D height. | |

basis2basis | change of basis. | |

changesupc | Change the conditions format. | |

colorR2B | Color scale from red to blue. | |

deriveP | Derivative of matrix V. | |

domain2 | Creates a two dimentional domain. | |

eorth | ith vector from orthogonal basis. | |

epow | ith vector from canonical basis. | |

evalonly | Evaluate selected variables in a string. | |

evalonlynewton | Evaluate selected variables in a string. | |

fc | Functional coefficient. | |

findint | Find integral terms. | |

findkernel | Find kernel of integral term. | |

fun2orth | Funtion to orthogonal polynomial basis. | |

isivp | Checks for initial value problems | |

matrixT | T matrix related with Ta=b. | |

maxerror | Maximal error between a function and its approximation. | |

mno | M N and O matrices. | |

myspy | Plots a 3D spy of matrix T | |

poly2str | Converts multivariate polynomials from arrays to strings. | |

prodcoef | Product of coefficients. | |

readcondcell | Read conditions. | |

readcondcellsys | Read conditions for systems. | |

reorg2poly | Reorganize to polynomial. | |

reorg2vec | Reorganize to vector. | |

solvingthesystem | Solve a system of linear algebraic equations. | |

spliteq | Split equation. | |

str2poly | Converts multivariate polynomials from strings to arrays. | |

tau | Creates (independent and dependent) tau variables. | |

tauc | Tau clearing. | |

tausettings | Tau settings. | |

tausystem | Matrix T and vector b. |