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# Symbolic Math Toolbox Release Notes

## R2014b

New Features, Bug Fixes, Compatibility Considerations

#### MATLAB solve function returning parameters and conditions in solutions

The MATLAB® symbolic solve function returns the parameters in a solution and the conditions under which a solution is valid when the ReturnConditions option is specified as true.

#### Compatibility Considerations

• Do not specify equations and variables as character strings. Instead of string inputs, declare symbolic variables using syms, and then pass them as input arguments separated by commas, or as a vector of input arguments. For example, replace

`solve('2*r = 1, r')`

with

```syms r
solve(2*r == 1, r)```

In a future release, string input arguments will be interpreted as option names to support shortcuts, such as ignorea for IgnoreAnalyticConstraints.

• solve warns when it calls the numerical solver and returns a numerical output.

• solve does not warn if provably no solution exists.

• When the list of equations is empty, solve throws an error instead of a warning.

#### Functions for analyzing and reducing systems of differential algebraic equations (DAEs), such as isLowIndexDAE and reduceDAEIndex

• Identify subsets (blocks) of equations that can be used to define subsets of variables.

• Identify high-index differential algebraic equations.

• Reduce high-index differential algebraic equations to systems of differential index 1 or 0.

#### MATLAB functions representing orthogonal polynomials: chebyshevT, chebyshevU, legendreP, laguerreL, hermiteH, jacobiP, and gegenbauerC

• The MATLAB symbolic chebyshevT and chebyshevU functions represent Chebyshev polynomials of the first and second kind, respectively.

• The MATLAB symbolic gegenbauerC function represents Gegenbauer polynomials.

• The MATLAB symbolic hermiteH function represents Hermite polynomials.

• The MATLAB symbolic jacobiP function represents Jacobi polynomials.

• The MATLAB symbolic legendreP function represents Legendre polynomials.

• The MATLAB symbolic laguerreL function represents Laguerre polynomials.

#### MATLAB pade function for computing Padé approximation

The MATLAB symbolic pade function calculates Padé approximations.

#### funm function for computing matrix functions

The MATLAB symbolic funm function and the MuPAD funm function represent a general matrix function. A matrix function is a scalar function that maps one matrix to another, for example, function f in B = f(A) maps matrix A to matrix B.

#### MATLAB kummerU function for computing confluent hypergeometric (Kummer U) function

The MATLAB symbolic kummerU function computes the value of the confluent hypergeometric function. This function is also known as the Kummer U function.

#### MATLAB polylog function for computing polylogarithms

The MATLAB symbolic polylog function computes polylogarithms.

#### MATLAB signIm function for computing signs of imaginary parts of complex numbers

The MATLAB symbolic signIm function returns a sign of the imaginary part of a complex number. For all complex numbers with a nonzero imaginary part, singIm(z) = sign(imag(z)). For real numbers, signIm(z) = -sign(z).

#### MATLAB in function for representing conditions on symbolic inputs

The MATLAB symbolic in function represents the condition that the input is of the specified type. The allowed types are integer, real and rational. The in function is used in the input and output of other functions such as solve to represent conditions on symbolic variables. If the input is a number of the specified type, the in function returns logical 1 (true), and if it is not of the specified type, the in function returns logical 0 (false).

#### MATLAB divisors function for finding divisors of integers and polynomials

The MATLAB symbolic divisors function computes divisors of integers and polynomials.

#### MATLAB functions nnz and nonzeros for finding nonzero elements in a symbolic array

The MATLAB symbolic nnz function computes the number of nonzero elements in a symbolic vector, matrix, or multidimensional array.

The MATLAB symbolic nonzeros function returns a column vector containing all nonzero elements of a symbolic vector, matrix, or multidimensional array.

#### MATLAB pochhammer function to calculate Pochhammer symbol

The MATLAB symbolic pochhammer function calculates the Pochhammer symbol.

#### MATLAB kroneckerDelta function for computing the Kronecker delta function

The MATLAB symbolic kroneckerDelta function calculates the Kronecker delta function.

#### MATLAB dirac function with two input arguments for computing derivatives of the Dirac delta function

The MATLAB symbolic dirac function with one input argument represents the Dirac delta function. dirac with two input arguments, dirac(n,x) represents the nth derivative of the Dirac delta function at x.

#### MATLAB isAlways function warns when returning false for undecidable inputs

The MATLAB symbolic isAlways function issues a warning when it returns logical 0 (false) for an undecidable input.

#### Compatibility Considerations

• In previous releases, isAlways did not issue a warning before returning logical 0 (false) for an undecidable input. To go back to this behavior, suppress the warning by specifying the Unknown option as false, as in isAlways(cond,'Unknown','false').

#### MuPAD generate::fortran function can use Fortran 90 as the target compiler

The MuPAD generate::fortran function can use Fortran 90 as the target compiler, in addition to Fortran 77.

#### MATLAB mod function for computing modulus after division

The MATLAB symbolic mod function finds the modulus after division as follows: mod(a,b) = a - b*floor(a/b).

#### Compatibility Considerations

The MATLAB symbolic mod function uses the same definition as the MuPAD modp function. Also, by default, the MuPAD mod operator and its functional form _mod are equivalent to the MuPAD modp function.

In previous releases, the MuPAD modp and mods functions computed the modulus after division according to these definitions:

• If a and b are integers, then modp(a,b) is an integer r, such that a = qb + r, 0 <= r < |b|, and q = a div b.

If b is an integer and a is a rational number, a = u/v, such that v and b are coprime integers, then modp(a,b) = modp(u*w,b). Here, w is an inverse of v mod b, that is, v*w1 (mod b).

• If a and b are integers, then mods(a,b) is an integer r, such that a = qb + r and -|b|/2 < r <= |b|/2.

If b is an integer and a is a rational number, a = u/v, such that v and b are coprime integers, then mods(a,b) = mods(u*w,b). Here, w is an inverse of v mod b, that is, v*w1 (mod b).

Now, MuPAD uses the same definitions for an integer or rational a. As in previous releases, b must be an integer.

• modp(a,b) = a - b*floor(a/b)

• mods(a,b) = a - b*round(a/b)

In MuPAD, to get the same results as in previous releases, use the MuPAD numlib::lincongruence function to find a modular inverse, and then use the MuPAD modulo operator or functions, for example:

```w := numlib::lincongruence(3, 1, 5)[1]:
22*w mod 5;
modp(22*w, 5);
mods(22*w, 5);
_mod(22*w, 5)```

In MATLAB, to get the same results as in previous releases, use the gcd function with three output arguments to find a modular inverse, and then use the mod function as follows. For example, for mod(a,b) = mod(u/v,b) use these commands:

```[~, A, ~] = gcd(sym(v), b);
mod(A*u, b)```

#### MATLAB gcd and lcm functions accept vectors and matrices

The MATLAB symbolic gcd and lcm functions accept vectors and matrices as input arguments. If A is a vector or a matrix, then gcd(A) and lcm(A) find the greatest common divisor and least common multiple of all elements of A. If A and B are vectors or matrices of the same size, then gcd(A,B) and lcm(A,B) find the greatest common divisor and least common multiple of the pairs of elements of A and B.

#### MATLAB rem function accepts vectors and matrices

The MATLAB symbolic rem function accepts vectors and matrices as input arguments. Also, the new MuPAD rem function lets you compute a remainder after division in a MuPAD notebook.

#### Compatibility Considerations

In previous release, rem accepted polynomials as its input arguments. For example, sym x; R = rem(x^2 + 2, x) returned 2. To perform polynomial division in this and later releases, use the quorem function, for example, sym x; [~, R] = quorem(x^2 + 2, x).

#### MATLAB factor function only accepts scalar inputs and returns vector of factors of input

The MATLAB symbolic factor function only accepts scalar inputs. The factor function returns a symbolic vector of irreducible factors of the input.

#### Compatibility Considerations

The factor function does not accept nonscalar inputs. The output is a symbolic vector and is not of type factored.

#### MuPAD Notebook app supports left and right double square brackets

The MuPAD Symbol::LeftDoubleBracket and Symbol::RightDoubleBracket functions insert left and right double square brackets. These symbols are also called white square brackets.

#### Functionality being removed or changed

FunctionalityWhat Happens When You Use It?Use This InsteadCompatibility Considerations
mfunStill runsAppropriate special function. For example, replace mfun('P',1,2,3,4) with jacobiP(1,2,3,4).Replace all instances of mfun with the appropriate function call. See Special Functions for the list of available special functions.
mfunlistStill runsSee Special Functions.See Special Functions for the list of available special functions.
simpleWarnssimplifyReplace all instances of simple(S) with simplify(S). There is no replacement for [r,how] = simple(S).
emlBlockErrorsmatlabFunctionBlockReplace all instances of emlBlock with matlabFunctionBlock.

## R2014a

New Features, Bug Fixes, Compatibility Considerations

#### MATLAB functions for computing special integrals, gamma functions, dilogarithm function, and number-theoretic functions

The following special functions are available:

• The MATLAB symbolic sinhint and coshint function compute the hyperbolic sine and cosine integral functions, respectively.

• The MATLAB symbolic ssinint function computes the shifted sine integral function.

• The MATLAB symbolic dawson function computes the Dawson integral.

• The MATLAB symbolic fresnelc and fresnels functions return the Fresnel cosine and sine integral functions respectively.

• The MATLAB symbolic logint function computes the logarithmic integral function. This function is also called the integral logarithm.

• The MATLAB symbolic gammaln function computes the logarithmic gamma function.

• The MATLAB symbolic igamma function computes the incomplete gamma function.

• The MATLAB symbolic dilog function computes the dilogarithm function.

• The MATLAB symbolic bernoulli function computes the Bernoulli numbers and polynomials.

• The MATLAB symbolic euler function computes the Euler numbers and polynomials.

• The MATLAB symbolic harmonic function computes the harmonic function. For positive integer arguments, the harmonic function produces harmonic numbers.

• The MATLAB symbolic catalan function represents the Catalan constant. To approximate the Catalan constant with the current precision set by digits, use vpa(catalan).

• The MATLAB symbolic eulergamma function represents the Euler-Mascheroni constant. To approximate the Euler-Mascheroni constant with the current precision set by digits, use vpa(eulergamma).

#### MATLAB function qr for computing symbolic QR factorization

The MATLAB symbolic qr function computes the QR factorization of a matrix. The result can be used to solve matrix equations.

#### MATLAB function combine for combining symbolic expressions with multiple calls to the same function

The MATLAB symbolic combine function applies rewriting rules to the input expression to combine multiple calls to a function, and returns the rewritten expression. The analytic constraints on applying rewriting rules can be optionally relaxed when the function is called.

#### MATLAB functions max and min for finding the largest and smallest elements of a symbolic array

The MATLAB symbolic max and min functions return the largest and the smallest element of a symbolic vector or matrix, all elements of which are convertible to floating-point numbers. For a symbolic matrix, these functions find the largest and smallest elements of each row or column.

#### vpasolve can use random starting points when searching for solutions

The MATLAB numeric solver vpasolve now uses random starting points when searching for solutions if you specify random. This enables the solver to find different solutions for nonpolynomial equations in subsequent calls.

#### Support for Unicode characters in MuPAD that includes using Asian language characters in strings and text

The toolbox provides support for Unicode® characters in MuPAD (including messages to print or display), variable names, file names, and external file content.

#### Compatibility Considerations

In previous releases, the MuPAD strmatch function used [^[] to match any characters excluding [. For example, the command strmatch("a[b", "[^[]", All) returned {"a", "b"}.

Now, use [^\\[] to match any characters excluding [. Thus, rewrite the example as follows: strmatch("a[b", "[^\\[]", All).

strmatch requires the same change for the closing parenthesis ].

#### Support for specifying encoding in MuPAD file operations

The MuPAD functions for file operations, such as finput, fopen, fprint, read, write and more, accept the option Encoding. This option lets you specify the following values for encoding.

 Big5 ISO-8859-1 windows-932 EUC-JP ISO-8859-2 windows-936 GBK ISO-8859-3 windows-949 KSC_5601 ISO-8859-4 windows-950 Macintosh ISO-8859-9 windows-1250 Shift_JIS ISO-8859-13 windows-1251 US-ASCII ISO-8859-15 windows-1252 UTF-8 windows-1253 windows-1254 windows-1257

#### Choice of right- or left-handed spherical coordinate system for the MuPAD vector analysis functions

The MuPAD vector analysis functions curl, divergence, gradient, laplacian, and linalg::ogCoordTab let you choose between right- and left-handed spherical coordinate systems. By default, these functions use the right-handed coordinate system with [radial, polar, azimuthal] coordinates. To switch to [radial, azimuthal, polar] coordinates, specify Spherical[LeftHanded].

#### Compatibility Considerations

In previous releases, the MuPAD vector analysis functions used the left-handed spherical coordinate system. To get the same results as in previous releases, use 'Spherical[LeftHanded]'. To use the right-handed spherical coordinate system and suppress the warning, use 'Spherical[RightHanded]'.

#### MATLAB special functions and functions for computing integral and Z-transforms accept several nonscalar arguments

The following MATLAB symbolic functions now accept more than one nonscalar argument:

#### MATLAB function erfc accepts two arguments

The MATLAB symbolic erfc function with one input argument represents the complementary error function. erfc with two input arguments represents the iterated integrals of the complementary error function, erfc(k,x) = int(erfc(k - 1, y), y, x, inf).

#### Functionality being removed or changed

FunctionalityWhat Happens When You Use It?Use This InsteadCompatibility Considerations
MuPAD linalg::curlStill runscurlReplace all instances of linalg::curl with curl.
MuPAD linalg::detStill runsdetReplace all instances of linalg::det with det.
MuPAD linalg::divergenceStill runsdivergenceReplace all instances of linalg::divergence with divergence.
MuPAD linalg::hessianStill runshessianReplace all instances of linalg::hessian with hessian.
MuPAD linalg::jacobianStill runsjacobianReplace all instances of linalg::jacobian with jacobian.
MuPAD linalg::laplacianStill runslaplacianReplace all instances of linalg::laplacian with laplacian.
MuPAD linalg::potentialStill runspotentialReplace all instances of linalg::potential with potential.
MuPAD linalg::vectorPotentialStill runsvectorPotentialReplace all instances of linalg::vectorPotential with vectorPotential.
simpleWarnssimplifyReplace all instances of simple(S) with simplify(S). There is no replacement for [r, how] = simple(S).
emlBlockErrorsmatlabFunctionBlockReplace all instances of emlBlock with matlabFunctionBlock.

## R2013b

New Features, Bug Fixes, Compatibility Considerations

#### MATLAB evaluateMuPADNotebook and allMuPADNotebooks functions to evaluate MuPAD notebooks and return list of open notebooks

The MATLAB symbolic evaluateMuPADNotebook function lets you evaluate a MuPAD notebook from MATLAB without leaving the MATLAB Command Window or MATLAB Editor. You also can interrupt an evaluation of a MuPAD notebook from MATLAB.

The MATLAB symbolic allMuPADNotebooks function identifies all currently open notebooks and returns a vector of handles to them. You can use this vector to evaluate all or some of the notebooks or close them. If you already created a MuPAD notebook without a handle or if you lost the handle to a notebook, allMuPADNotebooks helps you create a new handle without saving the notebook.

#### bernstein function for approximating functions using Bernstein polynomials, and bernsteinMatrix function for computing Bezier curves

The MATLAB symbolic bernstein function and the MuPAD bernstein function approximate symbolic expressions and functions by Bernstein polynomials. The MATLAB symbolic bernsteinMatrix function and the MuPAD bernsteinMatrix function serve for constructing Bezier curves.

#### MATLAB cumsum and cumprod functions for computing cumulative sums and products

The MATLAB symbolic cumsum and cumprod functions return cumulative sums and products of elements of symbolic vectors and matrices.

#### MATLAB isfinite, isinf, and isnan functions for testing for finite, infinite, and NaN elements in symbolic arrays

The MATLAB symbolic isfinite, isinf, and isnan functions test whether the elements of a symbolic array are finite, infinite, or NaNs.

#### ExclusiveConditions option that makes MuPAD piecewise function equivalent to an if-elif-end_if statement

The new ExclusiveConditions option of the MuPAD piecewise function fixes the order of branches in a piecewise expression. Thus, piecewise with ExclusiveConditions is almost equivalent to an if-elif-end_if statement, except that piecewise takes into account assumptions on identifiers. For example, if the condition in the first branch returns TRUE, then piecewise returns the expression from the first branch. If a true condition appears in any further branch, then piecewise returns the expression from that branch and removes all subsequent branches.

#### MATLAB mupadNotebookTitle function to find the window title of the MuPAD notebook

The MATLAB symbolic mupadNotebookTitle function returns a cell array containing the window title of the MuPAD notebook. This function lets you find the title of a particular notebook as well as all currently open notebooks.

#### MATLAB close function to close MuPAD notebooks from MATLAB

The MATLAB symbolic close function lets you close MuPAD notebooks without leaving the MATLAB Command Window. This function also accepts the 'force' flag suppressing the dialog box that prompts you to save changes.

#### diff supports mixed derivatives

The MATLAB symbolic diff function lets you compute mixed derivatives in one function call. For example, diff(S,x,y) differentiates the expression S with respect to the variables x, and then differentiates the result with respect to the variable y.

#### coeffs function extracts coefficients of multivariate polynomials

The MATLAB symbolic coeffs function returns coefficients of multivariate polynomials. You can specify polynomial variables as a vector of these variables. If you do not specify the polynomial variables, then coeffs regards all symbolic variables found in the polynomial expression as polynomial variables.

#### linspace, logspace, and compan functions for symbolic objects

The MATLAB linspace and logspace functions, which generate linearly and logarithmically spaced vectors, and the compan function, which finds the companion matrix, now accept symbolic numbers, variables, expressions, and functions.

#### Indexing uses lists, vectors, and matrices of indices

The MuPAD _index function and its equivalent [] now accept lists, vectors, and matrices as indices.

#### MuPAD lets you set assumptions on matrices

The MuPAD assume, assumeAlso, assuming, and assumingAlso functions let you set assumptions on matrices.

#### int, symprod, and symsum let you specify lower and upper bounds as vectors

The MATLAB symbolic int, symprod, and symsum functions accept integration, summation, and product intervals specified by row and column vectors. For example, int(expr,var,[a,b]), int(expr,var,[a b]), and int(expr,var,[a;b]) are equivalent to int(expr,var,a,b).

#### Functionality being removed or changed

FunctionalityWhat Happens When You Use It?Use This InsteadCompatibility Considerations
simpleStill runssimplifyReplace all instances of simple(S) with simplify(S). There is no replacement for [r, how] = simple(S).
emlBlockWarnsmatlabFunctionBlockReplace all instances of emlBlock with matlabFunctionBlock.
diff and int methods for inputs of the char typeErrorssymUse the sym method instead.
MuPAD factoring functions numlib::mpqs, numlib::pollard, and numlib::ecmErrorsifactorReplace all instances of numlib::mpqs, numlib::pollard, and numlib::ecm with ifactor.
MuPAD Dom::SparseMatrixF2 domainErrorsDom::Matrix(Dom::IntegerMod(2))Replace all instances of Dom::SparseMatrixF2 with Dom::Matrix(Dom::IntegerMod(2)).
MuPAD setuserinfoErrorsprog::trace or debug Try using prog::trace or debug instead of setuserinfo.

## R2013a

New Features, Bug Fixes, Compatibility Considerations

#### Linear algebra functions for computing matrix factorizations (lu, chol), pseudoinverse, orthogonal basis, and adjoint

• lu computes the LU factorization of a matrix. Permutation information can be returned as a matrix or as a row vector.

• chol computes the Cholesky factorization of a matrix. The result can be returned as an upper or lower triangular matrix. Permutation information can be returned as a matrix or as a row vector.

• pinv computes the Moore-Penrose pseudoinverse of a matrix.

• orth computes an orthonormal basis for the range of a symbolic matrix.

• adjoint computes the adjoint of a symbolic square matrix.

#### Verification of solutions of systems of equations and arbitrary symbolic function substitution in subs function

The MATLAB symbolic subs function lets you:

• Verify solutions of systems of equations by substituting the solutions returned by solve back into the systems

• Substitute elements of a symbolic expression with arbitrary symbolic functions

#### Compatibility Considerations

subs(s,old,new,0) will not accept 0 in a future release. Replace all instances of subs(s,old,new,0) with subs(s,old,new). The subs function does not switch old and new anymore.

subs does not return double-precision floating-point results anymore. Instead, it consistently returns symbolic results. To convert such results to double-precision numbers, use double.

#### Simplification for more types of trigonometric and hyperbolic expressions and expressions with nested roots

The MATLAB symbolic simplify function and the MuPAD simplify function achieve better simplification of trigonometric expressions and expressions with nested roots.

The MATLAB symbolic simplify function accepts the new Criterion option. This option lets you discourage simplify from returning results containing complex numbers.

The MuPAD simplify function accepts two new options:

• Steps specifies the number of internal simplification steps.

• Seconds limits the time allowed for the internal simplification process.

#### Compatibility Considerations

The default number of simplification steps used by the MATLAB symbolic simplify function and the MuPAD simplify function changed from 100 to 1.

The FinalValuation option used in MuPAD Simplify function calls is renamed. The new name is Criterion.

#### Special functions for computing polar angle, atan2 function, imaginary error function, and exponential and elliptic integrals

• angle computes the polar angle of a complex value.

• atan2 computes the four-quadrant inverse tangent (arctangent).

• erfi computes the imaginary error function.

• ei computes the one-argument exponential integral.

• expint computes the two-argument exponential integral.

The following new MATLAB symbolic functions compute elliptic integrals:

• ellipticK computes the complete elliptic integral of the first kind.

• ellipticF computes the incomplete elliptic integral of the first kind.

• ellipticE computes the complete and incomplete elliptic integrals of the second kind.

• ellipticCK computes the complementary complete elliptic integral of the first kind.

• ellipticCE computes the complementary complete elliptic integral of the second kind.

• ellipticPi computes the complete and incomplete elliptic integrals of the third kind.

• ellipticCPi computes the complementary complete elliptic integral of the third kind.

• ellipke computes the complete elliptic integrals of the first and second kinds simultaneously.

#### toeplitz function for creating Toeplitz matrices

The new MATLAB symbolic toeplitz function generates a symbolic Toeplitz matrix from two vectors that specify its first column and first row. This function can also generate a symmetric Toeplitz matrix from one vector.

The MuPAD linalg::toeplitz function now generates a Toeplitz matrix from two vectors that specify its first column and first row. (In MuPAD, vectors are created as 1-by-n or n-by-1 matrices.) linalg::toeplitz accepts the new syntaxes along with the existing syntaxes.

#### sqrtm function for computing square roots of matrices

The MATLAB symbolic sqrtm function computes the square root of a symbolic matrix.

#### sign function for computing signs of numbers

The MATLAB symbolic sign function returns signs of symbolic real and complex values. The sign of a complex value z is defined as z/abs(z).

#### Real option of the linalg::orthog function for avoiding complex conjugates

The MuPAD linalg::orthog function accepts the new Real option. This option lets you avoid using a complex scalar product in the orthogonalization process.

#### Real option of the linalg::factorCholesky function for avoiding complex conjugates

The MuPAD linalg::factorCholesky function accepts the new Real option. When you use this option, linalg::factorCholesky assumes that the input matrix is real and symmetric, and does not apply complex conjugation in the course of the algorithm.

#### Compatibility Considerations

linalg::factorCholesky can now compute the Cholesky factorization of a complex Hermitian positive definite matrix. In previous releases, linalg::factorCholesky required the input matrix to be symmetric even when working with complex entries. To get the same results as in previous releases for symmetric matrices, use the Real option.

#### New arguments of the svd function for computing the "economy size" singular value decomposition

svd accepts the new arguments 0 and 'econ' that let you compute the "economy size" singular value decomposition of a matrix.

#### isequaln function for testing equality of symbolic objects

The MATLAB isequaln function tests symbolic objects for equality, treating NaN values as equal.

#### Control over the order in which solve and vpasolve functions return solutions

The MATLAB symbolic solve and vpasolve functions now let you control the order in which they return solutions. To ensure the order of the returned solutions, explicitly specify the independent (input) variables. For example, the syntax [b,a] = solve(eqns,b,a) guarantees the order of the returned solutions, while the syntax [b,a] = solve(eqns) does not.

#### Functionality being removed or changed

FunctionalityWhat Happens When You Use It?Use This InsteadCompatibility Considerations
simpleStill runssimplifyReplace all instances of simple(S) with simplify(S). There is no replacement for [r, how] = simple(S).
emlBlockWarnsmatlabFunctionBlockReplace all instances of emlBlock with matlabFunctionBlock.
MuPAD factoring functions numlib::mpqs, numlib::pollard, and numlib::ecmWarnsifactorReplace all instances of numlib::mpqs, numlib::pollard, and numlib::ecm with ifactor.
MuPAD Dom::SparseMatrixF2 domainWarnsDom::Matrix(Dom::IntegerMod(2))Replace all instances of Dom::SparseMatrixF2 with Dom::Matrix(Dom::IntegerMod(2)).
MuPAD setuserinfoWarnsprog::trace or debug Try using prog::trace or debug instead of setuserinfo.
polyErrorscharpolyReplace all instances of poly with charpoly.
sqrt target of the MuPAD simplify functionErrorsMuPAD radsimp or simplifyRadicalReplace all instances of simplify function calls involving the sqrt target with radsimp or simplifyRadical. Alternatively, replace these calls with simplify function calls without targets.
cos, sin, exp, and ln targets of the MuPAD simplify function ErrorsMuPAD simplify without targets Replace all instances of simplify function calls involving these targets with simplify function calls without targets. This can lead to a better simplification for some expressions.
MuPAD transform::fourierErrorsMuPAD fourierReplace all instances of transform::fourier with fourier.
MuPAD transform::invfourierErrorsMuPAD ifourierReplace all instances of transform::invfourier with ifourier.
MuPAD transform::laplaceErrorsMuPAD laplaceReplace all instances of transform::laplace with laplace.
MuPAD transform::invlaplaceErrorsMuPAD ilaplaceReplace all instances of transform::invlaplace with ilaplace.
MuPAD transform::ztransErrorsMuPAD ztransReplace all instances of transform::ztrans with ztrans.
MuPAD transform::invztransErrorsMuPAD iztransReplace all instances of transform::invztrans with iztrans.
MuPAD prog::calledFromErrorscontext(hold(procname))Replace all instances of prog::calledFrom() with context(hold(procname)).
MuPAD prog::memuseErrorsprog::trace(Mem)or bytes() Use prog::trace(Mem) or bytes() instead of prog::memuse.
MuPAD prog::testfuncErrorsprint(Unquoted, "...") Use print(Unquoted, "...") instead of prog::testfunc.
MuPAD prog::testmethodErrorsprog::test(..., Method = myTestMethod)Use prog::test(..., Method = myTestMethod) instead of prog::testmethod.
Dynamic modules for MuPAD, including the module, external, and Pref::unloadableModules functions and all functions of the module libraryErrorsNothingNo replacement

## R2012b

New Features, Bug Fixes, Compatibility Considerations

#### MATLAB symbolic matrix analysis functions for characteristic (charpoly) and minimal (minpoly) polynomials and for norm (norm) and condition (cond) number

charpoly computes the characteristic polynomial of a matrix.

minpoly computes the minimal polynomial of a matrix.

norm computes the 2-norm (default), 1-norm, Frobenius norm, and infinity norm of a symbolic matrix. It also computes the P-norm, Frobenius norm, infinity norm, and negative infinity norm of a symbolic vector.

cond computes the corresponding condition numbers of a matrix.

#### poles function for determining the poles of an expression

The MATLAB poles function determines the poles of a symbolic expression or function. The poles function is also implemented in MuPAD.

#### vpasolve function for solving equations and systems using variable precision arithmetic

The MATLAB vpasolve function solves equations and systems of equations numerically.

#### Functions for converting linear systems of equations to matrix form AX=B (equationsToMatrix) and solving matrix equations (linsolve)

The MATLAB equationsToMatrix function converts a linear system of equations to the matrix form AX = B. The function returns the coefficient matrix A and the vector B that contains the right sides of the equations.

The MATLAB linsolve function solves linear systems of equations represented in the matrix form AX = B. The function also returns the reciprocal of the condition number of the square coefficient matrix A. If A is rectangular, linsolve returns the rank of A.

#### MATLAB symbolic functions for describing pulses: rectangularPulse and triangularPulse

rectangularPulse and triangularPulse compute the rectangular and triangular pulse functions, respectively.

In MuPAD, the new rectangularPulse and triangularPulse functions are equivalent to rectpulse and tripulse, respectively.

#### MuPAD functions for computing integral and Z-transforms

These new MuPAD functions compute integral and Z-transforms:

• fourier computes the Fourier transform. You can specify the parameters of the Fourier transform using the new Pref::fourierParameters function.

• ifourier computes the inverse Fourier transform. You can specify the parameters of the inverse Fourier transform using the new Pref::fourierParameters function.

• laplace computes the Laplace transform.

• ilaplace computes the inverse Laplace transform.

• ztrans computes the Z-transform.

• iztrans computes the inverse Z-transform.

#### MuPAD Pref::fourierParameters function for specifying Fourier parameters

The MuPAD Pref::fourierParameters function lets you specify parameters for Fourier and inverse Fourier transforms.

#### MuPAD functions for adding transform patterns

These new MuPAD functions add new patterns for integral and Z-transforms:

MuPAD does not save custom patterns permanently. The new patterns are available in the current MuPAD session only.

#### noFlatten option of the MuPAD proc function for preventing sequence flattening

The MuPAD proc function accepts the new noFlatten option. This option prevents flattening of sequences passed as arguments of the procedure.

#### testtype uses testtypeDom slot for overloading by the second argument

If in the call testtype(object, T) the argument T is a domain, then the method testtypeDom of T is called with the arguments object, T. If T is not a domain, then the method testtypeDom of T::dom is called with the arguments object, T.

#### Compatibility Considerations

In previous releases, testtype used the testtype slot for overloading by the second argument.

#### New upper limit on the number of digits in double

By default, the working precision for double is now limited to at most by 664 digits. You can explicitly specify a larger precision using digits.

#### Compatibility Considerations

Some results returned by double can differ from previous releases. For example, in previous releases double approximated the expression

`x = sym('400!*((exp(2000)+1)/(exp(2000) - 1) - 1)')`

by 3.2997. Now it approximates this expression by 0.

To get the same result as in previous releases, increase the precision of computations:

```digits(1000)
double(x)```
```ans =
3.2997```

#### New definition for real and imag

Starting in R2012a, real and imag are no longer defined via conj. They use the MuPAD Re and Im functions instead.

#### Compatibility Considerations

In R2011b and earlier, real and imag are defined via the conj function:

```syms z
real(z)
imag(z)```
```ans =
z/2 + conj(z)/2

ans =
- (z*i)/2 + (conj(z)*i)/2```

Therefore, real and imag can return results in a different form. Results returned by real and imag now are mathematically equivalent to the results returned in previous releases.

#### Functionality being removed or changed

FunctionalityWhat Happens When You Use It?Use This InsteadCompatibility Considerations
Old syntax of taylorErrorsNew calling syntaxUpdate all instances of taylor function calls using the new syntax.
char(A,d)Errorschar(A)Replace all instances of char(A,d) with char(A).
polyWarnscharpolyReplace all instances of poly with charpoly.
emlBlockWarnsmatlabFunctionBlockReplace all instances of emlBlock with matlabFunctionBlock.
Ability to create links from MuPAD notebooks to MuPAD documentation pagesNot availableNothingNo replacement
openmuphlpErrorsNothingNo replacement
MuPAD Help BrowserNot availableDocumentation CenterMuPAD documentation is now available in Documentation Center.
MuPAD EditorNot availableMATLAB EditorOpen and edit MuPAD program files (.mu files) in the MATLAB Editor. The MATLAB Editor supports syntax highlighting and smart indenting for these files.
psi(k0:k1,X)Errorspsi(k,X), where k is a scalar specifying the kth derivative of psi at the elements of X.Replace all instances of psi(k0:k1,X) with psi(k,X), where k is a scalar. To modify your code, loop through the values k0:k1. For example:
```for k = k0:k1
Y(:,k) = psi(k,X);
end
```
In a future release, size(Y) will be size(X). Modify any code that depends on size(Y).
diff and int methods for inputs of the char typeErrorssymUse the sym method instead.
sqrt target of the MuPAD simplify functionErrorsMuPAD radsimp or simplifyRadicalReplace all instances of simplify function calls involving the sqrt target with radsimp or simplifyRadical. Alternatively, replace these calls with simplify function calls without targets.
cos, sin, exp, and ln targets of the MuPAD simplify function ErrorsMuPAD simplify without targets Replace all instances of simplify function calls involving these targets with simplify function calls without targets. This can lead to a better simplification for some expressions.
MuPAD transform::fourierWarnsMuPAD fourierReplace all instances of transform::fourier with fourier.
MuPAD transform::invfourierWarnsMuPAD ifourierReplace all instances of transform::invfourier with ifourier.
MuPAD transform::laplaceWarnsMuPAD laplaceReplace all instances of transform::laplace with laplace.
MuPAD transform::invlaplaceWarnsMuPAD ilaplaceReplace all instances of transform::invlaplace with ilaplace.
MuPAD transform::ztransWarnsMuPAD ztransReplace all instances of transform::ztrans with ztrans.
MuPAD transform::invztransWarnsMuPAD iztransReplace all instances of transform::invztrans with iztrans.
MuPAD prog::calledFromWarnscontext(hold(procname))Replace all instances of prog::calledFrom() with context(hold(procname)).
MuPAD prog::memuseWarnsprog::trace(Mem)or bytes() Use prog::trace(Mem) or bytes() instead of prog::memuse.
MuPAD prog::testfuncWarnsprint(Unquoted, "...") Use print(Unquoted, "...") instead of prog::testfunc.
MuPAD prog::testmethodWarnsprog::test(..., Method = myTestMethod)Use prog::test(..., Method = myTestMethod) instead of prog::testmethod.
Dynamic modules for MuPAD, including the module, external, and Pref::unloadableModules functions and all functions of the module libraryWarnsNothingNo replacement

## R2012a

New Features, Bug Fixes, Compatibility Considerations

#### New Special Functions

The following special functions are available:

• airy computes the Airy functions of the first and the second kinds. It also computes the first derivatives of the Airy functions.

• beta computes the beta function.

• erfinv and erfcinv compute the inverse and inverse complementary error functions.

• factorial computes the factorial function.

• nchoosek computes binomial coefficients.

• whittakerM and whittakerW compute the Whittaker M and Whittaker W functions.

#### New Vector Analysis Functions

The following vector analysis functions are available:

• curl computes the curl of a vector field.

• divergence computes the divergence of a vector field.

• laplacian computes the laplacian of a scalar function.

• potential computes the scalar potential of a vector field.

• vectorPotential computes the vector potential of a three-dimensional vector field.

#### Computations with Symbolic Functions

The toolbox lets you create symbolic functions. For details, see Creating Symbolic Functions.

dsolve, ezplot, the new odeToVectorField function, and other Symbolic Math Toolbox™ functions now support computations with symbolic functions.

The toolbox also provides the following functions to support common operations on symbolic functions:

• argnames returns a symbolic array of all input variables of a symbolic function.

• formula returns a mathematical expression that defines the symbolic function.

#### Assumptions on Variables

You can set assumptions on symbolic variables by using these functions:

• assume sets assumptions on symbolic variables.

• assumeAlso adds assumptions on symbolic variables without erasing the previous assumptions.

• assumptions shows assumptions set on symbolic variables.

#### New Relational Operators Create Equations, Inequalities, and Relations

Use these relational operators to create symbolic equations, inequalities, and relations:

• == and its functional form eq create a symbolic equation. You can solve these equations using solve or dsolve, plot them using ezplot, set assumptions using assume or assumeAlso, or use them in logical expressions.

• ~= and its functional form ne create a symbolic inequality. You can use inequalities in assumptions and logical expressions.

• >, >=, <, <=, and their functional forms ge, gt, le, and lt create symbolic relations. You can use relations in assumptions and logical expressions.

#### Compatibility Considerations

In previous releases, eq evaluated equations and returned logical 1 or 0. Now it returns unevaluated equations letting you create equations that you can pass to solve, assume, and other functions. To obtain the same results as in previous releases, wrap equations in logical or isAlways. For example, use logical(A == B).

#### New Logical Operators Create Logical Expressions

Use these logical operations let you create logical expressions with symbolic subexpressions:

• & or its functional form and defines the logical conjunction (the logical AND) for symbolic expressions.

• | or its functional form or defines the logical disjunction (the logical OR) for symbolic expressions.

• ~ or its functional form not defines the logical negation (the logical NOT) for symbolic expressions.

• xor defines the logical exclusive disjunction (the logical XOR) for symbolic expressions.

If logical expressions are elements of a symbolic array, you can use these new functions to test the logical expressions:

• all tests whether all equations and inequalities represented as elements of a symbolic array are valid.

• any tests whether at least one of equations and inequalities represented as elements of a symbolic array is valid.

#### New Functions Test Validity of Symbolic Equations, Inequalities, and Relations

Use these functions to test symbolic equations, inequalities, and relations, including logical statements:

• isAlways checks whether an equation, inequality, or relation holds for all values of its variables.

• logical checks the validity of an equation, inequality, or relation. This function does not simplify or mathematically transform expressions that form an equation, inequality, or relation. It also typically ignores assumptions on variables.

#### New Functions Manipulate Symbolic Expressions

These functions provide more flexible options for manipulating symbolic expressions:

• The rewrite function rewrites expressions in terms of target functions. It returns a mathematically equivalent form of an expression using the specified target functions. For example, it can rewrite trigonometric expressions using the exponential function.

• children returns child subexpressions, or terms, of a symbolic expression.

#### New odeToVectorField Function Converts Higher-Order Differential Equations to Systems of First-Order Differential Equations

odeToVectorField converts second- and higher-order differential equations to systems of first-order differential equations. It returns a symbolic vector representing the resulting system of first-order differential equations. With matlabFunction you can generate a MATLAB function from this vector, and then use it as an input for the MATLAB numerical solvers ode23 and ode45.

In MuPAD, the new numeric::odeToVectorField function is equivalent to numeric::ode2vectorfield.

#### New Calling Syntax for the taylor Function

The taylor function that computes the Taylor series expansions has a new syntax and set of options.

#### Compatibility Considerations

The new syntax is not valid before Version 5.8. The old syntax is still supported, but will be removed in a future release. To update existing code that relies on the old syntax, make the following changes to the taylor function calls:

• Specify the truncation order using the name-value pair argument Order.

• Specify the expansion point using the name-value pair argument ExpansionPoint.

Alternatively, specify the expansion point as a third input argument. In this case, you must also specify the independent variable or the vector of variables as the second input argument.

For details and examples, see taylor.

#### New MuPAD Functions Compute Rectangular and Triangular Pulse Functions

The MuPAD rectpulse and tripulse functions compute the rectangular and triangular pulse functions, respectively.

#### MuPAD det, linalg::det, inverse, linsolve, and linalg::matlinsolve Functions Accept the New Normal Option

The MuPAD det, linalg::det, inverse, linsolve, and linalg::matlinsolve functions accept the new Normal option that guarantees normalization of the returned results. The _invert methods of the MuPAD Dom::Matrix(R) and Dom::DenseMatrix(R) domains also accept Normal.

#### MuPAD linalg::matlinsolve Function Accepts the New ShowAssumptions Option

The MuPAD linalg::matlinsolve function accepts the new ShowAssumptions option. This option lets you see internal assumptions on symbolic parameters that linalg::matlinsolve makes while solving a system of equations.

#### Enhanced MuPAD pdivide Function

Enhanced MuPAD pdivide function now performs pseudo-division of multivariate polynomials.

#### Improved MuPAD prog::remember Function

Improved MuPAD prog::remember function, which lets you use the remember mechanism in procedures streamlines such processes as debugging, profiling, and argument checking.

#### Functionality Being Removed or Changed

FunctionalityWhat Happens When You Use It?Use This InsteadCompatibility Considerations
Old syntax of taylorWarnsNew syntaxUpdate all instances of taylor function calls using the new syntax.
Default number of simplification steps in simplify has changed from 50 to 100.Uses the new default settingsimplify(S,'Steps',50)To terminate algebraic simplification after 50 steps, call simplify with the name-value pair argument 'Steps', 50.
char(A,d)Warnschar(A)Replace all instances of char(A,d) with char(A).
emlBlockWarnsmatlabFunctionBlockReplace all instances of emlBlock with matlabFunctionBlock.
psi(k0:k1,X)Warnspsi(k,X) where k is a scalar specifying the kth derivative of psi at the elements of X.Replace all instances of psi(k0:k1,X) with psi(k,X), where k is a scalar. To modify your code, loop through the values k0:k1. For example:
```for k = k0:k1
Y(:,k) = psi(k,X);
end
```
In the future, size(Y) will be size(X). Modify any code that depends on size(Y).
sqrt target of the MuPAD simplify functionWarnsMuPAD radsimp or simplifyRadicalReplace all instances of simplify function calls involving the sqrt target with radsimp or simplifyRadical. Alternatively, replace these calls with simplify function calls without targets.
cos, sin, exp, and ln targets of the MuPAD simplify function WarnsMuPAD simplify without targets Replace all instances of simplify function calls involving these targets with simplify function calls without targets. This can lead to a better simplification for some expressions.
MuPAD frame functionErrorsNothingNo replacement

## R2011b

New Features, Bug Fixes, Compatibility Considerations

#### MATLAB Editor Now Supports MuPAD Program Files

You can open and edit MuPAD program files (.mu files) in the MATLAB Editor. MATLAB Editor supports syntax highlighting and smart indenting for these files.

#### dsolve, expand, int, simple, simplify, and solve Accept More Options

dsolve now accepts the IgnoreAnalyticConstraints and MaxDegree options.

expand now accepts the ArithmeticOnly and IgnoreAnalyticConstraints options.

int now accepts the IgnoreAnalyticConstraints, IgnoreSpecialCases, and PrincipalValue options.

simple now accepts the IgnoreAnalyticConstraints option.

simplify now accepts the IgnoreAnalyticConstraints, Seconds, and Steps options.

solve now accepts the IgnoreAnalyticConstraints, IgnoreProperties, MaxDegree, PrincipalValue, and Real options.

#### New read Function Reads MuPAD Program Files in MATLAB

read simplifies using your own MuPAD procedures in MATLAB. See Before Calling a Procedure for details.

#### New symprod Function Computes Products of Series

symprod computes definite and indefinite products of symbolic series.

#### New hessian Function Computes Hessian Matrices

hessian computes the Hessian matrix of a scalar function.

#### New gradient Function Computes Vector Gradients

gradient computes the vector gradient of a scalar function in Cartesian coordinates. In MuPAD, the new linalg::gradient function is equivalent to linalg::grad.

#### New erfc Function Computes the Complementary Error Function

erfc computes the complementary error function.

#### New psi Function Computes the Digamma and Polygamma Functions

psi computes the digamma and polygamma functions.

#### New wrightOmega Function Computes the Wright omega Function

wrightOmega computes the Wright omega function.

#### New simplifyFraction Function Simplifies Expressions

simplifyFraction returns a simplified form of a fraction where both numerator and denominator are polynomials and their greatest common divisor is 1. In MuPAD, the new simplifyFraction function is equivalent to normal.

#### pretty Function Now Uses Abbreviations in Long Output Expressions for Better Readability

pretty uses abbreviations when presenting symbolic results in the MATLAB Command Window. This new format of presenting symbolic results enhances readability of long output expressions.

#### MuPAD normal Function Accepts the New Expand Option

The MuPAD normal function accepts the new Expand option that determines whether numerators and denominators of fractions are expanded.

#### Compatibility Considerations

In previous releases, normal returned a fraction with the expanded numerator and denominator by default. Now the default setting is that normal can return factored expressions in numerators and denominators. In explicit calls to normal, you can use the Expand option to get the same behavior as in previous releases.

If a function calls normal internally, then that function can return its results in a different form. These new results are mathematically equivalent to the results that you get in previous releases. Many MuPAD library functions can call normal.

#### Modified MuPAD groebner Library

All functions of the MuPAD groebner library now can accept and return polynomials with arbitrary arithmetical expressions.

#### MuPAD groebner::gbasis Function Accepts the New Factor and IgnoreSpecialCases Options

With the Factor option, groebner::gbasis returns a set of lists, such that:

• Each list is the Groebner basis of an ideal.

• The union of these ideals is a superset of the ideal given as input, and a subset of the radical of that ideal.

With the IgnoreSpecialCases option, groebner::gbasis handles all coefficients in all intermediate results as nonzero unless these coefficients are equal to 0 for all parameter values.

#### New MuPAD Functions for Computing Logarithms

The new MuPAD log2 and log10 functions compute logarithms to the bases 2 and 10, respectively. Also, in MuPAD log(x) is now an alias for ln(x).

#### Functionality Being Removed or Changed

FunctionalityWhat Happens When You Use It?Use This InsteadCompatibility Considerations
emlBlockWarnsmatlabFunctionBlockReplace all instances of emlBlock with matlabFunctionBlock.
Real and IgnoreProperties options in MuPAD ode::solveWarnsIgnoreSpecialCases or IgnoreAnalyticConstraintsTry using IgnoreSpecialCases or IgnoreAnalyticConstraints instead.

## R2011a

New Features, Bug Fixes, Compatibility Considerations

#### Expression Wrapping of Math Output in the MuPAD Notebook Interface

The new default format of presenting results enhances readability by wrapping long output expressions, including long numbers, fractions and matrices.

#### Symbolic Solver Handles More Non-Algebraic Equations

The enhanced rationalize function in MuPAD helps the symbolic solver to handle more systems of non-algebraic equations. In particular, this improvement enables the toolbox to solve more systems of trigonometric equations.

#### Improved Performance in the Ordinary Differential Equation Solver

The ordinary differential equation solver demonstrates better performance.

#### Improved Performance for Polynomial Arithmetic Operations

The MuPAD functions gcdex, partfrac, polylib::resultant, and solvelib::pdioe now demonstrate better performance.

#### New MuPAD polylib::subresultant Function Computes Subresultants of Polynomials

polylib::subresultant computes subresultants of two polynomials or polynomial expressions.

#### MuPAD partfrac Function Accepts the New List Option

With the new List option, partfrac returns a list consisting of the numerators and denominators of the partial fraction decomposition.

#### New MuPAD inverf and inverfc Functions Compute the Inverses of Error Functions

The inverf function computes the inverse of the error function.

The inverfc function computes the inverse of the complementary error function.

#### New MuPAD numlib::checkPrimalityCertificate Function Tests Primality Certificates

numlib::checkPrimalityCertificate tests primality certificates returned by numlib::proveprime. For information about proving primality of numbers, see "Proving Primality" in the MuPAD documentation.

#### New Demos

There are three new demos that show how to solve equations and compute derivatives and integrals:

To run the new demos, enter symeqndemo, symdiffdemo, or symintdemo in the MATLAB Command Window.

#### Functionality Being Removed or Changed

FunctionalityWhat Happens When You Use This Functionality?Use This InsteadCompatibility Considerations
MuPAD matchlib::analyzeErrorsMuPAD prog::exprtreeTo visualize expressions, use prog::exprtree.
Old syntax of MuPAD prog::getOptionsErrorsThe new syntaxUpdate all instances of prog::getOptions calls using the new syntax.
Old syntax of MuPAD prog::traceErrorsThe new syntaxUpdate all instances of prog::trace calls using the new syntax.

## R2010b

New Features, Bug Fixes, Compatibility Considerations

#### sym Function Creates Matrices of Symbolic Variables

The sym function now provides a shortcut for creating vectors and matrices of symbolic variables.

For more information, see Creating a Matrix of Symbolic Variables.

#### generate::Simscape Function Generates Simscape Equations from MuPAD Expressions

The new MuPAD function generate::Simscape converts MuPAD expressions to Simscape™ equations.

#### MuPAD Code Generation Functions Accept the New NoWarning Option

MuPAD functions generate::C, generate::fortran, generate::MATLAB, and generate::Simscape accept the new NoWarning option. The option suppresses all warnings issued by these functions.

Creating and editing links in MuPAD has become easier with the improved Hyperlink dialog box.

#### MuPAD Notebook Highlights Matched and Unmatched Delimiters

MuPAD Notebook now can notify you about matched and unmatched delimiters such as parentheses, brackets, and braces.

#### Improved Performance When Solving Linear Systems in a Matrix Form

MuPAD linalg::matlinsolve function, which solves linear systems of equations in a matrix form, demonstrates better performance.

#### MuPAD Solver for Ordinary Differential Equations Handles More Equation Types

Enhanced MuPAD solver handles more first-order nonlinear and third-order linear ordinary differential equations. The solver demonstrates improved performance.

#### New Syntax for the MuPAD prog::getOptions Function

The prog::getOptions function that collects and verifies options within a procedure has the new syntax.

#### Compatibility Considerations

The new syntax is not valid in MuPAD versions earlier than 5.5. The old syntax is supported in MuPAD 5.5, but will be removed in a future release.

#### New Syntax for the MuPAD prog::trace Function

The prog::trace function used for debugging has the new syntax. The function observes entering and exiting the MuPAD functions.

#### Compatibility Considerations

The new syntax is not valid in MuPAD versions earlier than 5.5. The old syntax is not supported in MuPAD 5.5.

#### Improved Interface for Arithmetical Operations on Polynomials

Improved interface for arithmetical operations between polynomials and arithmetical expressions. In previous releases, to perform an arithmetical operation on a polynomial and an arithmetical expression, you must explicitly convert that expression to a polynomials of the corresponding type. Now, when you operate on a polynomial and an arithmetical expression, MuPAD internally converts the arithmetical expression to a polynomial and performs the calculation.

#### MuPAD igcd Function Now Accepts Complex Numbers as Arguments

The MuPAD igcd function, which computes the greatest common divisor of integers, now accepts complex numbers. Both real and imaginary parts of accepted complex numbers must be integers or arithmetic expressions that represent integers.

#### Enhanced Solver For Factorable Polynomial Systems

The MuPAD solve function performs better on factorable polynomial systems.

#### MuPAD Now Evaluates Large Sums with Subtractions Faster

MuPAD performs evaluations of large sums that contain subtractions faster than in previous releases.

#### Compatibility Considerations

In MuPAD, the difference operator (–) no longer invokes the _subtract function. Instead, it invokes the _plus and _negate functions. For example, a - b is equivalent to _plus(a, _negate(b)).

#### MuPAD freeIndets Function Accepts the New All Option

The freeIndets function accepts the new All option. With this option, freeIndets does not exclude the 0th operand from the list of free identifiers.

#### Functionality Being Removed or Changed

FunctionalityWhat Happens When You Use This Functionality?Use This InsteadCompatibility Considerations
diff and int methods for inputs of the char typeWarnssymUse the sym method instead.
MuPAD matchlib::analyzeWarnsMuPAD prog::exprtreeTo visualize expressions, use prog::exprtree.
The following options in MuPAD prog::trace:
• All

• Backup

• Force

• Name

• Proc

• Plain

• Width

ErrorsNoneNo replacement. These options are not supported in the current release.
Global properties in MuPADErrorsAssumptions on each variableMake assumptions on each variable instead.

## R2010a

New Features, Bug Fixes, Compatibility Considerations

#### When Opening Notebook, MuPAD Can Jump to Particular Locations

The mupad command that opens a MuPAD notebook now supports references to particular places inside a notebook. You can create a link target inside a notebook and refer to it when opening a notebook.

#### simscapeEquation Function Generates Simscape Equations from Symbolic Expressions

The new simscapeEquation command represents symbolic expressions in the form of Simscape equations. For more information, see Generating Simscape Equations in the Symbolic Math Toolbox documentation.

#### New Calling Syntax for the sort Function

The sort function that sorts the element of symbolic arrays and polynomials has the new syntax and set of options.

#### Compatibility Considerations

In previous releases, the sort function flattened symbolic matrices to vectors before sorting the elements. Now the sort function sorts the elements of each column or each row of a symbolic matrix. If you want to obtain the same results as in the previous release, flatten the symbolic matrix before sorting it: sort(A(:)).

#### Changes in the symengine Function

The toolbox no longer supports the ability to choose an alternative symbolic engine.

#### 64-Bit GUI Support for Macintosh

MuPAD now supports 64-bit graphical user interfaces (such as notebooks and Editor and Debugger windows) for a 64-bit Macintosh operating system.

#### New MuPAD Print Preview Dialog

Adjusting MuPAD documents for printing is easier with the new Print Preview dialog. You can view one or several pages, zoom in and out, switch between page orientations, adjust the page settings without closing the dialog, and print the page or save it to PDF format

#### Improved Configure MuPAD Dialog Box

Specifying the default settings for graphical user interfaces, such as notebooks and Editor and Debugger windows, has become easier with the improved configuration dialog box.

#### MuPAD Support for Basic Arithmetic Operations for Lists

Basic arithmetic operations now work for lists.

#### Improved Performance When Operating on Matrices with Symbolic Elements

MuPAD demonstrates better performance when handling some linear algebra operations on matrices containing symbolic elements.

#### Enhanced MuPAD divide Function

Enhanced MuPAD divide function computes the quotient and remainder for division of multivariate polynomials.

#### Improved Performance for Operations on Polynomials

Improved performance for conversions involving polynomials. Improved performance for operations on polynomials including evaluation, multiplication, and division.

#### Compatibility Considerations

If the coefficients of a polynomial contain the variables of the polynomial itself, the form of results returned by the MuPAD poly function can differ from previous releases. In previous releases, the poly function converted such coefficients to monomials. Now the poly function can return the coefficients of the original expression as coefficients in the resulting polynomial. To get the same behavior as in previous releases, useexpr to convert an original polynomial into an expression, and then call the poly function. For example, the following call exercises the old behavior: poly(expr(p), [y, x]).

#### MuPAD coeff Function Accepts the New All Option

The coeff function accepts the newAll option. With this option, coeff returns all coefficients of a polynomial including those equal to 0.

#### MuPAD expand Function Accepts the New ArithmeticOnly Option

The expand function accepts the new ArithmeticOnly option. The option allows you to expand a sum without expanding trigonometric expressions and special functions in its terms. Technically, the option omits overloading the expand function for each term of the original expression.

#### MuPAD expand Function Now Expands Powers of Products

The expand function now expands powers of products such as (xy)n for positive x and y. When called with the IgnoreAnalyticConstraints option , the function expands the power of products for arbitrary terms.

#### New Calling Syntax for MuPAD rationalize Function

The rationalize function that transforms an arbitrary expression into a rational expression has the new syntax and set of options.

#### Compatibility Considerations

The new syntax is not valid in MuPAD versions earlier than 5.4. The old syntax is supported in MuPAD 5.4, but will be removed in a future release.

#### Enhanced MuPAD simplify and Simplify Functions

Enhanced simplification functions, simplify and Simplify, demonstrate better results for expressions involving trigonometric and hyperbolic functions, square roots, and sums over roots of unity.

#### MuPAD subs Function Accepts the New EvalChanges Option

The subs function now accepts the new EvalChanges option. By default, subs does not evaluate an expression after making substitutions. With this option, subs evaluates all subexpressions that contain substitutions.

#### MuPAD Solver for Ordinary Differential Equations Handles More Equation Types

Enhanced MuPAD solver handles more second-order linear and first-order nonlinear ordinary differential equations. The solver demonstrates improved performance.

#### Functionality Being Removed or Changed

FunctionalityWhat Happens When You Use This Functionality?Use This InsteadCompatibility Considerations
MuPAD Domain Dom::IdealErrorsgroebnerRepresent ideals as lists, and use functions of the groebner package instead.
MuPAD student libraryErrorsplot::Integral and linalgUse plot::Integral and the linalg package instead.
MuPAD relation option in simplifyErrorsNoneNo replacement
Global property WarnsAssumptions on each variableMake assumptions on each variable instead.
digits and vpa do not let you set the number of digits to 1.ErrorsErrorsIt is no longer possible to set the number of digits to 1 when using the digits and vpa functions. The Symbolic Math Toolbox software version number 4.9 and lower allowed you to set the number of digits to 1.

## R2009b

New Features, Bug Fixes, Compatibility Considerations

#### Support for Windows x64 and 64-Bit Macintosh

The toolbox now supports 64-bit Windows® and Macintosh operating systems. If you work in the MuPAD Notebook Interface on a 64-bit Macintosh operating system, MuPAD runs a 64-bit engine with 32-bit graphical user interfaces, such as notebooks and Editor and Debugger windows.

#### sym and syms Use Reserved Words as Variable Names

sym and syms commands now treat reserved MuPAD words, except pi, as variable names.

#### Compatibility Considerations

In previous releases, the reserved words returned MuPAD values. If your code uses the reserved words as MuPAD commands, modify your code and use the evalin command with the reserved word as a name. For example, use evalin(symengine,'beta').

#### Toolbox Now Displays Floating-Point Results with Their Original Precision

The toolbox now displays the floating-point results with the original precision with which the toolbox returned them.

#### Compatibility Considerations

In previous releases, the toolbox displayed floating-point results with the current precision. You must update the existing code that relies on the output precision for displaying floating-point numbers. Use digits to set the precision you need before computing such results. The toolbox displays the results with the same number of digits it used to compute the results. The toolbox also can increase the specified precision of calculations by several digits.

In previous releases, sym(A, 'f') represented numbers in the form (2^e + N*2^(e - 52)) or -(2^e + N*2^(e - 52)) , with integers for N and e, and N ≥ 0. Now sym(A, 'f') displays results in the rational form that actually represents the double-precision floating-point numbers.

#### New MuPAD Preference Pref::outputDigits Controls Floating-Point Outputs

New preference Pref::outputDigits controls the precision MuPAD uses to display floating-point results.

#### Solver for Ordinary Differential Equations Handles More Equation Types

Enhanced solvers handle more equation types of second-order homogeneous linear ordinary differential equations. The solver demonstrates improved performance.

#### MuPAD limit Function Supports Limits for Incomplete Gamma Function and Exponential Integral Function

Enhanced limit function now can compute limits for incomplete Gamma function and exponential integral function.

#### Enhanced Simplification Routines for MuPAD Special Functions

Enhanced simplification routines for MuPAD hypergeom, mejerG, and bessel special functions.

#### Enhanced MuPAD combine Function for Logarithms

Enhanced combine function demonstrates better performance for logarithms.

#### MuPAD normal Function Accepts New Options

The normal command now accepts the options NoGcd, ToCancel, Rationalize, Recursive, and Iterations. The options control costly operations, such as recognizing greatest common divisors and algebraic dependencies.

#### Functionality Being Removed or Changed

FunctionalityWhat Happens When You Use This Functionality?Use This InsteadCompatibility Considerations
MuPAD Domain Dom::IdealWarnsgroebnerRepresent ideals as lists, and use functions of the groebner package instead.
MuPAD student libraryWarnsplot::Integral and linalgUse plot::Integral and the linalg package instead.
d in char(A, d)WarnsNoneNo replacement
MuPAD relation option in simplifyWarnsNoneNo replacement

## R2009a

New Features, Bug Fixes, Compatibility Considerations

#### dsolve Accepts the New Option IgnoreAnalyticConstraints

The dsolve command now accepts the option IgnoreAnalyticConstraints. The option controls the level of mathematical rigor that the solver uses on the analytical constraints on the solution. By default, the solver ignores all analytical constraints.

#### Compatibility Considerations

The results of the dsolve command can differ from those returned in the previous release. If you want to obtain the same solutions as in the previous release, set the value of the option IgnoreAnalyticConstraints to none.

#### emlBlock Function Generates Embedded MATLAB Function Blocks from Symbolic Objects

The new emlBlock command converts symbolic expressions to Embedded MATLAB® Function Blocks. You can use these blocks in any Simulink® installation, even those without a Symbolic Math Toolbox license. For more information, see Generating Embedded MATLAB Blocks in the Symbolic Math Toolbox documentation.

#### matlabFunction Improves Control over Input and Output Parameters

matlabFunction now accepts multiple expressions and cell arrays of symbolic arrays as input parameters. The function now allows you to specify the names of the output parameters.

#### Compatibility Considerations

In previous releases, the default name of an output variable was RESULT. Now the default names of the output variables coincide with the names you use to call matlabFunction. You must update existing code that relies on the default output name RESULT. You can change your code using any of these methods:

• Define the name of an output variable as RESULT.

• Change the name of an input variable to RESULT.

• Throughout your code change the variable name from RESULT to the input name.

#### Enhancements to Object-Oriented Programming Capabilities

The Symbolic Math Toolbox product uses some object-oriented programming features to implement symbolic objects. Major enhancements to object-oriented programming capabilities enable easier development and maintenance of large applications and data structures. For a full description of object-oriented features, see the MATLAB Object-Oriented Programming documentation.

#### Compatibility Considerations

It is no longer possible to add methods to @sym by creating a @sym directory containing custom methods.

For an empty x, sym(x) returns a symbolic object of the same size as x. In previous releases, sym(x) returned a symbolic object of size 0-by-0 for an empty x.

#### generate::MATLAB Function Converts MuPAD Expressions to MATLAB Code

The new generate::MATLAB command converts MuPAD expressions, equations, and matrices to MATLAB formatted strings.

#### MuPAD IgnoreAnalyticConstraints Option Specifies That Core Functions Apply Common Algebraic Assumptions to Simplify Results

The new IgnoreAnalyticConstraints option allows the use of a set of simplified mathematical rules when solving equations, simplifying expressions, or integrating. For example, this option applies practical, but not generally correct rules for combining logarithms: $\mathrm{ln}\left(a\right)+\mathrm{ln}\left(b\right)=\mathrm{ln}\left(a\cdot b\right)$

#### MuPAD Outputs Contain Abbreviations for Better Readability

The new default format of presenting results enhances readability of long output expressions by using abbreviations.

#### MuPAD Solver for Ordinary Differential Equations Handles More Equation Types

The solver now can handle more than 200 additional types of second-order ordinary differential equations. The solver demonstrates improved performance.

#### MuPAD limit Function Now Can Compute Limits for Piecewise Functions

The enhanced limit function computes limits of piecewise functions including bidirectional and one-sided limits.

#### New and Improved MuPAD Special Functions

MuPAD includes the following new special functions:

• laguerreL represents Laguerre's L function.

• erfc(x,n) returns iterated integrals of the complementary error function.

• meijerG represents the Meijer G function.

The hypergeom special function demonstrates better performance.

#### New Calling Syntax for Test Report Function prog::tcov

The prog::tcov function that inspects the data collected during the code execution has the new syntax and set of options.

#### Compatibility Considerations

The new syntax is not valid in MuPAD versions earlier than 5.2. MuPAD 5.2 does not support the earlier syntax.

#### New Demos

To see new demos that use MuPAD Notebook Interface, type mupadDemo at the MATLAB command line or click MuPAD Notebooks DemoMuPAD Notebooks Demo.

Bug Fixes

Bug Fixes

## R2007b+

New Features, Bug Fixes, Compatibility Considerations

#### MuPAD Engine Replaces Maple Engine

The default Symbolic Math Toolbox engine is now the MuPAD engine. For more information, see the MuPAD in Symbolic Math Toolbox chapter in the Symbolic Math Toolbox User's Guide.

#### Compatibility Considerations

The new engine causes many computed results to differ from those returned by previous versions of Symbolic Math Toolbox software.

General Differences

• Many computations return in a permuted order (such as a + b instead of b + a).

• Some computations return in a different, mathematically equivalent form (such as (cos(x))^2 instead of 1 - (sin(x))^2).

• diff(dirac(t)) returns dirac(t,1) instead of dirac(1,t).

• sym(x,'f') no longer produces strings of the form hex digits*2^n. Instead the strings have the form (2^e+N*2^(e-52)), where N and e are integers.

• For toolbox calculations, some symbols can only be used as symbolic variables, and not in strings: E, I, D, O, beta, zeta, theta, psi, gamma, Ci, Si, and Ei. This is because those symbols represent MuPAD reserved words, and are interpreted as the MuPAD word if you pass them as strings. The words Ci, Si, Ei represent special mathematical functions: the cosine integral, sine integral, and exponential integral respectively.

• Error and warning message IDs may have changed.

• Performance of numerical integration is slower than in previous versions.

• Subexpressions, calculated by the subexpr function, may be different than in previous versions.

• The pretty function no longer uses partial subexpressions (with syntax %n).

Calculus

• Int no longer evaluates some integrals, including many involving Bessel functions.

• symsum(sin(k*pi)/k,0,n) no longer evaluates to pi.

Linear Algebra

• The output of colspace may differ from previous versions, but it is mathematically equivalent.

• The eig function may return eigenvalues in a different order than previous versions. Expressions returned by eig may be larger than in previous versions.

• The jordan function may return diagonal subblocks in a different order than previous versions.

• svd may return singular values in a different order than previous versions.

Simplification

• The coeffs function may return multivariable terms in a different order than in previous versions.

• The expand function may return some trig and exponential expressions differently than in previous versions.

• The simplify function involving radicals and powers make fewer assumptions on unknown symbols than in previous versions.

• The subexpr function may choose a different subexpression to be the common subexpression than in previous versions.

• Subexpressions no longer have partial subexpressions (previous syntax %n).

• The solve function returns solutions with higher multiplicity only when solving a single polynomial.

• acot(-x) = -acot(x) instead of pi - acot(x) as in previous versions.

• acoth(-x) = -acoth(x) instead of 2*acoth(0) - acoth(x) as in previous versions.

• The simple function has several differences:

• The 'how' value combine(trig) has been replaced with combine(sincos), combine(sinhcosh), and combine(ln).

• The 'how' values involving convert have been replaced with rewrite.

• A new 'how' value of mlsimplify(100) indicates the MuPAD function Simplify(...,Steps=100) simplified the expression.

• Simplifications such as (sin(x)^2)^(1/2) to sin(x) are no longer performed, since the MuPAD language is careful not to make assumptions about the sign of sin(x).

Conversion

• Arithmetic involving the vpa function uses the current number of digits of precision. Variable precision arithmetic may have different rounding behaviors, and answers may differ in trailing digits (trailing zeros are now suppressed).

• The char function returns strings using MuPAD syntax instead of Maple™ syntax.

• Testing equality does not compare strings as in previous versions; the symbolic engine equality test is used.

• Saving and loading symbolic expressions is compatible with previous versions, except when the symbolic contents use syntax or functions that differ between Maple or MuPAD engines. For example, suppose you save the symbolic object sym('transform::fourier(f,x,w)'), which has MuPAD syntax. You get a MATLAB error if you try to open the object while using a Maple engine.

• LaTeX output from the latex function may look different than before.

• C and Fortran code generated with the ccode and fortran functions may be different than before. In particular, generated files have intermediate expressions as "optimized" code. For more information, see the Generating C or Fortran Code section of the User's Guide.

• pretty output may look different than before.

Equation Solving

• solve returns solutions with higher multiplicity only when solving a single polynomial.

• solve may return a different number of solutions than before.

• Some calls to dsolve that used to return results involving lambertw now return no solution.

• dsolve can now use the variable C.

• Some dsolve results are more complete (more cases are returned).

• Some dsolve results are less complete (not all previous answers are found).

• finverse may be able to find inverses for different classes of functions than before.

• When finverse fails to find an explicit inverse, it produces different output than before.

Transforms

• Fourier and inverse Fourier transforms return the MuPAD form transform::fourier when they cannot be evaluated. For example,

```h = sin(x)/exp(x^2);
FF = fourier(h)

FF =
transform::fourier(sin(x)/exp(x^2), x, -w)```

The reason for this behavior is the MuPAD definition of Fourier transform and inverse Fourier transform differ from their Symbolic Math Toolbox counterparts by the sign in the exponent:

Symbolic Math Toolbox definitionMuPAD definition
Fourier transform$F\left(w\right)=\underset{-\infty }{\overset{\infty }{\int }}f\left(x\right){e}^{-iwx}dx$$F\left(w\right)=\underset{-\infty }{\overset{\infty }{\int }}f\left(x\right){e}^{iwx}dx$
Inverse Fourier transform$f\left(x\right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}F\left(w\right){e}^{iwx}dw$$f\left(x\right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}F\left(w\right){e}^{-iwx}dw$

• Several Fourier transforms can no longer be calculated, especially those involving Bessel functions.

• ztrans and iztrans may return more complicated expressions than before.

Special Mathematical Functions

• The three-parameter Riemann Zeta function is no longer supported.

• heaviside(0) = 0.5; in previous versions it was undefined.

maple

• The maple, mhelp, and procread functions error, unless a Maple engine is installed and selected with symengine.

#### New MuPAD Language and Libraries Supplant Extended Symbolic Math Toolbox Software

The functionality of the MuPAD language, together with the included libraries, goes far beyond that of the previous Symbolic Math Toolbox software. However, it is not identical to that of the previous Extended Symbolic Math Toolbox™ software. The differences between these software packages are beyond the scope of these release notes.

You can access the MuPAD language in several ways:

• To learn the commands, syntax, and functionality of the language, use the MuPAD Help browser, or read the Tutorial.

• Use a MuPAD notebook, which contains an integrated help system for the language syntax.

• Use the new evalin function or feval function to access the MuPAD language at the MATLAB command line. More detail is available in the Calling Built-In MuPAD Functions from the MATLAB Command Window section of the User's Guide.

#### New MuPAD Help Viewer (GUI)

The MuPAD help viewer contains complete documentation of the MuPAD language, and of the MuPAD Notebook Interface. For more information, see the Getting Help for MuPAD section of the User's Guide.

#### New MuPAD Notebook Interface (GUI)

A MuPAD notebook is an interface for performing symbolic math computations with embedded math notation, graphics, animations, and text. It also enables you to share, document, and publish your calculations and graphics. For example, the MuPAD help viewer is essentially a special MuPAD notebook. For more information, see the Calculating in a MuPAD Notebook section of the User's Guide.

#### New MuPAD Editor and Debugger (GUI)

The MuPAD Editor GUI enables you to write custom symbolic functions and libraries in the MuPAD language. The Debugger enables you to test your code. For more information, consult the MuPAD help viewer.

#### New Functionality for Communication Between MATLAB Workspace and MuPAD

FunctionUse
doc(symengine,...)Access the MuPAD Help browser.
evalin(symengine,...)Use MuPAD functionality in the MATLAB workspace.
feval(symengine,...)Use MuPAD functionality in the MATLAB workspace.
getVarCopy expressions residing in a MuPAD notebook into the MATLAB workspace.
reset(symengine,...)Clear the MuPAD engine for the MATLAB workspace.
setVarCopy expressions residing in the MATLAB workspace into a MuPAD notebook.
symvarProduce a list of symbolic objects in an expression.

For more information, see the Integration of MuPAD and MATLAB section of the User's Guide.

#### New symengine Command for Choosing a Maple Engine

If you own a compatible version of a Maple software, you can choose to have Symbolic Math Toolbox software use the Maple engine instead of a MuPAD engine. You might want to do this if you have existing Maple programs. Choose the engine by entering symengine at the MATLAB command line; this brings up a GUI for making your choice.

#### New matlabFunction Generates MATLAB Functions

The new matlabFunction generates MATLAB functions from symbolic expressions. matlabFunction writes the generated code to a file or creates a function handle. You can use the generated function handles and files in any MATLAB installation, even those without a Symbolic Math Toolbox license. For more information, see Generating MATLAB Functions in the User's Guide.

Bug Fixes

Bug Fixes

## R2007a

New Features, Bug Fixes

#### Maple10 Access Added for Linux 64–bit Processors and Intel Macintosh Platforms

MATLAB now supports Maple Version 10 on 32-bit Windows, 32- and 64-bit Linux® platforms, and the Intel® and PowerPC® Macintosh platforms.

## R2006b

New Features, Bug Fixes, Compatibility Considerations

#### Change in call to code generation package using the maple function

Calling a function in code generation package using Maple software now requires you to explicitly include the package name. For example,

`maple('codegen[fortran](x^2-4)');  `

The generated code output using these methods is unaffected by this change.

#### Compatibility Considerations

In previous versions, functions in the code generation package of Maple software were made automatically available using the Maple with command, and did not require the package name. For example

`maple('fortran(x^2-4)');`

This sometimes caused a conflict when assigning to Maple variables having the same name as a function in the code generation package.