`series`

function for computing Puiseux series expansionThe MATLAB^{®} `series`

function
approximates a symbolic expression or function with a Puiseux series
expansion.

`hermiteForm`

and `smithForm`

functions for computing Hermite and Smith normal forms of matricesThe MATLAB `smithForm`

and `hermiteForm`

functions compute the Smith
and Hermite normal forms of a matrix, respectively. Elements of a
matrix must be integers or polynomials. Both functions also can return
corresponding transformation matrices.

The MuPAD^{®} `linalg::smithForm`

and `linalg::hermiteForm`

functions
provide more functionality:

`linalg::smithForm`

returns transformation matrices along with the Smith form of a matrix.`linalg::hermiteForm`

computes the Hermite form of a matrix of polynomials.

`Sparse`

argument for `matlabFunction`

, `odeFunction`

, and `daeFunction`

for using sparse instead of dense
matrices in generated MATLAB functions`matlabFunction`

, `odeFunction`

, and `daeFunction`

accept the name-value pair
argument, `'Sparse',true`

that triggers the generated MATLAB functions
to represent symbolic matrices by sparse numeric matrices in the generated
code.

`has`

function for searching subexpressions in a symbolic expressionThe MATLAB `has`

function
checks if an expression contains specified subexpressions.

`root`

function for representing roots of polynomialsThe MATLAB `root`

function
represents roots of polynomials. Symbolically solving a high degree
polynomial for its roots can be complex or mathematically impossible.
In this case, Symbolic Math Toolbox™ uses the `root`

function
to represent the roots of the polynomial.

The following new examples demonstrate the functionality of Symbolic Math Toolbox:

Numerical Computations With High Precision. To run this example, enter

`NumericComputingWithHighPrecisionExample`

in the MATLAB Command Window.Decimal Digits of pi. To run this example, enter

`DigitsOfPiExample`

in the MATLAB Command Window.Prime Factorizations. To run this example, enter

`PrimesExample`

in the MATLAB Command Window.Palindromes and the 196 Problem. To run this example, enter

`PalindromeExample`

in the MATLAB Command Window.Rotations and Scaling with Matrices. To run this example, enter

`RotationExample`

in the MATLAB Command Window.Evaluation Points and Weights of Quadrature Rules. To run this example, enter

`QuadratureRulesExample`

in the MATLAB Command Window.Simulate a Stochastic Process by Feynman-Kac Formula. To run this example, enter

`FeynmanKacExample`

in the MATLAB Command Window.

The following examples are updated and renamed:

Integration. To run this example, enter

`IntExample`

in the MATLAB Command Window.Differentiation. To run this example, enter

`DiffExample`

in the MATLAB Command Window.

Functionality | What Happens When You Use It? | Use This Instead | Compatibility Considerations |
---|---|---|---|

Character strings inputs to `sym` and `vpa` are
restricted to numbers and valid variable names. | Still runs | When creating symbolic expressions, first create symbolic variables,
and then use operations on them. For example, use: `syms x; x + 1` instead of `sym('x + 1')` `exp(sym(pi))` instead of `sym('exp(pi)')` `syms f(var1,...varN)` instead of `f(var1,...varN) = sym('f(var1,...varN)')` `vpa((1 + sqrt(sym(5)))/2)` instead of `vpa('(1 + sqrt(5))/2')`
| Support of strings that are not valid variable names and do not define a number will be removed in a future release. To create symbolic expressions, first create symbolic variables, and then use operations on them. |

Character strings as inputs to `solve` | Warns | When specifying equations and variables, use symbolic equations
and variables instead of character strings. For example, replace ```
solve('2*r
= 1', 'r')
``` with `syms r; solve(2*r == 1, r)` . | Do not specify equations and variables as character strings.
Instead of string inputs, create symbolic variables using `syms` , and then pass them as input arguments
separated by commas, or as a vector of input arguments. |

`findsym` | Warns | `symvar` | Replace all instances of `findsym` with `symvar` . |

Values `All` and `None` of `IgnoreAnalyticConstraints` | Warns | `true` and `false` | Replace all instances of `'IgnoreAnalyticConstraints','All'` with `'IgnoreAnalyticConstraints',true` .
Replace all instances of `'IgnoreAnalyticConstraints','None'` with `'IgnoreAnalyticConstraints',false` . |

`mfun` | Warns | Appropriate 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. |

`mfunlist` | Warns | See Special Functions. | See Special Functions for the list of available special functions. |

`poly2sym(c, 'var')` will not accept a character
string `'var'` in a future release. | Warns | `syms var; poly2sym(c, var)` or ```
poly2sym(c,
sym('var'))
``` | Replace all instances of `poly2sym(c, 'var')` with ```
syms
var; poly2sym(c, var)
``` or `poly2sym(c, sym('var'))` |

`setVar(nb, 'MuPADvar')` | Errors | Three-argument version `setVar(nb, 'MuPADvar', MATLABexpr)` | Replace all instances of `setVar(nb,'MuPADvar')` with ```
setVar(nb,
'MuPADvar', MATLABexpr)
``` . |

`functionalDerivative`

function for finding derivatives of functionalsThe MATLAB `functionalDerivative`

function
finds the derivative of a symbolic expression with respect to functions.

`odeFunction`

for converting systems of algebraic expressions to MATLAB functions suitable
for `ode45`

and other ODE solversThe MATLAB `odeFunction`

converts
a system of symbolic algebraic expressions to MATLAB function
handle suitable for `ode45`

, `ode15s`

,
and other ODE solvers.

`partfrac`

function for computing partial fraction decompositionThe MATLAB `partfrac`

function
finds the partial fraction decomposition of a rational expression.
This function accepts the name-value pair argument `'FactorMode',mode`

that
lets you choose a factorization mode used to factor the denominator.
Here, `mode`

is one of the following strings: `'real'`

, `'complex'`

, `'full'`

,
or `'rational'`

. By default, `partfrac`

performs
factorization over the rational numbers.

`sympref`

function for specifying preferences for symbolic functions `fourier`

, `ifourier`

,
and `heaviside`

The MATLAB `sympref`

function
specifies preferences for symbolic functions `fourier`

, `ifourier`

,
and `heaviside`

. `sympref`

specifies
values of parameters in `fourier`

and `ifourier`

,
and the return value of `heaviside`

at `0`

.

`Optimize`

argument for controlling code optimization in generated MATLAB functions returned by `matlabFunction`

, `odeFunction`

,
and `daeFunction`

`matlabFunction`

, `odeFunction`

,
and `daeFunction`

accept
the name-value pair argument, `'Optimize',false`

that
disables code optimization when you write the resulting code to a
file.

`isolate`

function for rearranging an equation so that the variable or expression appears on the left
sideThe MuPAD `isolate`

function
rearranges an equation so that the variable or expression appears
on the left side of the equation. If `isolate`

cannot
isolate the variable or expression, it moves all terms containing
the variable or expression to the left side.

`FactorMode`

argument offering different modes of factorization returned by MATLAB `factor`

functionThe MATLAB `factor`

function now accepts
the name-value pair argument `'FactorMode',mode`

that
lets you choose a factorization mode. Here, `mode`

is
one of the following strings: `'real'`

, `'complex'`

, `'full'`

,
or `'rational'`

. By default, `factor`

performs
factorization over the rational numbers.

`chol`

, `lu`

, `qr`

, and `rank`

now return certain outputs as type double The MATLAB functions `chol`

, `lu`

, `qr`

,
and `rank`

now
return certain outputs as type `double`

instead of
symbolic objects.

The functions `chol`

, `lu`

,
and `qr`

now
return the permutation information as matrices or vectors of double-precision
values. The `rank`

function
returns the rank of a matrix as a double-precision value. In previous
releases, these output arguments were returned as symbolic objects.
For details, see the Output Arguments section on the respective reference
pages.

`assume`

, `assumeAlso`

, `assumptions`

, `sym`

, and `syms`

have changes to assumptions mechanism `syms`

creates a vector or a matrix of symbolic variables where each element of the vector or matrix belongs to`set`

using the syntax`sym(A,dim,set)`

. For example,`A = sym('A',[3 3],'rational')`

creates the 3-by-3 matrix`A`

where MATLAB assumes all elements of`A`

are`rational`

.`assume`

clears assumptions on a variable`var`

using the syntax`assume(var,'clear')`

. If`var`

is an expression,`assume`

clears all assumptions on all variables in`var`

.`assume`

and`assumeAlso`

set the assumption that a variable is positive using the option`'positive'`

. For example, assume`x`

is positive using`assume(x,'positive')`

.

`syms`

does not create variables with the following names:`clear`

,`integer`

,`positive`

,`rational`

, and`real`

. For example, in previous releases`syms integer`

created the symbolic variable`integer`

. To use these variable names now, use`sym`

. For example, to create the symbolic variable`integer`

, use`integer = sym('integer')`

.The syntax

`sym(x,set)`

for`x`

that already exists in the MATLAB workspace will be removed in a future release. Use`assume(x,set)`

instead.The syntax

`sym(x,'clear')`

will be removed in a future release. Use`assume(x,'clear')`

instead.

The following new examples demonstrate the functionality of Symbolic Math Toolbox.

Explore Single-Period Asset Arbitrage. To run this example, enter

`ArbitrageExample`

in the MATLAB Command Window.Compute Binomial Coefficients Exactly. To run this example, enter

`BinomialCoefficientExample`

in the MATLAB Command Window.Radiation of Two Charged Particles Moving in an Elliptical Orbit. To run this example, enter

`DipoleExample`

in the MATLAB Command Window.Analytical Coefficients of Harmonic Frequency Combinations. To run this example, enter

`HarmonicFrequencyExample`

in the MATLAB Command Window.Hilbert Matrices and Their Inverses. To run this example, enter

`HilbertMatrixExample`

in the MATLAB Command Window.Symbolic Analysis of a Markov Chain. To run this example, enter

`MarkovChainExample`

in the MATLAB Command Window.Solve a Second-Order Differential Equation Numerically. To run this example, enter

`OdeToVectorFieldExample`

in the MATLAB Command Window.Model Dead-Time Input Using Padé Approximants. To run this example, enter

`PadeApproximantExample`

in the MATLAB Command Window.

The Differentiation example
is updated and renamed. To run this example, enter `DiffExample`

in
the MATLAB Command Window.

To run the Differentiation example in previous releases, enter`symdiffdemo`

in
the MATLAB Command Window.

`solve`

function uses default `MaxDegree`

value of `2`

The MATLAB `solve`

function
uses a default `MaxDegree`

value of `2`

.
In previous releases, the default value of `MaxDegree`

was `3`

.

In previous releases, `solve`

automatically
returned explicit solutions for equations of degree `3`

.
To obtain the same results as in previous releases, set `MaxDegree`

to `3`

.
For example, `solve(a*x^3 + b*x^2 + c*x + 1,'MaxDegree',3)`

.

The MuPAD orthogonal polynomial functions return polynomial
expressions when the polynomials are evaluated with identifiers. This
applies to all functions in the MuPAD `orthpoly`

library.

In previous releases, the MuPAD orthogonal polynomial functions
returned objects of type `DOM_POLY`

when
the polynomials were evaluated with identifiers.

`sym`

treats `i`

in string input as a variable The MATLAB function `sym`

treats `i`

in
string input as a variable. For example, `sym('1 + i')`

returns
the symbolic expression `i+1`

.

In previous releases,

`sym`

treated`i`

in string input as an imaginary number. Now, it is treated as a variable`i`

. For example,`sym('1 + i')^2`

returns the symbolic expression`(i + 1)^2`

. To obtain the same results as in previous releases, specify the imaginary number`i`

as`1i`

. For example,`sym('1 + 1i')^2`

returns`2i`

.

Functionality | What Happens When You Use It? | Use This Instead | Compatibility Considerations |
---|---|---|---|

`findsym` | Still runs | `symvar` | Replace all instances of `findsym` with `symvar` . |

Values `All` and `None` of `IgnoreAnalyticConstraints` | Still runs | `true` and `false` | Replace all instances of `'IgnoreAnalyticConstraints','All'` with `'IgnoreAnalyticConstraints',true` .
Replace all instances of `'IgnoreAnalyticConstraints','None'` with `'IgnoreAnalyticConstraints',false` . |

`mfun` | Warns | Appropriate 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. |

`mfunlist` | Warns | See Special Functions. | See Special Functions for the list of available special functions. |

`setVar(nb, 'MuPADvar')` | Warns | Three-argument version `setVar(nb, 'MuPADvar', MATLABexpr)` | Replace all instances of `setVar(nb,'MuPADvar')` with ```
setVar(nb,
'MuPADvar', MATLABexpr)
``` |

Character strings as inputs to `solve` | Warns | When specifying equations and variables, use symbolic equations
and variables instead of character strings. For example, replace ```
solve('2*r
= 1, r')
``` with `syms r; solve(2*r == 1, r)` . | Do not specify equations and variables as character strings.
Instead of string inputs, create symbolic variables using `syms` ,
and then pass them as input arguments separated by commas, or as a
vector of input arguments. |

`poly2sym(c, 'var')` will not accept a character
string `'var'` in a future release. | Warns | `syms var; poly2sym(c, var)` or ```
poly2sym(c,
sym('var'))
``` | Replace all instances of `poly2sym(c, 'var')` with ```
syms
var; poly2sym(c, var)
``` or `poly2sym(c, sym('var'))` |

`simple` | Errors | `simplify` | Replace all instances of `simple(S)` with `simplify` `(S)` .
There is no replacement for `[r,how] = simple(S)` . |

In previous releases, `in(x, type)` in some
cases returned logical `1` if `x` belonged
to `type` and `0` otherwise. | Returns a symbolic expression of the form `in(x, type)` | `isAlways` | To obtain the same results as in previous releases, wrap such
expressions in `isAlways` .
For example, use `isAlways(in(sym(5), 'integer'))` . |

In previous releases, the symbolic relational
operators in some cases evaluated equations involving only
symbolic numbers and returned logical `1` or `0` . | Returns a symbolic equation or inequality | `isAlways` | To obtain the same results as in previous releases, wrap equations
in `isAlways` .
For example, use `isAlways(A < B)` . |

`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`

.

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.

`isLowIndexDAE`

and `reduceDAEIndex`

These MATLAB and MuPAD functions help you

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`

.

The MATLAB symbolic

`reduceDifferentialOrder`

function reduces higher-order differential equations to a system of first-order differential equations. The`daetools::reduceDifferentialOrder`

function provides the same functionality in MuPAD.`incidenceMatrix`

computes the incidence matrix of a system of differential algebraic equations. The`daetools::incidenceMatrix`

function provides the same functionality in MuPAD.`reduceRedundancies`

eliminates simple equations from a system of symbolic differential algebraic equations. The`daetools::reduceRedundancies`

function provides the same functionality in MuPAD.`findDecoupledBlocks`

searches for decoupled blocks in systems of equations. The`daetools::findDecoupledBlocks`

function provides the same functionality in MuPAD.`isLowIndexDAE`

tests if a first-order system of differential algebraic equations is of differential index`0`

or`1`

. The`daetools::isLowIndexDAE`

function provides the same functionality in MuPAD.`reduceDAEIndex`

converts a system of first-order differential algebraic equations to an equivalent system of differential index`1`

. The`daetools::reduceDAEIndex`

function provides the same functionality in MuPAD.`reduceDAEToODE`

reduces the differential index of a system of first-order semilinear differential algebraic equations to`0`

. The`daetools::reduceDAEToODE`

function provides the same functionality in MuPAD.`daeFunction`

converts a system of differential algebraic equations to a MATLAB function handle.`massMatrixForm`

extracts the mass matrix and the right sides of a semilinear system of symbolic differential algebraic equations. The`daetools::massMatrixForm`

function provides the same functionality in MuPAD.

`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.

`pade`

function for computing Padé approximationThe MATLAB symbolic `pade`

function
calculates Padé approximations.

`kummerU`

function for computing confluent hypergeometric (Kummer U) functionThe MATLAB symbolic `kummerU`

function
computes the value of the confluent hypergeometric function. This
function is also known as the Kummer U function.

`polylog`

function for computing polylogarithmsThe MATLAB symbolic `polylog`

function
computes polylogarithms.

`signIm`

function for computing signs of imaginary parts of complex numbersThe 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)
```

.

`in`

function for representing conditions on symbolic inputsThe 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`

).

`divisors`

function for finding divisors of integers and polynomialsThe MATLAB symbolic `divisors`

function
computes divisors of integers and polynomials.

`pochhammer`

function to calculate Pochhammer symbolThe MATLAB symbolic `pochhammer`

function
calculates the Pochhammer symbol.

`kroneckerDelta`

function for computing the Kronecker delta functionThe MATLAB symbolic `kroneckerDelta`

function
calculates the Kronecker delta function.

`dirac`

function with two input arguments for computing derivatives of the Dirac delta functionThe MATLAB symbolic `dirac`

function
with one input argument represents the Dirac delta function. `dirac`

with
two input arguments, `dirac(n,x)`

represents the `n`

th
derivative of the Dirac delta function at `x`

.

`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.

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')`

.

`generate::fortran`

function can use Fortran 90 as the target compilerThe MuPAD `generate::fortran`

function
can use Fortran 90 as the target compiler, in addition to Fortran
77.

`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)
```

.

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*w`

≡`1 (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*w`

≡`1 (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)

`gcd`

and `lcm`

functions accept vectors and matricesThe 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`

.

`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.

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)`

.

`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.

The `factor`

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

The MuPAD `Symbol::LeftDoubleBracket`

and `Symbol::RightDoubleBracket`

functions
insert left and right double square brackets. These symbols are also
called white square brackets.

Functionality | What Happens When You Use It? | Use This Instead | Compatibility Considerations |
---|---|---|---|

`mfun` | Still runs | Appropriate 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. |

`mfunlist` | Still runs | See Special Functions. | See Special Functions for the list of available special functions. |

`setVar(nb, 'MuPADvar')` | Warns | Three-argument version `setVar(nb, 'MuPADvar', MATLABexpr)` | Replace all instances of `setVar(nb, 'MuPADvar')` with ```
setVar(nb,
'MuPADvar', MATLABexpr)
``` |

`simple` | Warns | `simplify` | Replace all instances of `simple(S)` with `simplify` `(S)` .
There is no replacement for `[r,how] = simple(S)` . |

`emlBlock` | Errors | `matlabFunctionBlock` | Replace all instances of `emlBlock` with `matlabFunctionBlock` . |

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)`

.

`qr`

for computing symbolic QR factorizationThe MATLAB symbolic `qr`

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

`combine`

for combining symbolic expressions with multiple calls to the same functionThe 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.

`vpasolve`

can use random starting points when searching for solutionsThe 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.

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

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 `]`

.

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` |

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]`

.

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]'`

.

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

`airy`

representing the Airy function`besseli`

,`besselj`

,`bessely`

, and`besselk`

representing the Bessel functions of the first and second kind, and the modified Bessel functions`beta`

representing the beta function`ellipticE`

,`ellipticF`

,`ellipticPi`

, and`ellipticCPi`

representing the elliptic integrals`lambertw`

representing the Lambert W function`whittakerM`

and`whittakerW`

representing the Whittaker M and Whittaker W functions`psi`

representing the polygamma function`fourier`

and`ifourier`

representing the Fourier and inverse Fourier transforms`laplace`

and`ilaplace`

representing the Laplace and inverse Laplace transforms`ztrans`

and`iztrans`

representing the Z-transform and inverse Z-transform

`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 | What Happens When You Use It? | Use This Instead | Compatibility Considerations |
---|---|---|---|

MuPAD `linalg::curl` | Still runs | `curl` | Replace all instances of `linalg::curl` with `curl` . |

MuPAD `linalg::det` | Still runs | `det` | Replace all instances of `linalg::det` with `det` . |

MuPAD `linalg::divergence` | Still runs | `divergence` | Replace all instances of `linalg::divergence` with `divergence` . |

MuPAD `linalg::grad` | Still runs | `gradient` | Replace all instances of `linalg::grad` with `gradient` . |

MuPAD `linalg::gradient` | Still runs | `gradient` | Replace all instances of `linalg::gradient` with `gradient` . |

MuPAD `linalg::hessian` | Still runs | `hessian` | Replace all instances of `linalg::hessian` with `hessian` . |

MuPAD `linalg::jacobian` | Still runs | `jacobian` | Replace all instances of `linalg::jacobian` with `jacobian` . |

MuPAD `linalg::laplacian` | Still runs | `laplacian` | Replace all instances of `linalg::laplacian` with `laplacian` . |

MuPAD `linalg::potential` | Still runs | `potential` | Replace all instances of `linalg::potential` with `potential` . |

MuPAD `linalg::vectorPotential` | Still runs | `vectorPotential` | Replace all instances of `linalg::vectorPotential` with `vectorPotential` . |

`simple` | Warns | `simplify` | Replace all instances of `simple(S)` with `simplify` `(S)` .
There is no replacement for `[r, how] = simple(S)` . |

`emlBlock` | Errors | `matlabFunctionBlock` | Replace all instances of `emlBlock` with `matlabFunctionBlock` . |

`evaluateMuPADNotebook`

and `allMuPADNotebooks`

functions to evaluate MuPAD notebooks and return list of open notebooksThe 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 curvesThe 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.

`isfinite`

, `isinf`

, and `isnan`

functions for testing for finite, infinite,
and `NaN`

elements in symbolic arrays`ExclusiveConditions`

option that makes MuPAD `piecewise`

function equivalent to an `if-elif-end_if`

statementThe 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.

`mupadNotebookTitle`

function to find the window title of the MuPAD notebookThe 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.

`close`

function to close MuPAD notebooks from MATLABThe 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 derivativesThe 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 polynomialsThe 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.

The MuPAD `_index`

function
and its equivalent `[]`

now accept lists, vectors,
and matrices as indices.

The MuPAD `assume`

, `assumeAlso`

, `assuming`

,
and `assumingAlso`

functions
let you set assumptions on matrices.

Functionality | What Happens When You Use It? | Use This Instead | Compatibility Considerations |
---|---|---|---|

`simple` | Still runs | `simplify` | Replace all instances of `simple(S)` with `simplify` `(S)` .
There is no replacement for `[r, how] = simple(S)` . |

`emlBlock` | Warns | `matlabFunctionBlock` | Replace all instances of `emlBlock` with `matlabFunctionBlock` . |

`diff` and `int` methods
for inputs of the `char` type | Errors | `sym` | Use the `sym` method
instead. |

MuPAD factoring functions `numlib::mpqs` , `numlib::pollard` ,
and `numlib::ecm` | Errors | `ifactor` | Replace all instances of `numlib::mpqs` , `numlib::pollard` ,
and `numlib::ecm` with `ifactor` . |

MuPAD `Dom::SparseMatrixF2` domain | Errors | `Dom::Matrix(Dom::IntegerMod(2))` | Replace all instances of `Dom::SparseMatrixF2` with `Dom::Matrix` `(` `Dom::IntegerMod` `(2))` . |

MuPAD `userinfo` | Errors | `print` | Use `print` instead
of `userinfo` . |

MuPAD `setuserinfo` | Errors | `prog::trace` or `debug` | Try using `prog::trace` or `debug` instead
of `setuserinfo` . |

`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.

`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 systemsSubstitute elements of a symbolic expression with arbitrary symbolic functions

`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`

.

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.

`atan2`

function, imaginary error function, and exponential and elliptic integralsThe 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 matricesThe 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 matricesThe MATLAB symbolic `sqrtm`

function
computes the square root of a symbolic matrix.

`sign`

function for computing signs of numbersThe 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 conjugatesThe 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.

`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.

`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 objectsThe MATLAB `isequaln`

function
tests symbolic objects for equality, treating `NaN`

values
as equal.

`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 | What Happens When You Use It? | Use This Instead | Compatibility Considerations |
---|---|---|---|

`simple` | Still runs | `simplify` | Replace all instances of `simple(S)` with `simplify` `(S)` .
There is no replacement for `[r, how] = simple(S)` . |

`emlBlock` | Warns | `matlabFunctionBlock` | Replace all instances of `emlBlock` with `matlabFunctionBlock` . |

MuPAD factoring functions `numlib::mpqs` , `numlib::pollard` ,
and `numlib::ecm` | Warns | `ifactor` | Replace all instances of `numlib::mpqs` , `numlib::pollard` ,
and `numlib::ecm` with `ifactor` . |

MuPAD `Dom::SparseMatrixF2` domain | Warns | `Dom::Matrix(Dom::IntegerMod(2))` | Replace all instances of `Dom::SparseMatrixF2` with `Dom::Matrix` `(` `Dom::IntegerMod` `(2))` . |

MuPAD `userinfo` | Warns | `print` | Use `print` instead
of `userinfo` . |

MuPAD `setuserinfo` | Warns | `prog::trace` or `debug` | Try using `prog::trace` or `debug` instead
of `setuserinfo` . |

`poly` | Errors | `charpoly` | Replace all instances of `poly` with `charpoly` . |

`sqrt` target of the MuPAD `simplify` function | Errors | MuPAD `radsimp` or `simplifyRadical` | Replace 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 | Errors | MuPAD `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::fourier` | Errors | MuPAD `fourier` | Replace all instances of `transform::fourier` with `fourier` . |

MuPAD `transform::invfourier` | Errors | MuPAD `ifourier` | Replace all instances of `transform::invfourier` with `ifourier` . |

MuPAD `transform::laplace` | Errors | MuPAD `laplace` | Replace all instances of `transform::laplace` with `laplace` . |

MuPAD `transform::invlaplace` | Errors | MuPAD `ilaplace` | Replace all instances of `transform::invlaplace` with `ilaplace` . |

MuPAD `transform::ztrans` | Errors | MuPAD `ztrans` | Replace all instances of `transform::ztrans` with `ztrans` . |

MuPAD `transform::invztrans` | Errors | MuPAD `iztrans` | Replace all instances of `transform::invztrans` with `iztrans` . |

MuPAD `transform::fourier::addpattern` | Errors | MuPAD `fourier::addpattern` | Replace all instances of `transform::fourier::addpattern` with `fourier::addpattern` . |

MuPAD `transform::invfourier::addpattern` | Errors | MuPAD `ifourier::addpattern` | Replace all instances of `transform::invfourier::addpattern` with `ifourier::addpattern` . |

MuPAD `transform::laplace::addpattern` | Errors | MuPAD `laplace::addpattern` | Replace all instances of `transform::laplace::addpattern` with `laplace::addpattern` . |

MuPAD `transform::invlaplace::addpattern` | Errors | MuPAD `ilaplace::addpattern` | Replace all instances of `transform::invlaplace::addpattern` with `ilaplace::addpattern` . |

MuPAD `transform::ztrans::addpattern` | Errors | MuPAD `ztrans::addpattern` | Replace all instances of `transform::ztrans::addpattern` with `ztrans::addpattern` . |

MuPAD `transform::invztrans::addpattern` | Errors | MuPAD `iztrans::addpattern` | Replace all instances of `transform::invztrans::addpattern` with `iztrans::addpattern` . |

MuPAD `prog::calledFrom` | Errors | `context(hold(procname))` | Replace all instances of `prog::calledFrom()` with `context` `(` `hold` `(procname))` . |

MuPAD `prog::calltree` | Errors | `prog::trace` | Use `prog::trace` instead
of `prog::calltree` . |

MuPAD `prog::error` | Errors | `getlasterror` | Use `getlasterror` instead
of `prog::error` . |

MuPAD `prog::memuse` | Errors | `prog::trace(Mem)` or `bytes()` | Use `prog::trace` `(Mem)` or `bytes` `()` instead
of `prog::memuse` . |

MuPAD `prog::testfunc` | Errors | `print(Unquoted, "...")` | Use `print` ```
(Unquoted,
"...")
``` instead of `prog::testfunc` . |

MuPAD `prog::testmethod` | Errors | `prog::test(..., Method = myTestMethod)` | Use `prog::test` ```
(...,
Method = myTestMethod)
``` instead of `prog::testmethod` . |

MuPAD `prog::testnum` | Errors | Nothing | No replacement |

Dynamic modules for MuPAD, including the `module` , `external` ,
and `Pref::unloadableModules` functions and all functions
of the `module` library | Errors | Nothing | No replacement |

`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.

`vpasolve`

function for solving equations and systems using variable precision arithmeticThe MATLAB `vpasolve`

function
solves equations and systems of equations numerically.

`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`

.

`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.

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.

`Pref::fourierParameters`

function for specifying Fourier parametersThe MuPAD `Pref::fourierParameters`

function
lets you specify parameters for Fourier and inverse Fourier transforms.

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

`fourier::addpattern`

adds new patterns for the Fourier transform.`ifourier::addpattern`

adds new patterns for the inverse Fourier transform.`laplace::addpattern`

adds new patterns for the Laplace transform.`ilaplace::addpattern`

adds new patterns for the inverse Laplace transform.`ztrans::addpattern`

adds new patterns for the Z-transform.`iztrans::addpattern`

adds new patterns for the inverse Z-transform.

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 flatteningThe 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`

.

In previous releases, `testtype`

used the `testtype`

slot for overloading by the second
argument.

`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`

.

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

`real`

and `imag`

Starting in R2012a, `real`

and `imag`

are
no longer defined via `conj`

.
They use the MuPAD `Re`

and `Im`

functions
instead.

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 | What Happens When You Use It? | Use This Instead | Compatibility Considerations |
---|---|---|---|

Old syntax of `taylor` | Errors | New calling syntax | Update all instances of `taylor` function
calls using the new syntax. |

`char(A,d)` | Errors | `char(A)` | Replace all instances of `char(A,d)` with `char` `(A)` . |

`poly` | Warns | `charpoly` | Replace all instances of `poly` with `charpoly` . |

`emlBlock` | Warns | `matlabFunctionBlock` | Replace all instances of `emlBlock` with `matlabFunctionBlock` . |

Ability to create links from MuPAD notebooks to MuPAD documentation pages | Not available | Nothing | No replacement |

`openmuphlp` | Errors | Nothing | No replacement |

MuPAD Help Browser | Not available | Documentation Center | MuPAD documentation is now available in Documentation Center. |

MuPAD Editor | Not available | MATLAB Editor | Open 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)` | Errors | `psi(k,X)` , where `k` is
a scalar specifying the `k` th 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 `size(Y)` will
be `size(X)` . Modify any code that depends on `size(Y)` . |

`diff` and `int` methods
for inputs of the `char` type | Errors | `sym` | Use the `sym` method instead. |

`sqrt` target of the MuPAD `simplify` function | Errors | MuPAD `radsimp` or `simplifyRadical` | Replace 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 | Errors | MuPAD `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::fourier` | Warns | MuPAD `fourier` | Replace all instances of `transform::fourier` with `fourier` . |

MuPAD `transform::invfourier` | Warns | MuPAD `ifourier` | Replace all instances of `transform::invfourier` with `ifourier` . |

MuPAD `transform::laplace` | Warns | MuPAD `laplace` | Replace all instances of `transform::laplace` with `laplace` . |

MuPAD `transform::invlaplace` | Warns | MuPAD `ilaplace` | Replace all instances of `transform::invlaplace` with `ilaplace` . |

MuPAD `transform::ztrans` | Warns | MuPAD `ztrans` | Replace all instances of `transform::ztrans` with `ztrans` . |

MuPAD `transform::invztrans` | Warns | MuPAD `iztrans` | Replace all instances of `transform::invztrans` with `iztrans` . |

MuPAD `transform::fourier::addpattern` | Warns | MuPAD `fourier::addpattern` | Replace all instances of `transform::fourier::addpattern` with `fourier::addpattern` . |

MuPAD `transform::invfourier::addpattern` | Warns | MuPAD `ifourier::addpattern` | Replace all instances of `transform::invfourier::addpattern` with `ifourier::addpattern` . |

MuPAD `transform::laplace::addpattern` | Warns | MuPAD `laplace::addpattern` | Replace all instances of `transform::laplace::addpattern` with `laplace::addpattern` . |

MuPAD `transform::invlaplace::addpattern` | Warns | MuPAD `ilaplace::addpattern` | Replace all instances of `transform::invlaplace::addpattern` with `ilaplace::addpattern` . |

MuPAD `transform::ztrans::addpattern` | Warns | MuPAD `ztrans::addpattern` | Replace all instances of `transform::ztrans::addpattern` with `ztrans::addpattern` . |

MuPAD `transform::invztrans::addpattern` | Warns | MuPAD `iztrans::addpattern` | Replace all instances of `transform::invztrans::addpattern` with `iztrans::addpattern` . |

MuPAD `prog::calledFrom` | Warns | `context(hold(procname))` | Replace all instances of `prog::calledFrom()` with `context` `(` `hold` `(procname))` . |

MuPAD `prog::calltree` | Warns | `prog::trace` | Use `prog::trace` instead
of `prog::calltree` . |

MuPAD `prog::error` | Warns | `getlasterror` | Use `getlasterror` instead
of `prog::error` . |

MuPAD `prog::memuse` | Warns | `prog::trace(Mem)` or `bytes()` | Use `prog::trace` `(Mem)` or `bytes` `()` instead
of `prog::memuse` . |

MuPAD `prog::testfunc` | Warns | `print(Unquoted, "...")` | Use `print` ```
(Unquoted,
"...")
``` instead of `prog::testfunc` . |

MuPAD `prog::testmethod` | Warns | `prog::test(..., Method = myTestMethod)` | Use `prog::test` ```
(...,
Method = myTestMethod)
``` instead of `prog::testmethod` . |

MuPAD `prog::testnum` | Warns | Nothing | No replacement |

Dynamic modules for MuPAD, including the `module` , `external` ,
and `Pref::unloadableModules` functions and all functions
of the `module` library | Warns | Nothing | No replacement |

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.

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.

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:

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.

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.

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)`

.

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:

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.

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.

`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`

.

The `taylor`

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

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`

.

The MuPAD `rectpulse`

and `tripulse`

functions
compute the rectangular and triangular pulse functions, respectively.

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`

.

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 now
performs pseudo-division of multivariate polynomials.

Improved MuPAD `prog::remember`

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

Functionality | What Happens When You Use It? | Use This Instead | Compatibility Considerations |
---|---|---|---|

Old syntax of `taylor` | Warns | New syntax | Update 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 setting | `simplify(S,'Steps',50)` | To terminate algebraic simplification after 50 steps, call `simplify` with
the name-value pair argument `'Steps', 50` . |

`char(A,d)` | Warns | `char(A)` | Replace all instances of `char(A,d)` with `char(A)` . |

`emlBlock` | Warns | `matlabFunctionBlock` | Replace all instances of `emlBlock` with `matlabFunctionBlock` . |

`psi(k0:k1,X)` | Warns | `psi(k,X)` where `k` is a
scalar specifying the `k` th 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 `size(Y)` will be `size(X)` .
Modify any code that depends on `size(Y)` . |

`sqrt` target of the MuPAD `simplify` function | Warns | MuPAD `radsimp` or `simplifyRadical` | Replace 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 | Warns | MuPAD `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` function | Errors | Nothing | No replacement |

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`

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.

`read`

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

`symprod`

computes
definite and indefinite products of symbolic series.

`hessian`

computes
the Hessian matrix of a scalar function.

`gradient`

computes
the vector gradient of a scalar function in Cartesian coordinates.
In MuPAD, the new `linalg::gradient`

function
is equivalent to `linalg::grad`

.

`erfc`

computes
the complementary error function.

`psi`

computes
the digamma and polygamma functions.

`wrightOmega`

computes
the Wright omega function.

`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`

.

The new MuPAD `simplifyRadical`

function
is equivalent to the MuPAD `radsimp`

function.

`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.

The MuPAD `normal`

function accepts the
new `Expand`

option that determines whether numerators
and denominators of fractions are expanded.

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`

.

All functions of the MuPAD `groebner`

library
now can accept and return polynomials with arbitrary arithmetical
expressions.

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.

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 | What Happens When You Use It? | Use This Instead | Compatibility Considerations |
---|---|---|---|

`emlBlock` | Warns | `matlabFunctionBlock` | Replace all instances of `emlBlock` with `matlabFunctionBlock` . |

`Real` and `IgnoreProperties` options
in MuPAD `ode::solve` | Warns | `IgnoreSpecialCases` or `IgnoreAnalyticConstraints` | Try using `IgnoreSpecialCases` or `IgnoreAnalyticConstraints` instead. |

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

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.

The ordinary differential equation solver demonstrates better performance.

The MuPAD functions `gcdex`

, `partfrac`

, `polylib::resultant`

,
and `solvelib::pdioe`

now demonstrate better performance.

`polylib::subresultant`

computes subresultants
of two polynomials or polynomial expressions.

With the new `List`

option, `partfrac`

returns
a list consisting of the numerators and denominators of the partial
fraction decomposition.

The `inverf`

function computes the inverse
of the error function.

The `inverfc`

function computes the inverse
of the complementary error function.

`numlib::checkPrimalityCertificate`

tests primality
certificates returned by `numlib::proveprime`

. For
information about proving primality of numbers, see "Proving
Primality" in the MuPAD documentation.

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 | What Happens When You Use This Functionality? | Use This Instead | Compatibility Considerations |
---|---|---|---|

MuPAD `matchlib::analyze` | Errors | MuPAD `prog::exprtree` | To visualize expressions, use` prog::exprtree` . |

MuPAD `prog::testcall` | Errors | Nothing | No replacement |

MuPAD `prog::testerrors` | Errors | Nothing | No replacement |

Old syntax of MuPAD `prog::getOptions` | Errors | The new syntax | Update all instances of `prog::getOptions` calls
using the new syntax. |

Old syntax of MuPAD `prog::trace` | Errors | The new syntax | Update all instances of `prog::trace` calls
using the new syntax. |

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.

The new MuPAD function `generate::Simscape`

converts MuPAD expressions
to Simscape™ equations.

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

MuPAD `linalg::matlinsolve`

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

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

The `prog::getOptions`

function that collects
and verifies options within a procedure has the new syntax.

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.

The `prog::trace`

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

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 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.

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.

The MuPAD `solve`

function performs better
on factorable polynomial systems.

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

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))
```

.

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 | What Happens When You Use This Functionality? | Use This Instead | Compatibility Considerations |
---|---|---|---|

`diff` and `int` methods
for inputs of the `char` type | Warns | `sym` | Use the `sym` method instead. |

MuPAD `matchlib::analyze` | Warns | MuPAD `prog::exprtree` | To visualize expressions, use` prog::exprtree` . |

MuPAD `prog::testcall` | Warns | None | No replacement |

MuPAD `prog::testerrors` | Warns | None | No replacement |

The following options in MuPAD `prog::trace` : `All` `Backup` `Force` `Name` `Proc` `Plain` `Width`
| Errors | None | No replacement. These options are not supported in the current release. |

Global properties in MuPAD | Errors | Assumptions on each variable | Make assumptions on each variable instead. |

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.

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.

The `sort`

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

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(:))`

.

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

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

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

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.

Basic arithmetic operations now work for lists.

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

Enhanced MuPAD `divide`

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

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

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, use`expr`

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])
```

.

The `coeff`

function accepts the new`All`

option.
With this option, `coeff`

returns all coefficients
of a polynomial including those equal to 0.

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.

The `expand`

function now expands powers of
products such as (*x**y*)^{n} for
positive *x* and *y*. When called
with the `IgnoreAnalyticConstraints`

option , the
function expands the power of products for arbitrary terms.

The `rationalize`

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

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 simplification functions, `simplify`

and `Simplify`

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

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.

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

Functionality | What Happens When You Use This Functionality? | Use This Instead | Compatibility Considerations |
---|---|---|---|

MuPAD Domain `Dom::Ideal` | Errors | `groebner` | Represent ideals as lists, and use functions of the `groebner` package
instead. |

MuPAD `student` library | Errors | `plot::Integral` and `linalg` | Use `plot::Integral` and the `linalg` package
instead. |

MuPAD `relation` option
in `simplify` | Errors | None | No replacement |

Global property | Warns | Assumptions on each variable | Make assumptions on each variable instead. |

`digits` and `vpa` do not
let you set the number of digits to 1. | Errors | Errors | It 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. |

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.

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')`

.

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

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 preference `Pref::outputDigits`

controls
the precision MuPAD uses to display floating-point results.

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

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

Enhanced simplification routines for MuPAD `hypergeom`

, `mejerG`

,
and `bessel`

special functions.

Enhanced `combine`

function demonstrates better
performance for logarithms.

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 | What Happens When You Use This Functionality? | Use This Instead | Compatibility Considerations |
---|---|---|---|

MuPAD `Domain Dom::Ideal` | Warns | `groebner` | Represent ideals as lists, and use functions of the `groebner` package
instead. |

MuPAD `student` library | Warns | `plot::Integral` and `linalg` | Use `plot::Integral` and the `linalg` package
instead. |

`d` in `char` ```
(A,
d)
``` | Warns | None | No replacement |

MuPAD `relation` option
in `simplify` | Warns | None | No replacement |

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.

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`

.

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`

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.

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.

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.

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`

.

The new `generate::MATLAB`

command converts MuPAD expressions,
equations, and matrices to MATLAB formatted strings.

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: $$\mathrm{ln}(a)+\mathrm{ln}(b)=\mathrm{ln}(a\cdot b)$$

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

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

The enhanced `limit`

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

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.

The `prog::tcov`

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

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

To see new demos that use MuPAD Notebook Interface, type `mupadDemo`

at
the MATLAB command line or click MuPAD
Notebooks Demo.

Bug Fixes

Bug Fixes

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.

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 definition MuPAD definition Fourier transform $$F(w)={\displaystyle \underset{-\infty}{\overset{\infty}{\int}}f(x){e}^{-iwx}}dx$$ $$F(w)={\displaystyle \underset{-\infty}{\overset{\infty}{\int}}f(x){e}^{iwx}}dx$$ Inverse Fourier transform $$f(x)=\frac{1}{2\pi}{\displaystyle \underset{-\infty}{\overset{\infty}{\int}}F(w){e}^{iwx}dw}$$ $$f(x)=\frac{1}{2\pi}{\displaystyle \underset{-\infty}{\overset{\infty}{\int}}F(w){e}^{-iwx}dw}$$ 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`

.

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.

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.

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.

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.

Function | Use |
---|---|

`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. |

`getVar` | Copy expressions residing in a MuPAD notebook into the MATLAB workspace. |

`mupad` | Launch a MuPAD notebook . |

`mupadwelcome` | Access MuPAD GUIs . |

`reset` `(symengine,...)` | Clear the MuPAD engine for the MATLAB workspace. |

`setVar` | Copy expressions residing in the MATLAB workspace into a MuPAD notebook. |

`symvar` | Produce a list of symbolic objects in an expression. |

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

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.

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

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)');

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)');

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