Convert symbolic expressions to function handle for ODE solvers

uses
additional options specified by one or more `f`

= odeFunction(___,`Name,Value`

)`Name,Value`

pair
arguments.

Convert a system of symbolic differential algebraic
equations to a function handle suitable for the MATLAB ODE solvers.
Then solve the system by using the `ode15s`

solver.

Create the following second-order differential algebraic equation.

syms y(t); eqn = diff(y(t),t,2) == (1-y(t)^2)*diff(y(t),t) - y(t);

Use `reduceDifferentialOrder`

to rewrite that equation as a
system of two first-order differential equations. Here, `vars`

is a
vector of state variables of the system. The new variable `Dy(t)`

represents the first derivative of `y(t)`

with respect to
`t`

.

[eqs,vars] = reduceDifferentialOrder(eqn,y(t))

eqs = diff(Dyt(t), t) + y(t) + Dyt(t)*(y(t)^2 - 1) Dyt(t) - diff(y(t), t) vars = y(t) Dyt(t)

Set initial conditions for `y(t)`

and its derivative
`Dy(t)`

to `2`

and `0`

respectively.

initConditions = [2 0];

Find the mass matrix `M`

of the system and the right sides of the
equations `F`

.

[M,F] = massMatrixForm(eqs,vars)

M = [ 0, 1] [ -1, 0] F = - y(t) - Dyt(t)*(y(t)^2 - 1) -Dyt(t)

`M`

and `F`

refer to the form $$M\left(t,x\left(t\right)\right)\dot{x}\left(t\right)=F\left(t,x\left(t\right)\right).$$. To simplify further computations, rewrite the system in the form $$\dot{x}\left(t\right)=f\left(t,x\left(t\right)\right)$$.

f = M\F

f = Dyt(t) - Dyt(t)*y(t)^2 - y(t) + Dyt(t)

Convert `f`

to a MATLAB function handle by using `odeFunction`

. The resulting
function handle is input to the MATLAB ODE solver `ode15s`

.

odefun = odeFunction(f,vars); ode15s(odefun, [0 10], initConditions)

Convert a system of symbolic differential equations containing both state variables and symbolic parameters to a function handle suitable for the MATLAB ODE solvers.

Create the system of differential algebraic equations. Here, the symbolic functions
`x1(t)`

and `x2(t)`

represent the state variables of
the system. The system also contains constant symbolic parameters `a`

,
`b`

, and the parameter function `r(t)`

. These
parameters do not represent state variables. Specify the equations and state variables
as two symbolic vectors: equations as a vector of symbolic equations, and variables as a
vector of symbolic function calls.

syms x1(t) x2(t) a b r(t) eqs = [diff(x1(t),t) == a*x1(t) + b*x2(t)^2,... x1(t)^2 + x2(t)^2 == r(t)^2]; vars = [x1(t) x2(t)];

Find the mass matrix `M`

and vector of the right side
`F`

for this system. `M`

and `F`

refer to the form $$M\left(t,x\left(t\right)\right)\dot{x}\left(t\right)=F\left(t,x\left(t\right)\right).$$.

[M,F] = massMatrixForm(eqs,vars)

M = [ 1, 0] [ 0, 0] F = b*x2(t)^2 + a*x1(t) r(t)^2 - x1(t)^2 - x2(t)^2

Use `odeFunction`

to generate MATLAB function handles from `M`

and `F`

. The
function handle `F`

contains symbolic parameters.

M = odeFunction(M,vars) F = odeFunction(F,vars,a,b,r(t))

M = function_handle with value: @(t,in2)reshape([1.0,0.0,0.0,0.0],[2,2]) F = function_handle with value: @(t,in2,param1,param2,param3)[param1.*in2(1,:)+... param2.*in2(2,:).^2;param3.^2-in2(1,:).^2-in2(2,:).^2]

Specify the parameter values.

a = -0.6; b = -0.1; r = @(t) cos(t)/(1+t^2);

Create the reduced function handle `F`

.

F = @(t,Y) F(t,Y,a,b,r(t));

Specify consistent initial conditions for the DAE system.

t0 = 0; y0 = [-r(t0)*sin(0.1); r(t0)*cos(0.1)]; yp0 = [a*y0(1) + b*y0(2)^2; 1.234];

Create an option set that contains the mass matrix `M`

of
the system and vector `yp0`

of initial conditions for the
derivatives.

opt = odeset('mass',M,'InitialSlope',yp0);

Now, use `ode15s`

to solve the system of equations.

ode15s(F, [t0, 1], y0, opt)

Write the generated function handles to files by using the
`File`

option. When writing to files,
`odeFunction`

optimizes the code using intermediate variables named
`t0`

, `t1`

, .… Include comments the files by
specifying the `Comments`

option.

Define the system of differential equations. Find the mass matrix
`M`

and the right side `F`

.

syms x(t) y(t) eqs = [diff(x(t),t)+2*diff(y(t),t) == 0.1*y(t), ... x(t)-y(t) == cos(t)-0.2*t*sin(x(t))]; vars = [x(t) y(t)]; [M,F] = massMatrixForm(eqs,vars);

Write the MATLAB code for `M`

and `F`

to the files
`myfileM`

and `myfileF`

.
`odeFunction`

overwrites existing files. Include the comment
`Version: 1.1`

in the files You can open and edit the output
files.

M = odeFunction(M,vars,'File','myfileM','Comments','Version: 1.1');

function expr = myfileM(t,in2) %MYFILEM % EXPR = MYFILEM(T,IN2) % This function was generated by the Symbolic Math Toolbox version 7.3. % 01-Jan-2017 00:00:00 %Version: 1.1 expr = reshape([1.0,0.0,2.0,0.0],[2, 2]);

F = odeFunction(F,vars,'File','myfileF','Comments','Version: 1.1');

function expr = myfileF(t,in2) %MYFILEF % EXPR = MYFILEF(T,IN2) % This function was generated by the Symbolic Math Toolbox version 7.3. % 01-Jan-2017 00:00:00 %Version: 1.1 x = in2(1,:); y = in2(2,:); expr = [y.*(1.0./1.0e1);-x+y+cos(t)-t.*sin(x).*(1.0./5.0)];

Specify consistent initial values for `x(t)`

and
`y(t)`

and their first derivatives.

xy0 = [2; 1]; % x(t) and y(t) xyp0 = [0; 0.05*xy0(2)]; % derivatives of x(t) and y(t)

Create an option set that contains the mass matrix `M`

, initial
conditions `xyp0`

, and numerical tolerances for the numerical
search.

opt = odeset('mass', M, 'RelTol', 10^(-6),... 'AbsTol', 10^(-6), 'InitialSlope', xyp0);

Solve the system of equations by using `ode15s`

.

ode15s(F, [0 7], xy0, opt)

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

when
converting sparse symbolic matrices to MATLAB function handles.

Create the system of differential algebraic equations. Here, the symbolic functions
`x1(t)`

and `x2(t)`

represent the state variables of
the system. Specify the equations and state variables as two symbolic vectors: equations
as a vector of symbolic equations, and variables as a vector of symbolic function
calls.

syms x1(t) x2(t) a = -0.6; b = -0.1; r = @(t) cos(t)/(1 + t^2); eqs = [diff(x1(t),t) == a*x1(t) + b*x2(t)^2,... x1(t)^2 + x2(t)^2 == r(t)^2]; vars = [x1(t) x2(t)];

Find the mass matrix `M`

and vector of the right side
`F`

for this system. `M`

and `F`

refer to the form $$M\left(t,x\left(t\right)\right)\dot{x}\left(t\right)=F\left(t,x\left(t\right)\right).$$.

[M,F] = massMatrixForm(eqs,vars)

M = [ 1, 0] [ 0, 0] F = - (3*x1(t))/5 - x2(t)^2/10 cos(t)^2/(t^2 + 1)^2 - x1(t)^2 - x2(t)^2

Generate MATLAB function handles from `M`

and `F`

.
Because most of the elements of the mass matrix `M`

are zeros, use the
`Sparse`

argument when converting `M`

.

M = odeFunction(M,vars,'Sparse',true) F = odeFunction(F,vars)

M = function_handle with value: @(t,in2)sparse([1],[1],[1.0],2,2) F = function_handle with value: @(t,in2)[in2(1,:).*(-3.0./5.0)-in2(2,:).^2./1.0e+1;... cos(t).^2.*1.0./(t.^2+1.0).^2-in2(1,:).^2-in2(2,:).^2]

Specify consistent initial conditions for the DAE system.

t0 = 0; y0 = [-r(t0)*sin(0.1); r(t0)*cos(0.1)]; yp0= [a*y0(1) + b*y0(2)^2; 1.234];

Create an option set that contains the mass matrix `M`

of
the system and vector `yp0`

of initial conditions for the
derivatives.

opt = odeset('mass',M,'InitialSlope', yp0);

Solve the system of equations using `ode15s`

.

ode15s(F, [t0, 1], y0, opt)

`expr`

— System of algebraic expressionsvector of symbolic expressions

System of algebraic expressions, specified as a vector of symbolic expressions.

`vars`

— State variablesvector of symbolic functions | vector of symbolic function calls

State variables, specified as a vector of symbolic functions
or function calls, such as `x(t)`

.

**Example: **`[x(t),y(t)]`

or `[x(t);y(t)]`

`p1,...,pN`

— Parameters of systemsymbolic variables | symbolic functions | symbolic function calls | symbolic vector | symbolic matrix

Parameters of the system, specified as symbolic variables, functions,
or function calls, such as `f(t)`

. You can also specify
parameters of the system as a vector or matrix of symbolic variables,
functions, or function calls. If `expr`

contains
symbolic parameters other than the variables specified in `vars`

,
you must specify these additional parameters as `p1,...,pN`

.

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`odeFunction(expr,vars,'File','myfile')`

`'Comments'`

— Comments to include in file headercharacter vector | cell array of character vectors | string vector

Comments to include in the file header, specified as a character vector, cell array of character vectors, or string vector.

`'File'`

— Path to file containing generated codecharacter vector

Path to the file containing generated code, specified as a character
vector. The generated file accepts arguments of type `double`

,
and can be used without Symbolic Math
Toolbox™. If the value is
empty, `odeFunction`

generates an anonymous function.
If the character vector does not end in `.m`

, the
function appends `.m`

.

By default, `odeFunction`

with the `File`

argument
generates a file containing optimized code. Optimized means intermediate
variables are automatically generated to simplify or speed up the
code. MATLAB generates intermediate variables as a lowercase
letter `t`

followed by an automatically generated
number, for example `t32`

. To disable code optimization,
use the `Optimize`

argument.

`'Optimize'`

— Flag preventing optimization of code written to function file`true`

(default) | `false`

Flag preventing optimization of code written to a function file,
specified as `false`

or `true`

.

By default, `odeFunction`

with the `File`

argument
generates a file containing optimized code. Optimized means intermediate
variables are automatically generated to simplify or speed up the
code. MATLAB generates intermediate variables as a lowercase
letter `t`

followed by an automatically generated
number, for example `t32`

.

`odeFunction`

without the `File`

argument
(or with a file path specified by an empty character vector) creates
a function handle. In this case, the code is not optimized. If you
try to enforce code optimization by setting `Optimize`

to `true`

,
then `odeFunction`

throws an error.

`'Sparse'`

— Flag that switches between sparse and dense matrix generation`false`

(default) | `true`

Flag that switches between sparse and dense matrix generation,
specified as `true`

or `false`

.
When you specify `'Sparse',true`

, the generated function
represents symbolic matrices by sparse numeric matrices. Use `'Sparse',true`

when
you convert symbolic matrices containing many zero elements. Often,
operations on sparse matrices are more efficient than the same operations
on dense matrices. See Sparse Matrices.

`f`

— Function handle that is input to numerical MATLAB ODE solvers, except `ode15i`

MATLAB function handle

Function handle that can serve as input argument to all numerical MATLAB ODE
solvers, except for `ode15i`

, returned as a MATLAB function
handle.

`odeFunction`

returns a function handle suitable
for the ODE solvers such as `ode45`

, `ode15s`

, `ode23t`

,
and others. The only ODE solver that does not accept this function
handle is the solver for fully implicit differential equations, `ode15i`

.
To convert the system of equations to a function handle suitable for `ode15i`

,
use `daeFunction`

.

`daeFunction`

| `decic`

| `findDecoupledBlocks`

| `incidenceMatrix`

| `isLowIndexDAE`

| `massMatrixForm`

| `matlabFunction`

| `ode15i`

| `ode15s`

| `ode23t`

| `ode45`

| `reduceDAEIndex`

| `reduceDAEToODE`

| `reduceDifferentialOrder`

| `reduceRedundancies`

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