To simulate a dynamic system, you compute its states at successive time steps over a
specified time span. This computation uses information provided by a model of the system.
*Time steps* are time intervals when the computation happens. The
size of this time interval is called *step size*. The process of
computing the states of a model in this manner is known as *solving*
the model. No single method of solving a model applies to all systems. Simulink^{®} provides a set of programs called *solvers*. Each solver
embodies a particular approach to solving a model.

A solver applies a numerical method to solve the set of ordinary differential equations that represent the model. Through this computation, it determines the time of the next simulation step. In the process of solving this initial value problem, the solver also satisfies the accuracy requirements that you specify.

Simulink provides two main types of solvers —fixed-step and variable-step solvers. You
can see the solvers under each type in the **Solver** pane of model
configuration parameters.

Solvers are broadly classified using these criteria:

The type of step size used in the computation

Fixed-step solvers, as the name suggests, solve the model at fixed step sizes from the beginning to the end of the simulation. You can specify the step size or let the solver choose it. Generally, decreasing the step size increases the accuracy of the results and increases the time required to simulate the system.

Variable-step solvers vary the step size during the simulation. They reduce the step size to increase accuracy when the states of a model change rapidly and during zero-crossing events. They increase the step size to avoid taking unnecessary steps when the states of a model change slowly. Computing the step size adds to the computational overhead at each step. However, it can reduce the total number of steps, and hence the simulation time required to maintain a specified level of accuracy for models with piecewise continuous or rapidly changing states.

The nature of states in the model

Continuous solvers use numerical integration to compute continuous states of a model at the current time step based on the states at previous time steps and the state derivatives. Continuous solvers rely on individual blocks to compute the values of the discrete states of the model at each time step.

Discrete solvers are primarily for solving purely discrete models. They compute only the next simulation time step for a model. When they perform this computation, they rely on each block in the model to update its individual discrete state. They do not compute continuous states.

When choosing a solver for simulating a model, consider:

The dynamics of the system

The stability of the solution

The speed of computation

The robustness of the solver

A solver might not completely satisfy all of your requirements, so use an iterative approach when choosing one. Compare simulation results from several solvers and select one that offers the best performance with minimal tradeoffs.

There are two ways to select a solver for your model:

Use auto solver. New models have their solver selection set to auto solver by default. Auto solver recommends a fixed-step or variable-step solver for your model as well as maximum step size. For more information, see Use Auto Solver to Select a Solver

If you are not satisfied with simulation results using auto solver, select a solver in the

**Solver**pane in the model configuration parameters.

When you build and simulate a model, you can choose the solver based on the dynamics of your model. A variable-step solver is better suited for purely continuous models, like the dynamics of a mass spring damper system. A fixed-step solver is recommended for a model that contains several switches, like an inverter power system, due to the number of solver resets that would cause a variable-step solver to behave like a fixed-step solver.

When you deploy a model as generated code, you can use only a fixed-step solver. If you select a variable-step solver during simulation, use it to calculate the step size required for the fixed-step solver that you need at deployment.

This chart provides a broad classification of solvers in the Simulink library.

To tailor the selected solver to your model, see Check and Improve Simulation Accuracy.

Ideally, the solver you select should:

Solve the model successfully.

For variable-step solvers, provide a solution within the tolerance limits you specify.

Solve the model in a reasonable duration.

A single solver might not meet all of these requirements. Try simulating with several solvers before making a selection.

The Simulink library provides several solvers, all of which can work with the algebraic loop solver. For more information, see How the Algebraic Loop Solver Works.

Discrete | Continuous | Variable-Order | ||
---|---|---|---|---|

Fixed-Step | Explicit | Not Applicable | Fixed-Step Continuous Explicit Solvers | Not Applicable |

Implicit | Not Applicable | Fixed-Step Continuous Implicit Solvers | Not Applicable | |

Variable-Step | Explicit | Choose a Variable-Step Solver | Variable-Step Continuous Explicit Solvers | Single-Order Versus Variable-Order Continuous Solvers |

Implicit | Variable-Step Continuous Implicit Solvers | Single-Order Versus Variable-Order Continuous Solvers |

In the **Solver** pane of model configuration parameters, the
Simulink library of solvers is divided into two major types. See Fixed-Step Versus Variable-Step Solvers.

You can further categorize the solvers of each type:

Discrete Versus Continuous Solvers

Explicit Versus Implicit Continuous Solvers

One Step Versus Multistep Continuous Solvers

Single Order Versus Variable Order Continuous Solvers

Fixed-step and variable-step solvers compute the next simulation time as the sum
of the current simulation time and the step size. The
**Type** control on the **Solver**
configuration pane allows you to select the type of solver. With a fixed-step
solver, the step size remains constant throughout the simulation. With a
variable-step solver, the step size can vary from step to step, depending on the
model dynamics. In particular, a variable-step solver increases or reduces the step
size to meet the error tolerances that you specify.

The choice between these types depends on how you plan to deploy your model and the model dynamics. If you plan to generate code from your model and run the code on a real-time computer system, choose a fixed-step solver to simulate the model. You cannot map the variable-step size to the real-time clock.

If you do not plan to deploy your model as generated code, the choice between a variable-step and a fixed-step solver depends on the dynamics of your model. A variable-step solver might shorten the simulation time of your model significantly. A variable-step solver allows this saving because, for a given level of accuracy, the solver can dynamically adjust the step size as necessary. This approach reduces the number of steps required. The fixed-step solver must use a single step size throughout the simulation, based on the accuracy requirements. To satisfy these requirements throughout the simulation, the fixed-step solver typically requires a small step.

The `ex_multirate`

shows how a variable-step solver can
shorten simulation time for a multirate discrete model.

The model generates outputs at two different rates: every 0.5 s and
every 0.75 s. To capture both outputs, the fixed-step solver must take a time step
every 0.25 s (the *fundamental sample time* for the
model).

[0.0 0.25 0.5 0.75 1.0 1.25 1.5 ...]

By contrast, the variable-step solver has to take a step only when the model generates an output.

[0.0 0.5 0.75 1.0 1.5 ...]

This scheme significantly reduces the number of time steps required to simulate the model.

When you select a solver type, you can also select a specific solver. Both sets of solvers include two types: discrete and continuous. Discrete and continuous solvers rely on the model blocks to compute the values of any discrete states. Blocks that define discrete states are responsible for computing the values of those states at each time step. However, unlike discrete solvers, continuous solvers use numerical integration to compute the continuous states that the blocks define. When choosing a solver, determine first whether to use a discrete solver or a continuous solver.

If your model has no continuous states, then Simulink switches to either the fixed-step discrete solver or the variable-step discrete solver. If your model has only continuous states or a mix of continuous and discrete states, choose a continuous solver from the remaining solver choices based on the dynamics of your model. Otherwise, an error occurs.

The solver library contains two discrete solvers—a fixed-step discrete solver and a variable-step discrete solver. The fixed-step solver by default chooses a step size and simulation rate fast enough to track state changes in the fastest block in your model. The variable-step solver adjusts the simulation step size to keep pace with the actual rate of discrete state changes in your model. This adjustment can avoid unnecessary steps and shorten simulation time for multirate models (see Sample Times in Systems for more information.)

The fixed-step discrete solvers do not solve for discrete states. Each block calculates its discrete states independently of the solver.

You represent an explicit system by the system of equation

$$\dot{x}=f(x)$$

For any given value of *x*, you can compute $$\dot{x}$$ by substituting *x* in *f(x)*
and evaluating the equation.

Equations of the form

$$F(\dot{x},x)=0$$

are considered to be implicit. For any given value of $$x$$, you must solve this equation to calculate $$\dot{x}$$.

A linearly implicit system can be represented by the equation

$$M(x).\dot{x}=f(x)$$

*M*(*x*) is called the mass matrix and $$f(x)$$ is the forcing function. A system becomes linearly implicit when
you use physical modeling blocks in the model.

While you can apply an implicit or explicit continuous solver to solve all these systems, implicit solvers are designed specifically for solving stiff problems. Explicit solvers solve nonstiff problems. An ordinary differential equation problem is said to be stiff if the desired solution varies slowly, but there are closer solutions that vary rapidly. The numerical method must then take small time steps to solve the system. Stiffness is an efficiency issue. The more stiff a system, the longer it takes to for the explicit solver to perform a computation. A stiff system has both slowly and quickly varying continuous dynamics.

When compared to explicit solvers, implicit solvers provide greater stability for
oscillatory behavior. However, implicit solvers are also computationally more
expensive. They generate the Jacobian matrix and solve the set of algebraic
equations at every time step using a Newton-like method. To reduce this extra cost,
the implicit solvers offer a `Solver Jacobian method`

parameter
that allows you to improve the simulation performance of implicit solvers. See Choose a Jacobian Method for an Implicit Solver for more information.
Implicit solvers are more efficient than explicit solvers for solving a linearly
implicit system.

The Simulink solver library provides both one-step and multistep solvers. The
one-step solvers estimate using the solution at the immediately preceding time point, , and the values of the derivative at multiple points between
*t _{n}* and

Multistep solvers use the results at several preceding time steps to compute the
current solution. Simulink provides one explicit multistep solver, `ode113`

, and
one implicit multistep solver, `ode15s`

. Both are variable-step
solvers.

This distinction is based on the number of orders that the solver uses to solve
the system of equation. Two variable-order solvers, `ode15s`

and
`ode113`

, are part of the solver library. They use multiple
orders to solve the system of equations. Specifically, the implicit, variable-step
`ode15s`

solver uses first-order through fifth-order equations,
while the explicit, variable-step `ode113`

solver uses first-order
through thirteenth-order equations. For `ode15s`

, you can limit the
highest order applied via the `Maximum Order`

parameter. For more
information, see Maximum Order.

The fixed-step discrete solver computes the time of the next simulation step by
adding a fixed step size to the current time. The accuracy and the length of time of
the resulting simulation depends on the size of the steps taken by the simulation:
the smaller the step size, the more accurate the results are but the longer the
simulation takes. By default, Simulink chooses the step size or you can choose the step size yourself. If you
choose the default setting of `auto`

, and if the model has
discrete sample times, then Simulink sets the step size to the fundamental sample time of the model.
Otherwise, if no discrete rates exist, Simulink sets the size to the result of dividing the difference between the
simulation start and stop times by 50.

If you try to use the fixed-step discrete solver to update or simulate a model that has continuous states, an error message appears. Thus, selecting a fixed-step solver and then updating or simulating a model is a quick way to determine whether the model has continuous states.

The fixed-step continuous solvers, like the fixed-step discrete solver, compute the next simulation time by adding a fixed-size time step to the current time. For each of these steps, the continuous solvers use numerical integration to compute the values of the continuous states for the model. These values are calculated using the continuous states at the previous time step and the state derivatives at intermediate points (minor steps) between the current and the previous time step.

Simulink uses the fixed-step discrete solver for a model that contains no states or only discrete states, even if you specify a fixed-step continuous solver for the model.

Simulink provides two types of fixed-step continuous solvers — explicit and implicit.

The difference between these two types lies in the speed and the stability. An implicit solver requires more computation per step than an explicit solver but is more stable. Therefore, the implicit fixed-step solver that Simulink provides is more adept at solving a stiff system than the fixed-step explicit solvers. For more information, see Explicit Versus Implicit Continuous Solvers.

**Fixed-Step Continuous Explicit Solvers. **Explicit solvers compute the value of a state at the next time step as an
explicit function of the current values of both the state and the state
derivative. A fixed-step explicit solver is expressed mathematically as:

$$x(n+1)=x(n)+h\ast Dx(n)$$

where *x* is the state,
*Dx* is a solver-dependent function that estimates the
state derivative, *h* is the step size, and
*n* indicates the current time step.

Simulink provides a set of fixed-step continuous explicit solvers. The
solvers differ in the specific numerical integration technique that they use to
compute the state derivatives of the model. This table lists each solver and the
integration technique it uses. The table lists the solvers in order of the
computational complexity of the integration methods they use, from the least
complex (`ode1`

) to the most complex
(`ode8`

).

Solver | Integration Technique | Order of Accuracy |
---|---|---|

| Euler's Method | First |

| Heun's Method | Second |

| Bogacki-Shampine Formula | Third |

| Fourth-Order Runge-Kutta (RK4) Formula | Fourth |

| Dormand-Prince (RK5) Formula | Fifth |

| Dormand-Prince RK8(7) Formula | Eighth |

None of these solvers has an error control mechanism. Therefore, the accuracy and the duration of a simulation depend directly on the size of the steps taken by the solver. As you decrease the step size, the results become more accurate, but the simulation takes longer. Also, for any given step size, the higher the order of the solver, the more accurate the simulation results.

If you specify a fixed-step solver type for a model, then by default,
Simulink selects the `FixedStepAuto`

solver. Auto solver
then selects an appropriate fixed-step solver that can handle both continuous
and discrete states with moderate computational effort. As with the discrete
solver, if the model has discrete rates (sample times), then Simulink sets the step size to the fundamental sample time of the model by
default. If the model has no discrete rates, Simulink automatically uses the result of dividing the simulation total
duration by 50. Consequently, the solver takes a step at each simulation time at
which Simulink must update the discrete states of the model at its specified
sample rates. However, it does not guarantee that the default solver accurately
computes the continuous states of a model. Therefore, you may need to choose
another solver, a different fixed step size, or both to achieve acceptable
accuracy and an acceptable simulation time.

**Fixed-Step Continuous Implicit Solvers. **An implicit solver computes the state at the next time step as an implicit
function of the state at the current time step and the state derivative at the
next time step. In other words:

$$x(n+1)-x(n)-h\ast Dx(n+1)=0$$

Simulink provides one fixed-step implicit solver:
`ode14x`

. This solver uses a combination of Newton's method and
extrapolation from the current value to compute the value of a state at the next
time step. You can specify the number of Newton's method iterations and the
extrapolation order that the solver uses to compute the next value of a model
state (see Fixed-step size (fundamental sample time)). The more
iterations and the higher the extrapolation order that you select, the greater
the accuracy you obtain. However, you simultaneously create a greater
computational burden per step size.

Any of the fixed-step continuous solvers in the Simulink product can simulate a model to any desired level of accuracy, given a
small enough step size. Unfortunately, it is not possible or practical to decide
* without trial,* the combination of
solver and step size that will yield acceptable results for the continuous states in
the shortest time. Determining the best solver for a particular model generally
requires experimentation.

To select a fixed-step continuous solver,

Choose error tolerances. For more information, see Error Tolerances for Variable-Step Solvers.

Use one of the variable-step solvers to simulate your model to the level of accuracy that you want. Start with

`ode45`

. If your model runs slowly, your problem may be stiff and need an implicit solver. The results of this step give a good approximation of the correct simulation results and the appropriate fixed step size.Use

`ode1`

to simulate your model at the default step size for your model. Compare the simulation results for`ode1`

with the simulation for the variable-step solver. If the results are the same for the specified level of accuracy, you have found the best fixed-step solver for your model, namely`ode1`

. You arrive at this conclusion because`ode1`

is the simplest of the fixed-step solvers and hence yields the shortest simulation time for the current step size.If

`ode1`

does not give satisfactory results, repeat the preceding steps with the other fixed-step solvers until you find one that gives accurate results with the least computational effort. The most efficient way to perform this task is to use a binary search technique:Try

`ode3`

.If

`ode3`

gives accurate results, try`ode2`

. If`ode2`

gives accurate results, it is the best solver for your model; otherwise,`ode3`

is the best.If

`ode3`

does not give accurate results, try`ode5`

. If`ode5`

gives accurate results, try`ode4`

. If`ode4`

gives accurate results, select it as the solver for your model; otherwise, select`ode5`

.If

`ode5`

does not give accurate results, reduce the simulation step size and repeat the preceding process. Continue in this way until you find a solver that solves your model accurately with the least computational effort.

When you set the **Type** control of the **Solver** configuration pane to
`Variable-step`

, the **Solver** control
allows you to choose one of the variable-step solvers. As with fixed-step solvers, the
set of variable-step solvers comprises a discrete solver and a subset of continuous
solvers. However, unlike the fixed-step solvers, the step size varies dynamically based
on the local error.

The choice between the two types of variable-step solvers depends on whether the blocks in your model define states and, if so, the type of states that they define. If your model defines no states or defines only discrete states, select the discrete solver. If a model has no states or only discrete states, Simulink uses the discrete solver to simulate the model even if you specify a continuous solver. If the model has continuous states, the continuous solvers use numerical integration to compute the values of the continuous states at the next time step.

Variable-step solvers dynamically vary the step size during the simulation. Each of these solvers increases or reduces the step size using its local error control to achieve the tolerances that you specify. Computing the step size at each time step adds to the computational overhead. However, it can reduce the total number of steps, and the simulation time required to maintain a specified level of accuracy.

You can further categorize the variable-step continuous solvers as: one-step or multistep, single-order or variable-order, and explicit or implicit. (See Compare Solvers for more information.)

The variable-step explicit solvers are designed for nonstiff problems. Simulink provides four such solvers:

`ode45`

`ode23`

`ode113`

`odeN`

ODE Solver | One-Step Method | Multistep Method | Order of Accuracy | Method |
---|---|---|---|---|

`ode45` | X | Medium | Runge-Kutta, Dormand-Prince (4,5) pair | |

`ode23` | X | Low | Runge-Kutta (2,3) pair of Bogacki & Shampine | |

`ode113` | X | Variable, Low to High | PECE Implementation of Adams-Bashforth-Moulton | |

`odeN` | X | Refer to Order of
Accuracy in Fixed-Step Continuous Explicit Solvers | Refer to Integration
Technique in Fixed-Step Continuous Explicit Solvers |

A good heuristic for choosing the `odeN`

solver when
simulation speed is important:

The model contains lots of zero-crossings and/or solver resets

The

**Solver Profiler**does not detect any failed steps when profiling the model

If your problem is stiff, try using one of the variable-step implicit solvers:

`ode15s`

`ode23s`

`ode23t`

`ode23tb`

ODE Solver | One-Step Method | Multistep Method | Order of Accuracy | Solver Reset Method | Max. Order | Method |
---|---|---|---|---|---|---|

`ode15s` | X | Variable, Low to Medium | X | X | Numerical Differentiation Formulas (NDFs) | |

`ode23s` | X | Low | Second-order, modified Rosenbrock formula | |||

`ode23t` | X | Low | X | Trapezoidal rule using a “free” interpolant | ||

`ode23tb` | X | Low | X | TR-BDF2 |

**Solver Reset Method. **For three of the solvers for stiff problems — `ode15s`

,
`ode23t`

, and `ode23tb`

— a drop-down menu
for the **Solver reset method** appears on the
**Solver details** section of the Configuration pane.
This parameter controls how the solver treats a reset caused, for example, by a
zero-crossing detection. The options allowed are `Fast`

and `Robust`

. `Fast`

specifies
that the solver does not recompute the Jacobian for a solver reset, whereas
`Robust`

specifies that the solver does.
Consequently, the `Fast`

setting is computationally faster but
it may use a small step size in certain cases. To test for such cases, run the
simulation with each setting and compare the results. If there is no difference
in the results, you can safely use the `Fast`

setting
and save time. If the results differ significantly, try reducing the step size
for the fast simulation.

**Maximum Order. **For the `ode15s`

solver, you can choose the maximum order of
the numerical differentiation formulas (NDFs) that the solver applies. Since the
`ode15s`

uses first- through fifth-order formulas, the
`Maximum order`

parameter allows you to choose 1 through 5.
For a stiff problem, you may want to start with order 2.

**Tips for Choosing a Variable-Step Implicit Solver. **The following table provides tips relating to the application of variable-step
implicit solvers. For an example comparing the behavior of these solvers, see
sldemo_solvers.

For a *stiff* problem, solutions can
change on a time scale that is very small as compared to the interval of
integration, while the solution of interest changes on a much longer
time scale. Methods that are not designed for stiff problems are
ineffective on intervals where the solution changes slowly because these
methods use time steps small enough to resolve the fastest possible
change. For more information, see Shampine, L. F., *Numerical Solution of Ordinary
Differential Equations*, Chapman & Hall, 1994.

The variable-step discrete and continuous solvers use zero-crossing detection (see Zero-Crossing Detection) to handle continuous signals.

**Local Error. **The variable-step solvers use standard control techniques to monitor the local
error at each time step. During each time step, the solvers compute the state
values at the end of the step and determine the *local
error*—the estimated error of these state values. They then compare
the local error to the *acceptable error*, which is a
function of both the relative tolerance (*rtol*) and the
absolute tolerance (*atol*). If the local error is greater
than the acceptable error for *any one* state, the solver
reduces the step size and tries again.

The

*Relative tolerance*measures the error relative to the size of each state. The relative tolerance represents a percentage of the state value. The default, 1e-3, means that the computed state is accurate to within 0.1%.*Absolute tolerance*is a threshold error value. This tolerance represents the acceptable error as the value of the measured state approaches zero.The solvers require the error for the

`i`

th state,*e*, to satisfy:_{i}$${e}_{i}\le \mathrm{max}(rtol\times \left|{x}_{i}\right|,ato{l}_{i}).$$

The following figure shows a plot of a state and the regions in which the relative tolerance and the absolute tolerance determine the acceptable error.

**Absolute Tolerances. **Your model has a global absolute tolerance that you can set on the Solver pane
of the Configuration Parameters dialog box. This tolerance applies to all states
in the model. You can specify `auto`

or a real scalar. If you
specify `auto`

(the default), Simulink initially sets the absolute tolerance for each state based on the
relative tolerance. If the relative tolerance is larger 1e-3,
`abstol`

is initialized at 1e-6. However, for
`reltol`

smaller than 1e-3, `abstol`

for
the state is initialized at `reltol * 1e-3`

. As the simulation
progresses, the absolute tolerance for each state resets to the maximum value
that the state has assumed so far, times the relative tolerance for that state.
Thus, if a state changes from 0 to 1 and `reltol`

is 1e-3,
`abstol`

initializes at 1e-6 and by the end of the
simulation reaches 1e-3 also. If a state goes from 0 to 1000, then
`abstol`

changes to 1.

Now, if the state changes from 0 to 1 and `reltol`

is set at
1e-4, then `abstol`

initializes at 1e-7 and by the end of the
simulation reaches a value of 1e-4.

If the computed initial value for the absolute tolerance is not suitable, you
can determine an appropriate value yourself. You might have to run a simulation
more than once to determine an appropriate value for the absolute tolerance. You
can also specify if the absolute tolerance should adapt similar to its
`auto`

setting by enabling or disabling the
`AutoScaleAbsTol`

parameter. For more information, see
Auto scale absolute tolerance.

Several blocks allow you to specify absolute tolerance values for solving the model states that they compute or that determine their output:

The absolute tolerance values that you specify for these blocks override the global settings in the Configuration Parameters dialog box. You might want to override the global setting if, for example, the global setting does not provide sufficient error control for all of your model states because they vary widely in magnitude. You can set the block absolute tolerance to:

`auto`

–

`1`

(same as`auto`

)`positive scalar`

`real vector`

(having a dimension equal to the number of corresponding continuous states in the block)

**Tips. **If you do choose to set the absolute tolerance, keep in mind that too low of a
value causes the solver to take too many steps in the vicinity of near-zero
state values. As a result, the simulation is slower.

On the other hand, if you set the absolute tolerance too high, your results can be inaccurate as one or more continuous states in your model approach zero.

Once the simulation is complete, you can verify the accuracy of your results by reducing the absolute tolerance and running the simulation again. If the results of these two simulations are satisfactorily close, then you can feel confident about their accuracy.

For implicit solvers, Simulink must compute the *solver Jacobian*, which is a
submatrix of the Jacobian matrix associated with the continuous representation of a
Simulink model. In general, this continuous representation is of the
form:

$$\begin{array}{l}\dot{x}=f(x,t,u)\\ y=g(x,t,u).\end{array}$$

The Jacobian, *J*, formed from this system of equations
is:

$$J=\left(\begin{array}{cc}\frac{\partial f}{\partial x}& \frac{\partial f}{\partial u}\\ \frac{\partial g}{\partial x}& \frac{\partial g}{\partial u}\end{array}\right)=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right).$$

In turn, the solver Jacobian is the submatrix, $${J}_{x}$$.

$${J}_{x}=A=\frac{\partial f}{\partial x}.$$

**Sparsity of Jacobian. **For many physical systems, the solver Jacobian
*J _{x}* is

Consider the following system of equations:

$$\begin{array}{l}{\dot{x}}_{1}={f}_{1}({x}_{1},{x}_{3})\\ {\dot{x}}_{2}={f}_{2}({x}_{2})\\ {\dot{x}}_{3}={f}_{3}({x}_{2}).\end{array}$$

From this system, you can derive a sparsity pattern that reflects the structure of the equations. The pattern, a Boolean matrix, has a 1 for each$${x}_{i}$$ that appears explicitly on the right-hand side of an equation. Therefore, you attain:

$${J}_{x,pattern}=\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 0& 1& 0\end{array}\right)$$

As discussed in Full and Sparse Perturbation Methods and Full and Sparse Analytical Methods respectively, the Sparse Perturbation Method and the Sparse Analytical Method may be able to take advantage of this sparsity pattern to reduce the number of computations necessary and improve performance.

When you choose an implicit solver from the **Solver** pane of
the configuration parameters dialog box, a parameter called ```
Solver
Jacobian method
```

and a drop-down menu appear. This menu has five
options for computing the solver Jacobian:

**auto****Sparse perturbation****Full perturbation****Sparse analytical****Full analytical**

If you set **Automatic solver parameter selection** to
`error`

in the Solver Diagnostics pane, and you
choose a different solver method than Simulink, you may receive an error.

**Limitations. **The solver Jacobian methods have the following limitations associated with
them.

If you select an analytical Jacobian method, but one or more blocks in the model do not have an analytical Jacobian, then Simulink applies a perturbation method.

If you select sparse perturbation and your model contains data store blocks, Simulink applies the full perturbation method.

The default setting for the `Solver Jacobian method`

is
`auto`

. Selecting this choice causes Simulink to perform a heuristic to determine which of the remaining four
methods best suits your model. This algorithm is depicted in the following flow
chart.

Because sparse methods are beneficial for models having a large number of states,
if 50 or more states exist in your model, the heuristic chooses a sparse method. The
logic also leads to a sparse method if you specify `ode23s`

because, unlike other implicit solvers, `ode23s`

generates a new
Jacobian at every time step. A sparse analytical or a sparse perturbation method is,
therefore, highly advantageous. The heuristic also ensures that the analytical
methods are used only if every block in your model can generate an analytical
Jacobian.

The full perturbation method was the standard numerical method that Simulink used to solve a system. For this method, Simulink solves the full set of perturbation equations and uses LAPACK for linear algebraic operations. This method is costly from a computational standpoint, but it remains the recommended method for establishing baseline results.

The sparse perturbation method attempts to improve the run-time performance by taking mathematical advantage of the sparse Jacobian pattern. Returning to the sample system of three equations and three states,

$$\begin{array}{l}{\dot{x}}_{1}={f}_{1}({x}_{1},{x}_{3})\\ {\dot{x}}_{2}={f}_{2}({x}_{2})\\ {\dot{x}}_{3}={f}_{3}({x}_{2}).\end{array}$$

The solver Jacobian is:

$$\begin{array}{c}{J}_{x}=\left(\begin{array}{ccc}\frac{\partial {f}_{1}}{\partial {x}_{1}}& \frac{\partial {f}_{1}}{\partial {x}_{2}}& \frac{\partial {f}_{1}}{\partial {x}_{3}}\\ \frac{\partial {f}_{2}}{\partial {x}_{1}}& \frac{\partial {f}_{2}}{\partial {x}_{2}}& \frac{\partial {f}_{2}}{\partial {x}_{3}}\\ \frac{\partial {f}_{3}}{\partial {x}_{1}}& \frac{\partial {f}_{3}}{\partial {x}_{2}}& \frac{\partial {f}_{3}}{\partial {x}_{3}}\end{array}\right)\\ =\left(\begin{array}{ccc}\frac{{f}_{1}({x}_{1}+\Delta {x}_{1},{x}_{2},{x}_{3})-{f}_{1}}{\Delta {x}_{1}}& \frac{{f}_{1}({x}_{1},{x}_{2}+\Delta {x}_{2},{x}_{3})-{f}_{1}}{\Delta {x}_{2}}& \frac{{f}_{1}({x}_{1},{x}_{2},{x}_{3}+\Delta {x}_{3})-{f}_{1}}{\Delta {x}_{3}}\\ \frac{{f}_{2}({x}_{1}+\Delta {x}_{1},{x}_{2},{x}_{3})-{f}_{2}}{\Delta {x}_{1}}& \frac{{f}_{2}({x}_{1},{x}_{2}+\Delta {x}_{2},{x}_{3})-{f}_{2}}{\Delta {x}_{2}}& \frac{{f}_{2}({x}_{1},{x}_{2},{x}_{3}+\Delta {x}_{3})-{f}_{2}}{\Delta {x}_{3}}\\ \frac{{f}_{3}({x}_{1}+\Delta {x}_{1},{x}_{2},{x}_{3})-{f}_{3}}{\Delta {x}_{1}}& \frac{{f}_{3}({x}_{1},{x}_{2}+\Delta {x}_{2},{x}_{3})-{f}_{3}}{\Delta {x}_{2}}& \frac{{f}_{3}({x}_{1},{x}_{2},{x}_{3}+\Delta {x}_{3})-{f}_{3}}{\Delta {x}_{3}}\end{array}\right)\end{array}$$ |

It is, therefore, necessary to perturb each of the three states three times and to
evaluate the derivative function three times. For a system with *n*
states, this method perturbs the states *n* times.

By applying the sparsity pattern and perturbing states
*x*_{1} and *x
*_{2} together, this matrix reduces to:

$${J}_{x}=\left(\begin{array}{ccc}\frac{{f}_{1}({x}_{1}+\Delta {x}_{1},{x}_{2}+\Delta {x}_{2},{x}_{3})-{f}_{1}}{\Delta {x}_{1}}& 0& \frac{{f}_{1}({x}_{1},{x}_{2},{x}_{3}+\Delta {x}_{3})-{f}_{1}}{\Delta {x}_{3}}\\ 0& \frac{{f}_{2}({x}_{1}+\Delta {x}_{1},{x}_{2}+\Delta {x}_{2},{x}_{3})-{f}_{2}}{\Delta {x}_{2}}& 0\\ 0& \frac{{f}_{3}({x}_{1}+\Delta {x}_{1},{x}_{2}+\Delta {x}_{2},{x}_{3})-{f}_{3}}{\Delta {x}_{2}}& 0\end{array}\right)$$ |

The solver can now solve columns 1 and 2 in one sweep. While the sparse perturbation method saves significant computation, it also adds overhead to compilation. It might even slow down the simulation if the system does not have a large number of continuous states. A tipping point exists for which you obtain increased performance by applying this method. In general, systems having a large number of continuous states are usually sparse and benefit from the sparse method.

The sparse perturbation method, like the sparse analytical method, uses UMFPACK to perform linear algebraic operations. Also, the sparse perturbation method supports both RSim and Rapid Accelerator mode.

The full and sparse analytical methods attempt to improve performance by calculating the Jacobian using analytical equations rather than the perturbation equations. The sparse analytical method, also uses the sparsity information to accelerate the linear algebraic operations required to solve the ordinary differential equations.

For details on how to access and interpret the sparsity pattern in MATLAB^{®}, see sldemo_metro.

While the sparse perturbation method supports RSim, the sparse analytical method does not. Consequently, regardless of which sparse method you select, any generated code uses the sparse perturbation method. This limitation applies to Rapid Accelerator mode as well.