choosing a solver

8/8/2019 Choosing a Solver

http://slidepdf.com/reader/full/choosing-a-solver 1/15

1 / 15

Choosing a Solver

In this document…

What Is a Solver?

Choosing a Solver Type

Choosing a Fixed-Step Solver

Choosing a Variable-Step Solver

Choosing a Jacobian Method for an Implicit Solver

What Is a Solver?A solver is a component of the Simulink software. The Simulink product provides an extensive library of

solvers, each of which determines the time of the next simulation step and applies a numerical method

to solve the set of ordinary differential equations that represent the model. In the process of solving thisinitial value problem, the solver also satisfies the accuracy requirements that you specify. To help you

choose the solver best suited for your application, Choosing a Solver Type provides background on the

different types of solvers while Choosing a Fixed-Step Solver and Choosing a Variable-Step

Solver provide guidance on choosing a specific fixed-step or variable-step solver, respectively.

The following table summarizes the types of solvers in the Simulink library and provides links to specific

categories. All of these solvers can work with the algebraic loop solver.

Discrete Continuous Variable-Order

Fixed-Step Explicit Not Applicable Explicit Fixed-Step Continuous

Solvers

Not Applicable

Implicit Not Applicable Implicit Fixed-Step Continuous

Solvers

Not Applicable

Variable-

Step

Explicit Choosing a Variable-Step

Solver

Explicit Continuous Variable-

Step Solvers

Variable-Order

Solvers

Implicit Implicit Continuous Variable-

Step Solvers

Variable-Order

Solvers

Note

1. The fixed-step discrete solvers do not solve for discrete states; each block calculates its discretestates independent of the solver. For more information, see Discrete versus Continuous Solvers

2. Every solver in the Simulink library can perform on models that contain algebraic loops.

For information on tailoring the selected solver to your model, see Improving Simulation Accuracy.

Back to Top

http://www.mathworks.com/help/toolbox/simulink/ug/f11-8983.html



http://www.mathworks.com/help/toolbox/simulink/ug/f11-69449.html#top_of_page








2 / 15

Choosing a Solver TypeThe Simulink library of solvers is divided into two major types in the Solver Pane: fixed-step and variable-

step. You can further divide the solvers within each of these categories as: discrete or continuous,

explicit or implicit, one-step or multistep, and single-order or variable-order.

Fixed-Step versus Variable-Step Solvers

Both fixed-step and variable-step solvers compute the next simulation time as the sum of the current

simulation time and a quantity known as the step size. With a fixed-step solver, the step size remains

constant throughout the simulation. In contrast, 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 Type control on the

Simulink Solver configuration pane allows you to select either of these two types of solvers.

The choice between the two 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 because 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 savings because, for a

given level of accuracy, the solver can dynamically adjust the step size as necessary and thus reduce the

number of steps. Whereas the fixed-step solver must use a single step size throughout the simulation

based upon the accuracy requirements. To satisfy these requirements throughout the simulation, the

fixed-step solver might require a very small step.

The following model shows how a variable-step solver can shorten simulation time for a multi-rate

discrete model.

This 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 needs 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.

http://www.mathworks.com/help/toolbox/simulink/gui/bq7cmsp-1.html






3 / 15

If you wish to achieve evenly spaced steps, you must use the format 0.4*[0.0: 100.0] rather than

[0.0:0.4:40].

Discrete versus Continuous Solvers

When you set the Type control of the Solver configuration pane to fixed-step or to variable-step, theadjacent Solver control allows you to choose a specific solver. Both sets of solvers comprise 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. Therefore, when choosing a solver,

you must first determine whether you need 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 instead your model has continuous states, you must choose a

continuous solver from the remaining solver choices based on the dynamics of your model. Otherwise,

an error occurs.

Explicit versus Implicit Solvers

While you can apply either an implicit or explicit continuous solver, the implicit solvers are designed

specifically for solving stiff problems whereas explicit solvers are used to solve non-stiff problems. A

generally accepted definition of a stiff system is a system that has extremely different time scales.

Compared to the explicit solvers, the implicit solvers provide greater stability for oscillatory behavior,

but they 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 Choosing a Jacobian Method for an Implicit Solver for more

information.

One-Step versus Multistep Solvers

The Simulink solver library provides both one-step and multistep solvers. The one-step solvers

estimate y (t n) using the solution at the immediately preceding time point, y (t n-1), and the values of the

derivative at a number of points between t n and t n-1. These points are minor steps.

The 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.

Variable-Order Solvers

Two variable-order solvers, ode15s and ode113, are part of the solver library. These solvers 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. For ode15s, you can limit the highest order applied via the Maximum

Order parameter. For more information, see Maximum Order.

Back to Top

http://www.mathworks.com/help/toolbox/simulink/ug/f11-69449.html#bsfglhx











4 / 15

Choosing a Fixed-Step Solver

About the Fixed-Step Discrete Solver

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 sizeof the steps taken by the simulation: the smaller the step size, the more accurate the results are but the

longer the simulation takes. You can allow the Simulink software to choose the size of the step (the

default) 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.

Note 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.

About Fixed-Step Continuous Solvers

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. The fixed-step

continuous solvers can, therefore, handle models that contain both continuous and discrete states.

Note In theory, a fixed-step continuous solver can handle models that contain no continuous states.

However, that would impose an unnecessary computational burden on the simulation. Consequently,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.

Two types of fixed-step continuous solvers that Simulink provides are: explicit and implicit. (See Explicit

versus Implicit Solvers for more information). 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.

Explicit Fixed-Step Continuous 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. Expressed mathematically for a fixed-step explicit

solver:

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.



5 / 15

Simulink provides a set of explicit fixed-step continuous solvers. The solvers differ in the specific

numerical integration technique that they use to compute the state derivatives of the model. The

following table lists each solver and the integration technique it uses.

Solver Integration Technique Order of Accuracy

ode1Euler's Method

First

ode2Heun's Method

Second

ode3Bogacki-Shampine Formula

Third

ode4Fourth-Order Runge-Kutta (RK4) Formula

Fourth

ode5 Dormand-Prince (RK5) Formula Fifth

ode8Dormand-Prince RK8(7) Formula

Eighth

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).

None of these solvers has an error control mechanism. Therefore, the accuracy and the duration of a

simulation depends 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

more computationally complex the solver is, the more accurate are the simulation results.

If you specify a fixed-step solver type for a model, then by default, Simulink selects the ode3 solver,

which 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 might need to choose another solver, a different fixed

step size, or both to achieve acceptable accuracy and an acceptable simulation time.

Implicit Fixed-Step Continuous Solvers

An implicit fixed-step 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:



6 / 15

Simulink provides one implicit fixed-step 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.

Process for Choosing a Fixed-Step Continuous Solver

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 generally is not possible, or at least

not practical, to decide a priori which combination of solver and step size will yield acceptable results for

the continuous states in the shortest time. Determining the best solver for a particular model generally

requires experimentation.

Following is the most efficient way to choose the best fixed-step solver for your model experimentally.

1. Choose error tolerances. For more information, see Specifying Variable-Step Solver Error Tolerances.

2. Use one of the variable-step solvers to simulate your model to the level of accuracy that you desire.

Start with ode45. If your model runs slowly, your problem might 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.

3. 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 can draw this conclusion because ode1 is the simplest of the fixed-step solvers and hence yields

the shortest simulation time for the current step size.

4. If ode1 does not give satisfactory results, repeat the preceding steps with each of the other fixed-

step solvers until you find the 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:a. Try ode3.

b. 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.

c. 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.

d. 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.

Back to Top

Choosing a Variable-Step SolverWhen 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

http://www.mathworks.com/help/toolbox/simulink/gui/bq7cmsp-1.html#bq994cq-1











7 / 15

only discrete states, select the discrete solver. In fact, 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.

About Variable-Step Continuous Solvers

The variable-step solvers in the Simulink product 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 but 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 Choosing a Solver Type for more information.)

Explicit Continuous Variable-Step Solvers

The explicit variable-step solvers are designed for non-stiff problems. Simulink provides three such

solvers: ode45, ode23, and ode113.

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 toHigh

PECE Implementation of Adams-Bashforth-Moutlon

ODE

Solver

Tips on When to Use

ode45 In general, the ode45 solver is the best to apply as a first try for most problems. For this

reason, ode45 is the default solver for models with continuous states. This Runge-Kutta (4,5)

solver is a fifth-order method that performs a fourth-order estimate of the error. This solver

also uses a fourth-order "free" interpolant, which allows for event location and smoother

plots.

The ode45 is more accurate and faster than ode23. If the ode45 is slow computationally, yourproblem may be stiff and thus in need of an implicit solver.

ode23 The ode23 can be more efficient than the ode45 solver at crude error tolerances and in the

presence of mild stiffness. This solver provides accurate solutions for "free" by applying a

cubic Hermite interpolation to the values and slopes computed at the ends of a step.

ode113 For problems with stringent error tolerances or for computationally intensive problems, the

Adams-Bashforth-Moulton PECE solver can be more efficient than ode45.



8 / 15

Implicit Continuous Variable-Step Solvers

If your problem is stiff, try using one of the implicit variable-step solvers: ode15s, ode23s, ode23t,

or ode23tb.

ODESolver

One-StepMethod

MultistepMethod

Order of Accuracy

Solver ResetMethod

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 stiff solvers—ode15s, ode23t, and ode23tb—a drop-down menu for the Solver reset

method appears on the Solver 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. The former

setting specifies that the solver does not recompute the Jacobian for a solver reset, whereas the latter

setting specifies that the solver does. Consequently, the Fast setting is faster computationally but might

use a small step size for certain cases. To test for such cases, run the simulation with each setting and

compare the results. If there is no difference, 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 ademonstration comparing the behavior of these solvers, see sldemo_solvers.

ODE

Solver

Tips on When to Use

ode15s ode15s is a variable-order solver based on the numerical differentiation formulas (NDFs).

NDFs are related to, but are more efficient than the backward differentiation formulas (BDFs),

http://www.mathworks.com/help/toolbox/simulink/ug/rmvd_matlablink__99.html






9 / 15

ODE

Solver

Tips on When to Use

which are also known as Gear's method. The ode15s solver numerically generates the

Jacobian matrices. If you suspect that a problem is stiff, or if ode45 failed or was highly

inefficient, tryode15s. As a rule, start by limiting the maximum order of the NDFs to 2.ode23s ode23s is based on a modified Rosenbrock formula of order 2. Because it is a one-step solver,

it can be more efficient thanode15s at crude tolerances. Like ode15s, ode23s numerically

generates the Jacobian matrix for you. However, it can solve certain kinds of stiff problems

for which ode15s is not effective.

ode23t The ode23t solver is an implementation of the trapezoidal rule using a "free" interpolant. Use

this solver if your model is only moderately stiff and you need a solution without numerical

damping. (Energy is not dissipated when you model oscillatory motion.)

ode23tb ode23tb is an implementation of TR-BDF2, an implicit Runge-Kutta formula with two stages.

The first stage is a trapezoidal rule step while the second stage uses a backward

differentiation formula of order 2. By construction, the method uses the same iterationmatrix in evaluating both stages. Like ode23s, this solver can be more efficient than ode15s at

crude tolerances.

Note 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.

Support for Zero-Crossing Detection

Both the variable-step discrete and continuous solvers use zero-crossing detection (see Zero-Crossing

Detection) to handle continuous signals.

Specifying Variable-Step Solver Error Tolerances

The solvers use standard local error control techniques to monitor the 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 thelocal error is greater than the acceptable error for any 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.

http://www.mathworks.com/help/toolbox/simulink/ug/f7-8243.html#f7-9506








10 / 15

The solvers require the error for the ith

state, ei , to satisfy:

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.

If you specify auto (the default), Simulink initially sets the absolute tolerance for each state to 1e-6. As

the simulation progresses, the absolute tolerance for each state is reset to the maximum value that the

state has assumed thus far, times the relative tolerance for that state. Thus, if a state changes from 0 to

1 and reltol is 1e-3, then by the end of the simulation, abstol is set to 1e-3 also. If a state goes from 0 to1000, then abstol is set to 1.

If the computed setting is not suitable, you can determine an appropriate setting yourself. You might

have to run a simulation more than once to determine an appropriate value for the absolute tolerance.

The Integrator, Second-Order Integrator, Second-Order Integrator Limited, Variable Time Delay, Variable

Transport Delay, Transfer Fcn, State-Space, and Zero-Pole 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 in this way. For example, if the

global setting does not provide sufficient error control for all of your model states because they vary

widely in magnitude, then you might want to set the value yourself.Back to Top

Choosing a Jacobian Method for an Implicit Solver

About the Solver Jacobian

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:

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

http://www.mathworks.com/help/toolbox/simulink/slref/integrator.html



http://www.mathworks.com/help/toolbox/simulink/slref/secondorderintegratorlimited.html



http://www.mathworks.com/help/toolbox/simulink/slref/variabletimedelay.html




http://www.mathworks.com/help/toolbox/simulink/slref/transferfcn.html



http://www.mathworks.com/help/toolbox/simulink/slref/statespace.html



http://www.mathworks.com/help/toolbox/simulink/slref/zeropole.html

















11 / 15

In turn, the solver Jacobian is the sub-matrix, .

Sparsity of Jacobian

For many physical systems, the solver Jacobian J x is sparse, meaning that many of the elements of J x are

zero.

Consider the following system of equations:

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 that appears explicitly on the right-hand side of an

equation. You thereby attain:

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 thereby

improve performance.

Solver Jacobian Methods

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, and Full

analytical.



12 / 15

Note If you set Automatic solver parameter selection to warning or error in the Solver Diagnostics

pane, and you choose a different solver method than Simulink, you might receive a warning or 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.

Heuristic 'auto' 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 flowchart.



13 / 15



14 / 15

Because the sparse methods are beneficial for models having a large number of states, the heuristic

chooses a sparse method if more than 50 states exist in your model. 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.

Full and Sparse Perturbation Methods

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,

The solver Jacobian is:

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:



15 / 15

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.

Full and Sparse Analytical Methods

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.

Sparsity Pattern

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

Support for Code Generation

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.





choosing a solver

Documents