Best Practices

George E. Hrabovsky

MAST

Introduction

In this lesson we will examine useful tips and ideas to make life, and code, better in MMA/WL. I will try to organize these by what I consider the most important ideas first. Unfortunately this is a personal choice on my part and nothing can be done about that.

The Most Important Things

To Do Right Away Before Starting Anything Else

Go to the Edit Menu. Then go to Preferences.

Uncheck the Show suggestions bar after last output. This is responsible for throwing lots of errors.

Activate the button Automatically insert matching delimiters while typing. In version 12,3 the default is unchecked, so you should activate it. In version 13 this is the default state.

No Autosave

Save your work often. There is no autosave.

Use the ?

If you forget how to use a command type a new cell with ?command to get a description of it. If you type ??command you will also get a list of function attributes.

Referring to Previous Output

Never, never, NEVER, use % to refer to previous output. Assign a symbol to any output so you can refer to it later.

Aborting a Command

You can either go to the Evaluation Menu and select Abort Evaluation, or hold down the [ALT] key and type . .

If you can’t abort the command, then you will have to go to the Evaluation Menu, then Quit Kernel, then choose Local. This will stop all evaluations and reset the system. It will not save your work. If MMA/WL becomes unstable and you can’t do anything with it, then use the system in your operating system to kill the process. This will destroy all unsaved work. Save your work often.

Important Things

Avoid Iterative Loops

Avoid iterative programming, use functional recursions instead. For example.

Look Up the Syntax

Get to know the syntax of the WL. In the documentation system go to the Guide Wolfram Language Syntax. From there there are two things to get to know: the first is the associated Guide Wolfram Language Syntax Characters. The second is a link to the tutorial Textual Input and Output. You can also type into the Search Bar Operator Input Forms, this links to an included Tech note of the same name that lists all of these things.

Using Transpose on Datasets

Say that you have two data sets that you need to merge for a function, say a ListPlot. Use Transpose to link them element by element. Here is an example.

Make sure that the two original data files are the same length.

Weak Typing

MMA/WL have what is called “weak typing” that is it does not enforce hard rules for establishing symbol typing. This is an advantage most of the time, as it lends flexibility. It can cause problems if you change code, as you might inadvertently change the type of a variable in your mind without remembering to treat the symbol that way in the code.

A Note About Associations

Associations are memory hungry.

Navigating Code

Use [CTRL]. to select groups of symbols.

Use [CTRL][SHIFT]→ or [CTRL][SHIFT]← to extend your selection by one token in either direction.

Use [CTRL]↑ of [CTRL]↓ to jump up or down three line while the cursor stays in its current location.

Autocompletion

Take advantage of autocompletion where possible, you can enter this manually by using [CTRL]k

Contexts

If you ever want a list of variables and functions you have defined in your session use the names under the notebook context. You can find the name for the context by using the Context command.

or we can use this.

For a more in-depth look, we can write this.

From this we see that the notebook has introduced a context and that the system has also introduced a context.

To find the names under the notebook context write this.

If you want to clear all of the names used in the context write this.

If you want to keep the names, but clear all definitions and attributes use this command.

The Remove command should resolve most variable conflicts.

Contexts are the names of work environments produced by the MMA/WL system. Each package you load will introduce a new context. You can create new contexts.

Turning Off System Complaints

If you are doing something you know is correct, but it will generate warnings, end the code with //Quiet.

Things

Indexed Variables

If you must use indexed variables it is best to use an association as this allows you to bind many values to a variable name by position.

To display indices you can use DownValues to assign the value as a rule to a function head.

Function Naming Convention

The names of all Mathematica commands begin with a capital letter. I recommend that your user-defined functions do not, that makes it easier to tell the difference.

Never use an underscore in a name, it is reserved for patterns.

Avoid using single capital letters as names.

If You Must Perform a Procedural Iteration

Using Which is much more efficient than nested If statements.

Shadowing

If you load a package and a name is duplicated it can cause a shadowing error. This can happen is you use [Get[context`], or its short form <<Context`. You can avoid this by using Needs[Context`].

Less Important Things

Wrapping Forms Instead of Including Them

Don’t use TableForm or MatrixForm in table or matrix definitions. Wrap the definition in these commands instead. That way  the object is not defined in the formatted output form.

Note that we call the // symbol piping.

Finding Options

There are many functions that have hidden options or methods. This code is modified from the web page https://www.12000.org/my_notes/faq/mma_notes/MMA.htm.

#	Option	Options to this option
1	NDSolve`Adams	MaxDifferenceOrder→12 VariableStepCoefficients→Automatic
2	NDSolve`AnyEdge	{}
3	NDSolve`Auxiliary	Method→Automatic AuxiliaryTerms→Automatic DampingFactor→0
4	NDSolve`BaderSequenceFunction	{}
5	NDSolve`BDF	ImplicitSolver→NDSolve`Newton MaxDifferenceOrder→5 VariableStepCoefficients→Automatic
6	NDSolve`BootstrapDenseOutput	{}
7	NDSolve`BulirschSequenceFunction	{}
8	NDSolve`Chasing	Method→Automatic ExtraPrecision→0 ChasingType→LinearChasing
9	NDSolve`ChebyshevCoefficients	{}
10	NDSolve`CheckMethodProperty	{}
11	NDSolve`Composition	Coefficients→Automatic DifferenceOrder→Automatic Method→None
12	NDSolve`ComputeDerivatives	CollocationDirection→Centered MaxIterations→100 SecondDerivative→False Verbose→False ReturnResidualValue→False
13	NDSolve`ConsistentCoefficientsQ	{}
14	NDSolve`Continuity	{}
15	NDSolve`CountableCrossingEvent	{}
16	NDSolve`CountableCrossingValue	{}
17	NDSolve`CreateMethodData	{}
18	NDSolve`CreateWorkspace	{}
19	NDSolve`CreateWorkspaceList	{}
20	NDSolve`CrossDiscontinuity	{}
21	NDSolve`CubicHermite	{}
22	NDSolve`DAEInitialize	CollocationDirection→Automatic BasisType→Chebyshev CollocationPoints→{Automatic,ExtraCollocationPoints→Automatic} Verbose→False DefaultStartingValue→Automatic MaxIterations→100 CollocationRange→Automatic
23	NDSolve`DAEInitializeForProjection	SolvingMethod→LinearSolve InitialConditionWeight→10 MaxIterations→Automatic DefaultStartingValue→Automatic Verbose→False Method→Automatic
24	NDSolve`DAEInitializeReduced	Method→Automatic DefaultStartingValue→Automatic MaxIterations→Automatic
25	NDSolve`DAESensitivityInitialize	{}
26	NDSolve`DCayleyInverse	{}
27	NDSolve`DDEFunction	{}
28	NDSolve`DelayDESteps	Method→Automatic DerivativeEvaluation→Interpolation
29	NDSolve`DeltaDiscontinuity	{}
30	NDSolve`Dethread	{}
31	NDSolve`DExpInverse	{}
32	NDSolve`DExpInverseCoefficients	{}
33	NDSolve`DiscontinuityEvaluate	{}
34	NDSolve`DiscontinuityExclusion	{}
35	NDSolve`DiscontinuityReplace	{}
36	NDSolve`DiscontinuityScan	Domain→Automatic MaxContinuity→∞ Replacements→False
37	NDSolve`DiscontinuitySurface	{}
38	NDSolve`DiscontinuityValues	{}
39	NDSolve`DoubleStep	LocalExtrapolation→True Method→None StepSizeRatioBounds→Automatic StepSizeSafetyFactors→Automatic StiffnessTest→Automatic
40	NDSolve`EmbeddedExplicitRungeKuttaCoefficients	{}
41	NDSolve`EstimateStartingStep	{}
42	NDSolve`EvaluateJacobianWithSolutionData	{}
43	NDSolve`EvaluateWithSolutionData	{}
44	NDSolve`EventData	{}
45	NDSolve`EventLocator	Direction→All Event→None EventAction:→StopIntegration EventCondition→True EventLocationMethod→Automatic IncludeDerivatives→Automatic CheckDerivatives→False Method→Automatic
46	NDSolve`EventOrderBackward	{}
47	NDSolve`EventOrderForward	{}
48	NDSolve`ExplicitEuler	{}
49	NDSolve`ExplicitMidpoint
50	NDSolve`ExplicitModifiedMidpoint
51	NDSolve`ExplicitRungeKutta	Coefficients→EmbeddedExplicitRungeKuttaCoefficients DifferenceOrder→Automatic EmbeddedDifferenceOrder→Automatic StepSizeControlParameters→Automatic StepSizeSafetyFactors→Automatic StiffnessTest→Automatic
52	NDSolve`Extrapolation	ExtrapolationSequence→Automatic MaxDifferenceOrder→Automatic Method→ExplicitModifiedMidpoint MinDifferenceOrder→Automatic OrderSafetyFactors→Automatic StartingDifferenceOrder→Automatic StepSizeRatioBounds→Automatic StepSizeSafetyFactors→Automatic StiffnessTest→Automatic
53	NDSolve`FallingEdge	{}
54	NDSolve`FinalizeMethod	{}
55	NDSolve`FiniteDifferenceDerivative	DifferenceOrder→4 PeriodicInterpolation→False
56	NDSolve`FiniteDifferenceDerivativeFunction	{}
57	NDSolve`FiniteDifferenceWeights	{}
58	NDSolve`FiniteElement	AccuracyGoal→Automatic BoundaryTolerance→Automatic ConstraintMethod→Automatic InitializePDECoefficientsOptions→Automatic IntegrationOrder→Automatic InterpolationOrder→Automatic LinearSolveMethod→Automatic MeshOptions→Automatic PDESolveOptions→Automatic PrecisionGoal→Automatic PrecomputeGeometryData→True
59	NDSolve`FixedPoint	{}
60	NDSolve`FixedStep	Method→None StepSize→Automatic
61	NDSolve`GetDefaultDifferenceOrder	{}
62	NDSolve`GetDenseOutput	{}
63	NDSolve`GetMethodOptionValues	{}
64	NDSolve`GMRES	Preconditioner→Automatic OrthogonalizationType→ModifiedGramSchmidt MaxKrylovSubspaceDimension→Automatic MaxKrylovRestarts→Automatic
65	NDSolve`HairerOstermannSequenceFunction	{}
66	NDSolve`HarmonicSequenceFunction	{}
67	NDSolve`IDA	MaxDifferenceOrder→5 ImplicitSolver→Newton
68	NDSolve`ImplicitRungeKutta	Coefficients→ImplicitRungeKuttaGaussCoefficients DifferenceOrder→Automatic ImplicitSolver→Newton StepSizeControlParameters→Automatic StepSizeSafetyFactors→Automatic
69	NDSolve`ImplicitRungeKuttaAdjointCoefficients	{}
70	NDSolve`ImplicitRungeKuttaCoefficients	Solve→Row
71	NDSolve`ImplicitRungeKuttaGaussCoefficients	{}
72	NDSolve`ImplicitRungeKuttaLobattoIIIACoefficients	{}
73	NDSolve`ImplicitRungeKuttaLobattoIIIBCoefficients	{}
74	NDSolve`ImplicitRungeKuttaLobattoIIICCoefficients	{}
75	NDSolve`ImplicitRungeKuttaRadauIACoefficients	{}
76	NDSolve`ImplicitRungeKuttaRadauIIACoefficients	{}
77	NDSolve`IndexReductionVisualizer	IndexGoal→Automatic
78	NDSolve`InitialHistory	{}
79	NDSolve`InitializeMethod	{}
80	NDSolve`InitializeSubmethod	{}
81	NDSolve`InitializeSubmethods	{}
82	NDSolve`IntraStepEvaluationData	{}
83	NDSolve`InvokeMethod	{}
84	NDSolve`iScaledVectorNorm	{}
85	NDSolve`Iterate	{}
86	NDSolve`iVectorNorm	{}
87	NDSolve`iVectorNormWeights	{}
88	NDSolve`iWeightedVectorNorm	{}
89	NDSolve`KrylovIteration	BasisSize→Automatic MaxIterations→Automatic
90	NDSolve`LinearlyImplicitEuler	LinearSolveMethod→Automatic StabilityTest→True
91	NDSolve`LinearlyImplicitMidpoint	LinearSolveMethod→Automatic StabilityTest→True
92	NDSolve`LinearlyImplicitModifiedMidpoint	LinearSolveMethod→Automatic StabilityTest→True
93	NDSolve`LinearStabilityBoundary	{}
94	NDSolve`LocalInterpolation	{}
95	NDSolve`LocallyExact	SimplificationFunction→None
96	NDSolve`LocalSeries	{}
97	NDSolve`LSODA	LinearSolveMethod→Automatic MaxDifferenceOrder→12
98	NDSolve`MakeStateSpaceNumericalFunction	{}
99	NDSolve`MassMatrix	Method→Automatic
100	NDSolve`MethodData	{}
101	NDSolve`MethodOfLines	DifferentiateBoundaryConditions→Automatic DiscretizedMonitorVariables→False ExpandEquationsSymbolically→False SpatialDiscretization→Automatic TemporalVariable→Automatic
102	NDSolve`MethodSymbol	{}
103	NDSolve`MultistepControllerQ	{}
104	NDSolve`NDSolveFunction	{}
105	NDSolve`NDSolveMessage	{}
106	NDSolve`NDSolveParametricFunction	{}
107	NDSolve`NewMethodNorm	{}
108	NDSolve`Newton	LinearSolveMethod→Automatic
109	NDSolve`NonstiffTest	MaxRepetitions→{2,∞}
110	NDSolve`NonstiffTestData	{}
111	NDSolve`NumericalFunctionToEquations	{}
112	NDSolve`ODEInitialize	MaxIterations→100
113	NDSolve`ODESensitivityInitialize	{}
114	NDSolve`OptimalRoundingSequenceFunction	{}
115	NDSolve`OrthogonalChebyshevCoefficients	{}
116	NDSolve`OrthogonalProjection	Dimensions→{} MaxIterations→Automatic Method→StiffnessSwitching
117	NDSolve`ParametricPlugInFunction	{}
118	NDSolve`Parareal	MaxTimeUnit→10 ParallelMethod→None SerialMethod→None
119	NDSolve`PararealInitialize	{}
120	NDSolve`PararealParallelStep	{}
121	NDSolve`PararealParallelSteps	{}
122	NDSolve`PararealStateData	{}
123	NDSolve`PredicateBoundaryFunction	{}
124	NDSolve`ProcessEquations	AccuracyGoal→Automatic Compiled→Automatic DependentVariables→Automatic DiscreteVariables→{} EvaluationMonitor→None InitialSeeding→{} InterpolationOrder→Automatic MaxSteps→Automatic MaxStepSize→Automatic Method→Automatic NormFunction→Automatic PrecisionGoal→Automatic StartingStepSize→Automatic StepMonitor→None WorkingPrecision→MachinePrecision
125	NDSolve`ProcessSolutions	{}
126	NDSolve`Projection	AccuracyGoal→Automatic Invariants→None IterationSafetyFactor→1 LinearSolveMethod→Cholesky MaxIterations→Automatic Method→StiffnessSwitching PrecisionGoal→Automatic
127	NDSolve`RealLine	{}
128	NDSolve`ReduceIndex	IndexGoal→Automatic ConstraintMethod→Automatic Method→Automatic
129	NDSolve`Reinitialize	AccuracyGoal→Automatic EvaluationMonitor→None InitialSeeding→{} InterpolationOrder→Automatic MaxSteps→Automatic MaxStepSize→Automatic NormFunction→Automatic PrecisionGoal→Automatic StartingStepSize→Automatic StepMonitor→None
130	NDSolve`ReinitializeVector	AccuracyGoal→Automatic EvaluationMonitor→None InitialSeeding→{} InterpolationOrder→Automatic MaxSteps→Automatic MaxStepSize→Automatic NormFunction→Automatic PrecisionGoal→Automatic StartingStepSize→Automatic StepMonitor→None
131	NDSolve`RisingEdge	{}
132	NDSolve`RombergSequenceFunction	{}
133	NDSolve`SameDiscontinuities	{}
134	NDSolve`SamePrecisionQ	{}
135	NDSolve`ScaledVectorNorm	{}
136	NDSolve`Self	{}
137	NDSolve`SensitivityMethod	Method→Automatic
138	NDSolve`SetSolutionDataComponent	{}
139	NDSolve`SetState	{}
140	NDSolve`Shooting	DefaultStartingValue→0 ImplicitSolver→Automatic MaxIterations→Automatic StartingInitialConditions→Automatic
141	NDSolve`SimplifySystem	{}
142	NDSolve`SlidingModeFunction	{}
143	NDSolve`SolutionData	{}
144	NDSolve`SolutionDataComponent	{}
145	NDSolve`SolutionDataIndex	{}
146	NDSolve`SolutionDataListQ	{}
147	NDSolve`SolutionDataTemplate	{}
148	NDSolve`SpectralRadius	MaxIterations→50 SampleFrequency→Automatic
149	NDSolve`SpectralRadiusOptionValues	{}
150	NDSolve`Splitting	Coefficients→Automatic DifferenceOrder→Automatic Equations→{} Method→None
151	NDSolve`StabilizedRungeKutta	Coefficients→OrthogonalChebyshevCoefficients DifferenceOrder→Automatic MaxStages→Automatic SpectralRadius→Automatic StepSizeControlParameters→Automatic StepSizeSafetyFactors→Automatic
152	NDSolve`StateData	{}
153	NDSolve`StateDataQ	{}
154	NDSolve`StateSpace	Method→Automatic BlockLowerTriangular→Automatic StepControl→None
155	NDSolve`StateSpaceFunction	{}
156	NDSolve`StateSpaceFunctionQ	{}
157	NDSolve`StepSizeControlData	{}
158	NDSolve`StiffnessSwitching	NonstiffTest→Automatic Method→{Automatic,Automatic}
159	NDSolve`StiffnessTest	MaxRepetitions→{3,5}
160	NDSolve`StiffnessTestData	{}
161	NDSolve`Streamline	Mesh→None Method→Automatic SinkTest→5
162	NDSolve`StructuralIncidenceArray	{}
163	NDSolve`StructuralIndex	IndexGoal→Automatic
164	NDSolve`SubspaceIteration	BasisSize→Automatic MaxIterations→Automatic
165	NDSolve`SwitchingVariable	{}
166	NDSolve`SymmetricCompositionCoefficients	{}
167	NDSolve`SymmetricCompositionSymmetricMethodCoefficients	{}
168	NDSolve`SymplecticPartitionedRungeKutta	Coefficients→SymplecticPartitionedRungeKuttaCoefficients DifferenceOrder→Automatic PositionVariables→{}
169	NDSolve`SymplecticPartitionedRungeKuttaCoefficients	{}
170	NDSolve`TestImplicitSolver	{}
171	NDSolve`ToSymbol	{}
172	NDSolve`ValidNumericalFunctionQ	{}
173	NDSolve`ValidScaledVectorNormQ	{}
174	NDSolve`ValidStateDataQ	{}
175	NDSolve`VariableData	{}
176	NDSolve`VODE	{}
177	NDSolve`$EventSolution	{}
178	{NDSolve`ImplicitRungeKuttaGaussCoefficients,NDSolve`ImplicitRungeKuttaLobattoIIIACoefficients,NDSolve`ImplicitRungeKuttaLobattoIIIBCoefficients,NDSolve`ImplicitRungeKuttaLobattoIIICCoefficients,NDSolve`ImplicitRungeKuttaRadauIACoefficients,NDSolve`ImplicitRungeKuttaRadauIIACoefficients}	{}
179	5	{}