# Documentation

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## Characteristics of Lookup Table Data

### Sizes of Breakpoint Data Sets and Table Data

The following constraints apply to the sizes of breakpoint data sets and table data associated with lookup table blocks:

• The memory limitations of your system constrain the overall size of a lookup table.

• Lookup tables must use consistent dimensions so that the overall size of the table data reflects the size of each breakpoint data set.

To illustrate the second constraint, consider the following vectors of input and output values that create the relationship in the plot.

```Vector of input values: [-3 -2 -1 0 1 2 3] Vector of output values: [-3 -1 0 -1 0 1 3]```

In this example, the input and output data are the same size (1-by-7), making the data consistently dimensioned for a 1-D lookup table.

The following input and output values define the 2-D lookup table that is graphically shown.

```Row index input values: [1 2 3] Column index input values: [1 2 3 4] Table data: [11 12 13 14; 21 22 23 24; 31 32 33 34]```

In this example, the sizes of the vectors representing the row and column indices are 1-by-3 and 1-by-4, respectively. Consequently, the output table must be of size 3-by-4 for consistent dimensions.

### Monotonicity of Breakpoint Data Sets

The first stage of a table lookup operation involves relating inputs to the breakpoint data sets. The search algorithm requires that input breakpoint sets be strictly monotonically increasing, that is, each successive element is greater than its preceding element. For example, the vector

```A = [0 0.5 1 1.9 2.1 3] ```

is a valid breakpoint data set as each element is larger than its predecessors.

 Note:   Although a breakpoint data set is strictly monotonic in `double` format, it might not be so after conversion to a fixed-point data type.

### Representation of Discontinuities in Lookup Tables

You can represent discontinuities in lookup tables that have monotonically increasing breakpoint data sets. To create a discontinuity, repeat an input value in the breakpoint data set with different output values in the table data. For example, these vectors of input (x) and output (y) values associated with a 1-D lookup table create the step transitions depicted in the plot that follows.

```Vector of input values: [-2 -1 -1 0 0 1 1 2] Vector of output values: [-1 -1 -2 -2 2 2 1 1]```

This example has discontinuities at x = –1, 0, and +1.

When there are two output values for a given input value, the block chooses the output according to these rules:

• If the input signal is less than zero, the block returns the output value associated with the last occurrence of the input value in the breakpoint data set. In this example, if the input is –1, y is –2, marked with a solid circle.

• If the input signal is greater than zero, the block returns the output value associated with the first occurrence of the input value in the breakpoint data set. In this example, if the input is 1, y is 2, marked with a solid circle.

• If the input signal is zero and there are two output values specified at the origin, the block returns the average of those output values. In this example, if the input is 0, y is 0, the average of the two output values –2 and 2 specified at x = 0.

When there are three points specified at the origin, the block generates the output associated with the middle point. The following example demonstrates this special rule.

```Vector of input values: [-2 -1 -1 0 0 0 1 1 2] Vector of output values: [-1 -1 -2 -2 1 2 2 1 1]```

In this example, three points define the discontinuity at the origin. When the input is 0, y is 1, the value of the middle point.

You can apply this same method to create discontinuities in breakpoint data sets associated with multidimensional lookup tables.

### Formulation of Evenly Spaced Breakpoints

You can represent evenly spaced breakpoints in a data set by using one of these methods.

FormulationExampleWhen to Use This Formulation
`[first_value:spacing:last_value]``[10:10:200]`The lookup table does not use `double` or `single`.
`first_value + spacing * [0:(last_value-first_value)/spacing]``1 + (0.02 * [0:450])`The lookup table uses `double` or `single`.

Because floating-point data types cannot precisely represent some numbers, the second formulation works better for `double` and `single`. For example, use `1 + (0.02 * [0:450])` instead of `[1:0.02:10]`. For a list of lookup table blocks that support evenly spaced breakpoints, see Summary of Lookup Table Block Features.

Among other advantages, evenly spaced breakpoints can make the generated code division-free and reduce memory usage. For more information, see:

 Tip:   Do not use the MATLAB® `linspace` function to define evenly spaced breakpoints. Simulink uses a tighter tolerance to check whether a breakpoint set has even spacing. If you use `linspace` to define breakpoints for your lookup table, Simulink considers the breakpoints to be unevenly spaced.