Sqrt

Calculate square root, signed square root, or reciprocal of square root

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Libraries:
Simulink / Math Operations
HDL Coder / HDL Floating Point Operations
HDL Coder / Math Operations

Description

The Sqrt block calculates the square root, signed square root, or reciprocal of square root on the input signal. Select one of the following functions from the Function parameter list.

Function	Description	Mathematical Expression	MATLAB® Equivalent
sqrt	Square root of the input	u0.5	sqrt
signedSqrt	Square root of the absolute value of the input, multiplied by the sign of the input	sign(u)*|u|0.5	—
rSqrt	Reciprocal of the square root of the input	u-0.5	—

The block icon changes to match the function.

Examples

Square Root of Negative Values

This example shows how to compute the square root of a negative-valued input signal as complex-valued output.

By setting the Function to sqrt and Output signal type to complex, the block produces the correct result of 0 + 10i for an input of -100. If you change the Output signal type to auto or real, the block outputs NaN.

Signed Square Root of Negative Values

This example shows how to compute the signed square root of a negative-valued input signal.

When the block input is negative and you set the Function to signedSqrt, the Sqrt block output is the same for any setting of the Output signal type parameter. By setting the Numerica display format of the first Display block to decimal (Stored Integer), you can see the value of the imaginary part for the complex output.

rSqrt of Floating-Point Inputs

This example shows how to compute the rSqrt of a floating-point input signal. The Sqrt block has the following settings:

• Method = Newton-Raphson

• Number of iterations = 1

• Intermediate results data type = Inherit: Inherit from input

After one iteration of the Newton-Raphson algorithm, the block output is within 0.0004 of the final value (0.4834).

rSqrt of Fixed-Point Inputs

This example shows how to compute the rSqrt of a fixed-point input signal. The Sqrt block has the following settings:

• Method = Newton-Raphson

• Number of iterations = 1

• Intermediate results data type = Inherit: Inherit from input

After one iteration of the Newton-Raphson algorithm, the block output is within 0.0459 of the final value (0.4834).

Ports

Input

Port_1 — Input signal

scalar | vector | matrix

Input signal to the block to calculate the square root, signed square root, or reciprocal of square root. The sqrt function accepts real or complex inputs, except for complex fixed-point signals. signedSqrt and rSqrt do not accept complex inputs. The input signal must be a floating point number.

This table summarizes the support for complex types and negative values for floating point, integer, and fixed-point data types for sqrt, rSqrt, and signedSqrt functions.

Function	Data Type	Complex		Negative Values
		Input	Output
sqrt	Floating point	Yes	Yes	Yes
	Integer and fixed-point	No	No	No
rSqrt	Floating point	No	No	Yes
	Integer and fixed-point	No	No	No
signedSqrt	Floating point	No	Yes	Yes
	Integer and fixed-point	No	No	No

If the input is negative, set the Output signal to complex for all functions except signedSqrt.

Data Types: single | double | half | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point

Output

Port_1 — Output signal

scalar | vector | matrix

Output signal that is the square root, signed square root, or reciprocal of square root of the input signal. When the input is an integer or fixed-point type, the output must be floating point.

Data Types: single | double | half | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point

Parameters

Main

Function — Function the block performs

sqrt (default) | signedSqrt | rSqrt

Specify the mathematical function that the block calculates. The block icon changes to match the function you select.

Function	Block Icon
sqrt
signedSqrt
rSqrt

Dependency

When this parameter is set to signedSqrt, the Intermediate results data type parameter is disabled.

Programmatic Use

Block Parameter: Operator

Type: character vector

Values: 'sqrt' | 'signedSqrt' | 'rSqrt'

Default: 'sqrt'

Output signal type — Output signal type

auto (default) | real | complex

Specify the output signal type of the block.

Function	Input Signal Type	Output Signal Type
		Auto	Real	Complex
sqrt	real	real for nonnegative inputs NaN for negative inputs	real for nonnegative inputs NaN for negative inputs	complex
	complex	complex	error	complex
signedSqrt	real	real	real	complex
	complex	error	error	error
rSqrt	real	real	real	error
	complex	error	error	error

Programmatic Use

Block Parameter: OutputSignalType

Type: character vector

Values: 'auto' | 'real' | 'complex'

Default: 'auto'

Sample time (-1 for inherited) — Interval between samples

-1 (default) | scalar | vector

Specify the time interval between samples. To inherit the sample time, set this parameter to -1. For more information, see Specify Sample Time.

Dependencies

This parameter is visible only if you set it to a value other than -1. To learn more, see Blocks for Which Sample Time Is Not Recommended.

Programmatic Use

Block Parameter: SampleTime

Type: string scalar or character vector

Default: "-1"

Algorithm

Method — Method to compute reciprocal of square root

Exact (default) | Newton-Raphson

Specify the method for computing the reciprocal of a square root. This parameter is only valid for the rSqrt function.

Method	Data Types Supported	When to Use This Method
Exact	Floating point	You do not want an approximation. Note The input or output must be floating point.
Newton-Raphson	Floating-point, fixed-point, and built-in integer types	You want a fast, approximate calculation.

The Exact method provides results that are consistent with MATLAB computations.

Note

The algorithms for sqrt and signedSqrt are always of Exact type, no matter what selection appears on the block dialog box.

Programmatic Use

Block Parameter: AlgorithmType

Type: character vector

Values: 'Exact' | 'Newton-Raphson'

Default: 'Exact'

Number of iterations — Number of iterations used for Newton Raphson algorithm

3 (default) | integer

Specify the number of iterations to perform the Newton-Raphson algorithm. This parameter is valid with the rSqrt function and the Newton-Raphson value for Method.

Note

If you enter 0, the block output is the initial guess of the Newton-Raphson algorithm.

Programmatic Use

Block Parameter: Iterations

Type: character vector

Values: integer

Default: '3'

Data Types

The Data Type Assistant helps you set data attributes. To use the Data Type Assistant, click . For more information, see Specify Data Types Using Data Type Assistant.

Intermediate results data type — Data type of intermediate results

Inherit:Inherit via internal rule (default) | Inherit: Inherit from input | Inherit: Inherit from output | double | single | int8 | uint8 | int16 | uint16 | int32 | uint32 | int64 | uint64 | fixdt(1,16,,0) | fixdt(1,16,2^0,0) | <data type expression>

Specify the data type for intermediate results when you set Function to sqrt or rSqrt on the Main pane.

The type can be inherited, specified directly, or expressed as a data type object such as Simulink.NumericType.

Note

To avoid overflow, the intermediate data type must be larger than or equal to a data type that can contain the square of the output data type.

Follow these guidelines on setting an intermediate data type explicitly for the square root function, sqrt:

Input and Output Data Types	Intermediate Data Type
Input or output is double.	Use double.
Input or output is single, and any non-single data type is not double.	Use single or double.
Input and output are fixed point.	Use fixed point.

Follow these guidelines on setting an intermediate data type explicitly for the reciprocal square root function, rSqrt:

Input and Output Data Types	Intermediate Data Type
Input is double and output is not single.	Use double.
Input is not single and output is double.	Use double.
Input and output are fixed point.	Use fixed point.

Caution

Do not set Intermediate results data type to Inherit:Inherit from output when:

• You select Newton-Raphson to compute the reciprocal of a square root.

• The input data type is floating point.

• The output data type is fixed point.

Under these conditions, selecting Inherit:Inherit from output yields suboptimal performance and produces an error.

To avoid this error, convert the input signal from a floating-point to fixed-point data type. For example, insert a Data Type Conversion block in front of the Sqrt block to perform the conversion.

Dependencies

This parameter is disabled when the Function parameter is set to signedSqrt.

Programmatic Use

Block Parameter: IntermediateResultsDataTypeStr

Type: character vector

Values: 'Inherit: Inherit via internal rule' | 'Inherit: Inherit from input' | 'Inherit: Inherit from output' | 'double' | 'single', 'int8', 'uint8', int16, 'uint16', 'int32', 'uint32', 'int64', 'uint64', fixdt(1,16,0), fixdt(1,16,2^0,0). '<data type expression>'

Default: 'Inherit: Inherit via internal rule'

Output — Output data type

Inherit: Same as first input (default) | Inherit: Inherit via internal rule | Inherit: Inherit via back propagation | double | single | half | int8 | int32 | uint32 | int64 | uint64 | fixdt(1,16,2^0,0) | <data type expression> | ...

Specify the output data type. The type can be inherited, specified directly, or expressed as a data type object such as Simulink.NumericType.

Dependencies

When input is a floating-point data type smaller than single precision, the Inherit: Inherit via internal rule output data type depends on the setting of the Inherit floating-point output type smaller than single precision configuration parameter. Data types are smaller than single precision when the number of bits needed to encode the data type is less than the 32 bits needed to encode the single-precision data type. For example, half and int16 are smaller than single precision.

Programmatic Use

Block Parameter: OutDataTypeStr

Type: character vector

Values: 'Inherit: Inherit via internal rule' | 'Inherit: Inherit via back propagation' | 'Inherit: Same as first input' | 'double' | 'single' | 'half' | 'int8' | 'uint8' | int16 | 'uint16' | 'int32' | 'uint32' | 'int64' | 'uint64' | fixdt(1,16,0) | fixdt(1,16,2^0,0) | fixdt(1,16,2^0,0) | '<data type expression>'

Default: 'Inherit: Same as first input'

Minimum — Minimum output value for range checking

[] (default) | scalar

Specify the lower value of the output range that Simulink^® checks as a finite, real, double, scalar value.

Note

If you specify a bus object as the data type for this block, do not set the minimum value for bus data on the block. Simulink ignores this setting. Instead, set the minimum values for bus elements of the bus object specified as the data type. For information on the Minimum parameter for a bus element, see Simulink.BusElement.

Simulink uses the minimum to perform:

• Parameter range checking (see Specify Minimum and Maximum Values for Block Parameters) for some blocks.

• Simulation range checking (see Specify Signal Ranges and Enable Simulation Range Checking).

• Automatic scaling of fixed-point data types.

• Optimization of the code that you generate from the model. This optimization can remove algorithmic code and affect the results of some simulation modes such as SIL or external mode. For more information, see Optimize using the specified minimum and maximum values (Embedded Coder).

Note

Output minimum does not saturate or clip the actual output signal. Use the Saturation block instead.

Programmatic Use

Block Parameter: OutMin

Type: character vector

Values: scalar

Default: '[ ]'

Maximum — Maximum output value for range checking

[] (default) | scalar

Specify the upper value of the output range that Simulink checks as a finite, real, double, scalar value.

Note

If you specify a bus object as the data type for this block, do not set the maximum value for bus data on the block. Simulink ignores this setting. Instead, set the maximum values for bus elements of the bus object specified as the data type. For information on the Maximum parameter for a bus element, see Simulink.BusElement.

Simulink uses the maximum value to perform:

• Parameter range checking (see Specify Minimum and Maximum Values for Block Parameters) for some blocks.

• Simulation range checking (see Specify Signal Ranges and Enable Simulation Range Checking).

• Automatic scaling of fixed-point data types.

• Optimization of the code that you generate from the model. This optimization can remove algorithmic code and affect the results of some simulation modes such as SIL or external mode. For more information, see Optimize using the specified minimum and maximum values (Embedded Coder).

Note

Output maximum does not saturate or clip the actual output signal. Use the Saturation block instead.

Programmatic Use

Block Parameter: OutMax

Type: character vector

Values: scalar

Default: '[ ]'

Integer rounding mode — Rounding mode for fixed-point operations

Floor (default) | Ceiling | Convergent | Nearest | Round | Simplest | Zero

Specify the rounding mode for fixed-point operations. For more information, see Rounding (Fixed-Point Designer).

Programmatic Use

Block Parameter: RndMeth

Type: character vector

Values: 'Ceiling' | 'Convergent' | 'Floor' | 'Nearest' | 'Round' | 'Simplest' | 'Zero'

Default: 'Floor'

Lock output data type setting against changes by the fixed-point tools — Prevent fixed-point tools from overriding data types

off (default) | on

Select to lock the output data type setting of this block against changes by the Fixed-Point Tool and the Fixed-Point Advisor. For more information, see Use Lock Output Data Type Setting (Fixed-Point Designer).

Programmatic Use

Block Parameter: LockScale

Type: character vector

Values: 'off' | 'on'

Default: 'off'

Saturate on integer overflow — Choose the behavior when integer overflow occurs

off (default) | on

Action	Reasons for Taking This Action	What Happens for Overflows	Example
Select this check box.	Your model has possible overflow, and you want explicit saturation protection in the generated code.	Overflows saturate to either the minimum or maximum value that the data type can represent.	The maximum value that the int8 (signed, 8-bit integer) data type can represent is 127. Any block operation result greater than this maximum value causes overflow of the 8-bit integer. With the check box selected, the block output saturates at 127. Similarly, the block output saturates at a minimum output value of -128.
Do not select this check box.	You want to optimize efficiency of your generated code. You want to avoid overspecifying how a block handles out-of-range signals. For more information, see Troubleshoot Signal Range Errors.	Overflows wrap to the appropriate value that is representable by the data type.	The maximum value that the int8 (signed, 8-bit integer) data type can represent is 127. Any block operation result greater than this maximum value causes overflow of the 8-bit integer. With the check box cleared, the software interprets the overflow-causing value as int8, which can produce an unintended result. For example, a block result of 130 (binary 1000 0010) expressed as int8, is -126.

When you select this check box, saturation applies to every internal operation on the block, not just the output or result. Usually, the code generation process can detect when overflow is not possible. In this case, the code generator does not produce saturation code.

Programmatic Use

Block Parameter: DoSatur

Type: character vector

Value: 'off' | 'on'

Default: 'off'

Block Characteristics

Data Types	double | fixed point | half | integer | single
Direct Feedthrough	yes
Multidimensional Signals	yes
Variable-Size Signals	yes
Zero-Crossing Detection	no

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

HDL Code Generation

Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.

HDL Coder™ provides additional configuration options that affect HDL implementation and synthesized logic.

Sqrt Function Modes

For the Sqrt block with Function set to sqrt, the code generator supports various architectures and data types.

The sqrtfunction architecture supports code generation for fixed-point types and floating-point types. When you use floating-point types, set the floating point IP library parameters to Native Floating Point. You can specify the LatencyStrategy and CustomLatency HDL properties to choose from a range of frequency values when targeting your design on the hardware platform.

Use the UseMultiplier HDL block property in combination with the LatencyStrategy and CustomLatency properties to specify whether to compute the square root by using a pipelined shift and add or multiplication algorithm

Native Floating Point Settings for Different Modes

For this architecture, you can specify the HandleDenormals and LatencyStrategy settings from the Native Floating Point tab in the HDL Block Properties dialog box.

Architecture	Fixed-Point	Native Floating-Point	HandleDenormals	LatencyStrategy
sqrtfunction	✓	✓	✓	✓
sqrtnewton	✓	—	—	—
sqrtnewtonsinglerate	✓	—	—	—
recipsqrtnewton	✓	—	—	—
recipsqrtnewtonsinglerate	✓	—	—	—

HDL Architecture

This block has multi-cycle implementations that introduce additional latency in the generated code. To see the added latency, view the generated model or validation model. See Generated Model and Validation Model (HDL Coder).

Architecture	Parameter	Additional cycles of latency	Description
SqrtFunction (default)	UseMultiplier LatencyStrategy CustomLatency	Depends on parameter choices, output word length, and input and output fraction lengths.	To specify this architecture, set Function to sqrt or rSqrt. Compute the square root by using a pipelined shift/addition algorithm or multiplication-based algorithm. The SqrtFunction architecture has a maximum latency that is determined by the output word length, and input and output fraction lengths, for fixed-point data. For example, the maximum latency is 11 for output word length of 17. To see the latency calculation, see HDLMathLib Sqrt (HDL Coder). Improve design frequency and reduce resource utilization by setting the UseMultiplier to off and LatencyStrategy to inherit .
SqrtNewton	Iterations	Iterations + 3	To specify this architecture, set Function to sqrt. Use the iterative Newton method. Select this option to optimize area. The default value for Iterations is 3. The recommended value for Iterations is from 2 through 10. If Iterations is outside the recommended range, HDL Coder generates a message.
SqrtNewtonSingleRate	Iterations	(Iterations * 4) + 6	To specify this architecture, set Function to sqrt. Use the single rate pipelined Newton method. Select this option to optimize speed, or if you want a single rate implementation. The default value for Iterations is 3. The recommended value for Iterations is from 2 through 10. If Iterations is outside the recommended range, the coder generates a message.
RecipSqrtNewton	Iterations	Iterations + 2	To specify this architecture, set Function to rSqrt. Use the iterative Newton method. Select this option to optimize area.
RecipSqrtNewtonSingleRate	Iterations	(Iterations * 4) + 5	To specify this architecture, set Function to rSqrt. Use the single rate pipelined Newton method. Select this option to optimize speed, or if you want a single rate implementation.

The Newton-Raphson iterative method:

$$x_{i+1}=x_i-\frac{f(x_i)}{f\text{'}(x_i)}=x_i(1.5-0.5a{x}_i{}^2)$$

ReciprocalRsqrtBasedNewton and ReciprocalRsqrtBasedNewtonSingleRate implement the Newton-Raphson method with:

$$f(x)=\frac{1}{x^2}-1$$

HDL Block Properties

General
ConstrainedOutputPipeline	Number of registers to place at the outputs by moving existing delays within your design. Distributed pipelining does not redistribute these registers. The default is 0. For more details, see ConstrainedOutputPipeline (HDL Coder).
Iterations	Number of iterations for SqrtNewton or SqrtNewtonSingleRate implementation.
InputPipeline	Number of input pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see InputPipeline (HDL Coder).
OutputPipeline	Number of output pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see OutputPipeline (HDL Coder).
UseMultiplier	Select algorithm for SqrtFunction implementation. By default, this property is set to off, and the block computes square root by using the shift-add algorithm. When the property is set to on, the square root is computed by using multipliers. Increase design frequency and reduce resource utilization by setting the UseMultiplier to off and LatencyStrategy to inherit .
LatencyStrategy	Specify whether to map the blocks in your design to MAX, MIN, CUSTOM, or ZERO latency for fixed-point and floating-point types. The default is MAX. The block settings for floating-point types overrides the model-level settings. For fixed-point types, the block-level setting determines the latency strategy. See also LatencyStrategy (HDL Coder) . Increase design frequency and reduce resource utilization by setting the UseMultiplier to off and LatencyStrategy to inherit .
CustomLatency	When LatencyStrategy is set to CUSTOM and if you use fixed-point types, use this property to specify a custom latency value between ZERO and MAX for fixed-point types. For floating-point types, this value becomes the custom latency of the operator. See also NFPCustomLatency (HDL Coder).

Native Floating Point
HandleDenormals	Specify whether you want HDL Coder to insert additional logic to handle denormal numbers in your design. Denormal numbers are numbers that have magnitudes less than the smallest floating-point number that can be represented without leading zeros in the mantissa. The default is inherit. See also HandleDenormals (HDL Coder).

Restrictions

• Input must be an unsigned scalar value.

• Output is a fixed-point scalar value.

PLC Code Generation
Generate Structured Text code using Simulink® PLC Coder™.

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.

Version History

Introduced in R2010a