Sqrt
Calculate square root, signed square root, or reciprocal of square root
Libraries:
Simulink /
Math Operations
HDL Coder /
HDL Floating Point Operations
HDL Coder /
Math Operations
Alternative Configurations of Sqrt Block:
Signed Sqrt  Reciprocal Sqrt  Square Root  Signed Square Root  Reciprocal Square Root
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 

sqrt

signedSqrt
 Square root of the absolute value of the input, multiplied by the sign of the input 
 — 
rSqrt
 Reciprocal of the square root of the input 
 — 
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 negativevalued input signal as complexvalued 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 negativevalued 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 FloatingPoint Inputs
This example shows how to compute the rSqrt of a floatingpoint input signal. The Sqrt block has the following settings:
Method =
NewtonRaphson
Number of iterations =
1
Intermediate results data type =
Inherit: Inherit from input
After one iteration of the NewtonRaphson algorithm, the block output is within 0.0004 of the final value (0.4834).
rSqrt of FixedPoint Inputs
This example shows how to compute the rSqrt of a fixedpoint input signal. The Sqrt block has the following settings:
Method =
NewtonRaphson
Number of iterations =
1
Intermediate results data type =
Inherit: Inherit from input
After one iteration of the NewtonRaphson 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 fixedpoint 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 fixedpoint data types for
sqrt
, rSqrt
, and
signedSqrt
functions.
Function  Data Type  Complex  Negative Values  

Input  Output  
sqrt  Floating point  Yes  Yes  Yes 
Integer and fixedpoint  No  No  No  
 Floating point  No  No  Yes 
Integer and fixedpoint  No  No  No  
signedSqrt  Floating point  No  Yes  Yes 
Integer and fixedpoint  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 fixedpoint 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
 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

The default value depends on the block configuration.
sqrt
default — Sqrt and Square Root blockssignedSqrt
default — Signed Sqrt and Signed Square Root blocksrSqrt
default — Reciprocal Sqrt and Reciprocal Square Root blocks
Programmatic Use
Block Parameter:
Operator 
Type: character vector 
Values:
'sqrt'  'signedSqrt'
 'rSqrt' 
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  








 








 









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
To set the block parameter value programmatically, use
the set_param
function.
Parameter:  SampleTime 
Values:  "1" (default)  scalar or vector in quotes 
Algorithm
Method — Method to compute reciprocal of square root
Exact
(default)  NewtonRaphson
Specify the method for computing the reciprocal of a square root.
Method  Data Types Supported  When to Use This Method 

Exact
 Floating point  You do not want an approximation. The input or output must be floating point. 
NewtonRaphson
 Floatingpoint, fixedpoint, and builtin integer types  You want a fast, approximate calculation. 
The Exact
method provides results that are
consistent with MATLAB computations.
Dependencies
To enable this parameter, set Function to
rSqrt
.
When Function is set to
sqrt
or
signedSqrt
, this parameter is set to
Exact
.
Programmatic Use
Block Parameter:
AlgorithmType 
Type: character vector 
Values:
'Exact' 
'NewtonRaphson' 
Default:
'Exact' 
Number of iterations — Number of iterations used for Newton Raphson algorithm
3
(default)  integer
Specify the number of iterations to perform the NewtonRaphson
algorithm. This parameter is valid with the rSqrt
function and the NewtonRaphson
value for
Method.
If you enter 0
, the block output is the initial
guess of the NewtonRaphson 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
.
The type can be inherited, specified directly, or expressed as a data
type object such as a Simulink.NumericType
object.
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 nonsingle 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
NewtonRaphson
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 floatingpoint to fixedpoint data type. For example, insert a Data Type Conversion block in front of the Sqrt block to perform the conversion.
Dependencies
To enable this parameter, set Function to
sqrt
or
rSqrt
.
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 a
Simulink.NumericType
object.
Dependencies
When input is a floatingpoint data type smaller than single
precision, the Inherit: Inherit via internal
rule
output data type depends on the setting of
the Inherit floatingpoint 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 singleprecision 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 the software checks as a finite, real, double, scalar value.
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. The software ignores
this setting. Instead, set the minimum values for bus elements of the
bus object specified as the data type. For more information, see Simulink.BusElement
.
The software 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 fixedpoint 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).
Tips
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 the software checks as a finite, real, double, scalar value.
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. The software ignores
this setting. Instead, set the maximum values for bus elements of the
bus object specified as the data type. For more information, see Simulink.BusElement
.
The software 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 fixedpoint 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).
Tips
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 fixedpoint operations
Floor
(default)  Ceiling
 Convergent
 Nearest
 Round
 Simplest
 Zero
Specify the rounding mode for fixedpoint operations. For more information, see Rounding Modes (FixedPoint 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 fixedpoint tools — Prevent fixedpoint tools from overriding data types
off
(default)  on
Select to lock the output data type setting of this block against changes by the FixedPoint Tool and the FixedPoint Advisor. For more information, see Use Lock Output Data Type Setting (FixedPoint Designer).
Programmatic Use
Block Parameter:
LockScale 
Type: character vector 
Values:
'off' 
'on' 
Default:
'off' 
Saturate on integer overflow — Method of overflow action
on
(default)  off
Specify whether overflows saturate or wrap.
on
— Overflows saturate to either the minimum or maximum value that the data type can represent.off
— Overflows wrap to the appropriate value that the data type can represent.
For example, the maximum value that the signed 8bit integer int8
can represent is 127. Any block operation result greater than this maximum value causes
overflow of the 8bit integer.
With this parameter selected, the block output saturates at 127. Similarly, the block output saturates at a minimum output value of 128.
With this parameter cleared, the software interprets the overflowcausing value as
int8
, which can produce an unintended result. For example, a block result of 130 (binary 1000 0010) expressed asint8
is 126.
Tips
Consider selecting this parameter when your model has a possible overflow and you want explicit saturation protection in the generated code.
Consider clearing this parameter when you want to optimize efficiency of your generated code. Clearing this parameter also helps you to avoid overspecifying how a block handles outofrange signals. For more information, see Troubleshoot Signal Range Errors.
When you select this parameter, saturation applies to every internal operation on the block, not just the output or result.
In general, the code generation process can detect when overflow is not possible. In this case, the code generator does not produce saturation code.
Programmatic Use
To set the block parameter value programmatically, use
the set_param
function.
Parameter:  SaturateOnIntegerOverflow 
Values:  'on' (default)  'off' 
Block Characteristics
Data Types 

Direct Feedthrough 

Multidimensional Signals 

VariableSize Signals 

ZeroCrossing Detection 

Alternative Configurations
Signed Sqrt — Calculate signed square root
The Signed Sqrt block sets Function
to signedSqrt
.
Libraries:
Simulink /
Math Operations
Reciprocal Sqrt — Calculate reciprocal of square root
The Reciprocal Sqrt block sets
Function to
rSqrt
.
Libraries:
Simulink /
Math Operations
HDL Coder /
HDL Floating Point Operations
HDL Coder /
Math Operations
Square Root — Calculate square root
The Square Root block differs from the Sqrt block in name only.
Libraries:
Simulink /
Quick Insert /
Math Operations
Signed Square Root — Calculate signed square root
The Signed Square Root block sets
Function to
signedSqrt
.
The Signed Square Root block differs from the Signed Sqrt block in name only.
Libraries:
Simulink /
Quick Insert /
Math Operations
Reciprocal Square Root — Calculate reciprocal of square root
The Reciprocal Square Root block sets
Function to
rSqrt
.
The Reciprocal Square Root block differs from the Reciprocal Sqrt block in name only.
Libraries:
Simulink /
Quick Insert /
Math Operations
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.
Function  Architecture  Description 

 SqrtFunction (default)  Compute the square root by using a pipelined shiftaddition algorithm or
multiplicationbased algorithm. The
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. Improve design
frequency and reduce resource utilization by setting the
UseMultiplier to

sqrt  SqrtNewton  Use the iterative Newton method to optimize area. 
sqrt  SqrtNewtonSingleRate  Use the single rate pipelined Newton method. Select this option to optimize speed, or if you want a single rate implementation. 
rSqrt  RecipSqrtNewton  Use the iterative Newton method to optimize area. To specify this architecture, set
Method to

rSqrt  RecipSqrtNewtonSingleRate  Use the single rate pipelined Newton method. Select this option to optimize speed, or if you want a single rate implementation. To specify this architecture,
set Method to

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

Iterations  Number of iterations for

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

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

UseMultiplier  Select algorithm for Increase design frequency and
reduce resource utilization by setting the
UseMultiplier to

LatencyStrategy  Specify whether to map the blocks in your design to Increase design frequency and reduce
resource utilization by setting the
UseMultiplier to

CustomLatency  When LatencyStrategy is set to

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 floatingpoint
number that can be represented without leading zeros in the mantissa. The default is

The Sqrt block supports these data types for HDL code generation:
Input Port  Dimension  FixedPoint  FloatingPoint  Builtin Integers  Bus  Boolean  Complex Signal 

Port_1  Scalar Vector Matrix (up to 2D)  Yes  Single Double  Yes  Yes  No  No 
This block has multicycle 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).
FloatingPoint Latency
Architecture  FloatingPoint Type  Latency Strategy  Latency (in cycles)  Custom Latency Support 

SqrtFunction  Single  Min  16  Yes 
Max  28  
Double  Min  33  
Max  58 
FixedPoint Latency
Architecture  Maximum Latency (in cycles)  Description  Latency Strategy Support  Custom Latency Support 

SqrtFunction  To calculate the latency for fixedpoint data types, see Algorithm in Sqrt (HDL Coder).  The maximum latency is determined by the output word length, and input and output fraction lengths, for fixedpoint data.  Yes  Yes 
SqrtNewton  Iterations + 3  The default value for Iterations HDL block property is 3. The recommended value for Iterations is from 2 through 10. If Iterations is outside the recommended range, HDL Coder generates a message.  No  No 
SqrtNewtonSingleRate  (Iterations* 4) + 6  
RecipSqrtNewton  Number of iterations + 2  The default value for Number of iterations block parameter is 3. The recommended value for Number of iterations is from 2 through 10. If Number of iterations is outside the recommended range, HDL Coder generates a message.  
RecipSqrtNewtonSingleRate  (Number of iterations* 4) + 5 
These block parameter configurations are incompatible with HDL code generation.
Block Parameter  Limitations and Considerations 

Output Signal Type  Parameter value Complex is not
supported. 
Method  When you set Function parameter to
rSqrt , use
Exact for floatingpoint data
types and Newton Raphson for
fixedpoint data types. 
Intermediate results data type  Parameter value other than Inherit: Inherit
via internal rule is not supported 
Output  Choose an output port data type that is same as the input. 
Integer rounding mode  Parameter value Convergent ,
Round is not supported for
fixedpoint types. 
You can use these HDL Coder optimizations to optimize the speed, area, and I/Os.
Area Optimization
Optimization  Description 

Resource Sharing (HDL Coder)  Resource sharing is an area optimization in which HDL Coder identifies multiple functionally equivalent resources and replaces them with a single resource. 
Streaming (HDL Coder)  Streaming is an area optimization in which HDL Coder transforms a vector data path to a scalar data path (or to several smallersized vector data paths). 
Speed Optimization
Optimization  Description 

Distributed Pipelining (HDL Coder)  Distributed pipelining, or register retiming, is a speed optimization that moves existing delays in a design to reduce the critical path while preserving functional behavior. For the Sqrt block, HDL Coder distributes pipeline registers around the blocks instead of within them. 
ClockRate Pipelining (HDL Coder)  Clockrate pipelining is an optimization framework in HDL Coder that allows other speed and area optimizations to introduce latency at the clock rate. 
Critical Path Estimation (HDL Coder)  To quickly identify the most likely critical path in your design, use Critical Path Estimation. Critical path estimation speeds up the iterative process of finding the critical path. To know blocks that are characterized in critical path estimation, see Characterized Blocks (HDL Coder). 
I/O Optimization
Optimization  Description 

Frame to Sample Conversion (HDL Coder)  To optimize the I/O needed for your design, use frametosample conversion. This optimization converts framebased vector or matrix inputs to smallersized samples or pixels for HDL code generation to target streambased hardware and reduce the FPGA I/O needed to handle large input and output signals. 
HDL Coder implements reciprocal squareroot operation in
ReciprocalRsqrtBasedNewton
and
ReciprocalRsqrtBasedNewtonSingleRate
architecture
by using NewtonRaphson iterative method, which is given by the equation:
$${x}_{i+1}={x}_{i}\frac{f({x}_{i})}{f\text{'}({x}_{i})}={x}_{i}(1.50.5a{x}_{i}{}^{2})$$
$$f(x)=\frac{1}{{x}^{2}}1$$
NewtonRaphsonbased implementation is incompatible with HDL code generation
in native floatingpoint mode. To generate HDL code in native floatingpoint
mode, select the SqrtFunction
architecture in the HDL
Block properties.
Use SqrtFunction Architecture for Square Root Block (HDL Coder)
PLC Code Generation
Generate Structured Text code using Simulink® PLC Coder™.
FixedPoint Conversion
Design and simulate fixedpoint systems using FixedPoint Designer™.
Version History
Introduced in R2010a
See Also
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