cordicsqrt

CORDIC-based approximation of square root

Syntax

y=cordicsqrt(u)

y=cordicsqrt(u, niters)

y=cordicsqrt(___,
'ScaleOutput', B)

Description

y=cordicsqrt(u) computes the square root of u using a CORDIC algorithm implementation.

example

y=cordicsqrt(u, niters) computes the square root of u by performing niters iterations of the CORDIC algorithm.

example

y=cordicsqrt(___, 'ScaleOutput', B) scales the output depending on the Boolean value of B.

example

Examples

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Calculate the CORDIC Square Root

Open Live Script

Find the square root of fi object x using a CORDIC implementation.

x = fi(1.6,1,12);
y = cordicsqrt(x)

y = 
    1.2646

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 12
        FractionLength: 10

Because you did not specify niters, the function performs the maximum number of iterations, x.WordLength - 1.

Compute the difference between the results of the cordicsqrt function and the double-precision sqrt function.

err = abs(sqrt(double(x))-double(y))

err = 
1.0821e-04

Calculate the CORDIC Square Root with a Specified Number of Iterations

Open Live Script

Compute the square root of x with three iterations of the CORDIC kernel.

x = fi(1.6,1,12);
y = cordicsqrt(x,3)

y = 
    1.2646

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 12
        FractionLength: 10

Compute the difference between the results of the cordicsqrt function and the double-precision sqrt function.

err = abs(sqrt(double(x))-double(y))

err = 
1.0821e-04

Calculate the CORDIC Square Root Without Scaling the Output

Open Live Script

x = fi(1.6,1,12);
y = cordicsqrt(x, 'ScaleOutput', 0)

y = 
    1.0479

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 12
        FractionLength: 10

The output, y, was not scaled by the inverse CORDIC gain factor.

Compare Results of cordicsqrt and sqrt Functions

Open Live Script

Compare the results produced by 10 iterations of the cordicsqrt algorithm to the results of the double-precision sqrt function.

% Create 500 points between [0, 2)
stepSize = 2/500;
XDbl = 0:stepSize:2;
XFxp = fi(XDbl, 1, 12);    % signed, 12-bit fixed-point
sqrtXRef = sqrt(double(XFxp));   % reference results

% Use 12-bit quantized inputs and set the number
% of iterations to 10.
% Compare the  fixed-point CORDIC results to the
% double-precision sqrt function results.

niters = 10;
cdcSqrtX  = cordicsqrt(XFxp,  niters);
errCdcRef = sqrtXRef - double(cdcSqrtX);
figure
hold on
axis([0 2 -.5 1.5])
plot(XFxp, sqrtXRef,  'b')
plot(XFxp, cdcSqrtX,  'g')
plot(XFxp, errCdcRef, 'r')
ylabel('Sqrt(x)')
gca.XTick = 0:0.25:2;
gca.XTickLabel = {'0','0.25','0.5','0.75','1','1.25','1.5','1.75','2'};
gca.YTick = -.5:.25:1.5;
gca.YTickLabel = {'-0.5','-0.25','0','0.25','0.5','0.75','1','1.25','1.5'};
ref_str = 'Reference: sqrt(double(X))';
cdc_str = sprintf('12-bit CORDIC square root; N = %d', niters);
err_str = sprintf('Error (max = %f)', max(abs(errCdcRef)));
legend(ref_str, cdc_str, err_str, 'Location', 'southeast')

Figure contains an axes object. The axes object with ylabel Sqrt(x) contains 3 objects of type line. These objects represent Reference: sqrt(double(X)), 12-bit CORDIC square root; N = 10, Error (max = 0.002009).

Input Arguments

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`u` — Data input array
scalar | vector | matrix | multidimensional array

Data input array, specified as a positive scalar, vector, matrix, or multidimensional array of fixed-point or built-in data types. When the input array contains values between 0.5 and 2, the algorithm is most accurate. A pre- and post-normalization process is performed on input values outside of this range. For more information on this process, see Pre- and Post-Normalization.

`niters` — Number of iterations
scalar

The number of iterations that the CORDIC algorithm performs, specified as a positive, integer-valued scalar. If you do not specify niters, the algorithm uses a default value. For fixed-point inputs, the default value of niters is u.WordLength - 1. For floating-point inputs, the default value of niters is 52 for double precision; 23 for single precision.

`ScaleOutput` — Whether to scale the output
true (default) | false

Boolean value that specifies whether to scale the output by the inverse CORDIC gain factor. If you set ScaleOutput to true or 1, the output values are multiplied by a constant, which incurs extra computations. If you set ScaleOutput to false or 0, the output is not scaled.

Data Types: logical

Output Arguments

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`y` — Output array
scalar | vector | matrix | multidimensional array

Output array, returned as a scalar, vector, matrix, or multidimensional array.

More About

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CORDIC

CORDIC is an acronym for COordinate Rotation DIgital Computer. The Givens rotation-based CORDIC algorithm is one of the most hardware-efficient algorithms available because it requires only iterative shift-add operations (see References). The CORDIC algorithm eliminates the need for explicit multipliers. Using CORDIC, you can calculate various functions such as sine, cosine, arc sine, arc cosine, arc tangent, and vector magnitude. You can also use this algorithm for divide, square root, hyperbolic, and logarithmic functions.

Increasing the number of CORDIC iterations can produce more accurate results, but doing so increases the expense of the computation and adds latency.

Algorithms

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Signal Flow Diagrams

For further details on the pre- and post-normalization process, see Pre- and Post-Normalization.

CORDIC Hyperbolic Kernel

X is initialized to u'+.25, and Y is initialized to u'-.25, where u' is the normalized function input.

With repeated iterations of the CORDIC hyperbolic kernel, X approaches $A_{N} \sqrt{u'}$ , where A_N represents the CORDIC gain. Y approaches 0.

Pre- and Post-Normalization

For input values outside of the range of [0.5, 2) a pre- and post-normalization process occurs. This process performs bitshifts on the input array before passing it to the CORDIC kernel. The result is then shifted back into the correct output range during the post-normalization stage. For more details on this process see “Overcoming Algorithm Input Range Limitations” in Compute Square Root Using CORDIC.

fimath Propagation Rules

CORDIC functions discard any local fimath attached to the input.

The CORDIC functions use their own internal fimath when performing calculations:

OverflowAction—Wrap
RoundingMethod—Floor

The output has no attached fimath.

References

[1] Volder, JE. “The CORDIC Trigonometric Computing Technique.” IRE Transactions on Electronic Computers. Vol. EC-8, September 1959, pp. 330–334.

[2] Andraka, R. “A survey of CORDIC algorithm for FPGA based computers.” Proceedings of the 1998 ACM/SIGDA sixth international symposium on Field programmable gate arrays. Feb. 22–24, 1998, pp. 191–200.

[3] Walther, J.S. “A Unified Algorithm for Elementary Functions.” Hewlett-Packard Company, Palo Alto. Spring Joint Computer Conference, 1971, pp. 379–386. (from the collection of the Computer History Museum). www.computer.org/csdl/proceedings/afips/1971/5077/00/50770379.pdf

[4] Schelin, Charles W. “Calculator Function Approximation.” The American Mathematical Monthly. Vol. 90, No. 5, May 1983, pp. 317–325.

Extended Capabilities

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

Usage notes and limitations:

Variable-size signals are not supported.
The number of iterations the CORDIC algorithm performs, niters, must be a constant.

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

The cordicsqrt function also supports MATLAB^® to High-Level Synthesis (HLS) code generation.

Version History

Introduced in R2014a

cordicsqrt

Syntax

Description

Examples

Calculate the CORDIC Square Root

Calculate the CORDIC Square Root with a Specified Number of Iterations

Calculate the CORDIC Square Root Without Scaling the Output

Compare Results of cordicsqrt and sqrt Functions

Input Arguments

`u` — Data input array
scalar | vector | matrix | multidimensional array

`niters` — Number of iterations
scalar

`ScaleOutput` — Whether to scale the output
true (default) | false

Output Arguments

`y` — Output array
scalar | vector | matrix | multidimensional array

More About

CORDIC

Algorithms

Signal Flow Diagrams

Pre- and Post-Normalization

fimath Propagation Rules

References

Extended Capabilities

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

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

Version History

See Also

Topics

cordicsqrt

Syntax

Description

Examples

Calculate the CORDIC Square Root

Calculate the CORDIC Square Root with a Specified Number of Iterations

Calculate the CORDIC Square Root Without Scaling the Output

Compare Results of cordicsqrt and sqrt Functions

Input Arguments

u — Data input array scalar | vector | matrix | multidimensional array

niters — Number of iterations scalar

ScaleOutput — Whether to scale the output true (default) | false

Output Arguments

y — Output array scalar | vector | matrix | multidimensional array

More About

CORDIC

Algorithms

Signal Flow Diagrams

Pre- and Post-Normalization

fimath Propagation Rules

References

Extended Capabilities

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

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

Version History

See Also

Topics

`u` — Data input array
scalar | vector | matrix | multidimensional array

`niters` — Number of iterations
scalar

`ScaleOutput` — Whether to scale the output
true (default) | false

`y` — Output array
scalar | vector | matrix | multidimensional array

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

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