# Square Jacobi SVD HDL Optimized

**Libraries:**

Fixed-Point Designer HDL Support /
Matrices and Linear Algebra /
Matrix Factorizations

## Description

Use the Square Jacobi SVD HDL Optimized block to perform singular value
decomposition (SVD) on square matrices using the two-sided Jacobi algorithm. Given a square
matrix *A*, the Square Jacobi SVD HDL Optimized block uses the
two-sided Jacobi method to produce a vector *s* of nonnegative elements and
unitary matrices *U* and *V* such that

.*A* =
*U**diag(*s*)**V*'

**Note**

For non-square matrices, use the Non-Square Jacobi SVD HDL Optimized block.

## Examples

## Extended Examples

## Ports

### Input

### Output

## Parameters

## Tips

The Square Jacobi SVD HDL Optimized block computes the singular value decomposition in place. Set the fixed-point data type of the input

`n`

-by-`n`

matrix`A`

with enough precision and enough headroom to avoid overflow.First, use the

`fixed.singularValueUpperBound`

function to determine the upper bound on the singular values. Then define the integer length based on the value of the upper bound, with one additional bit for the sign, another additional bit for intermediate CORDIC growth, and one more bit for intermediate growth to compute the Jacobi rotations. Compute the fraction length based on the integer length and the desired word length.svdUpperBound = fixed.singularValueUpperBound(n,n,max(abs(A(:)))) additionalBitGrowth = 3; integerLength = ceil(log2(svdUpperBound)) + additionalBitGrowth wordLength = 16 fractionLength = wordLength - integerLength

The behavior of the Square Jacobi SVD HDL Optimized block is equivalent to

`[U,s,V] = fixed.jacobiSVD(A)`

. The`fixed.JacobiSVD`

function uses the same algorithm as the Square Jacobi SVD HDL Optimized, with the same output data types. However, there may be small numerical differences in the least-significant bit between the function and the block.The behavior of the Square Jacobi SVD HDL Optimized block is equivalent to

`[U,s,V] =`

when`fixed.svd`

(A,'econ','vector')`A`

is a square matrix.

## Algorithms

## References

[1] Arm Developer. "AMBA AXI and ACE Protocol Specification Version E." https://developer.arm.com/documentation/ihi0022/e/AMBA-AXI3-and-AXI4-Protocol-Specification/Single-Interface-Requirements/Basic-read-and-write-transactions/Handshake-process.

[2] Jacobi, Carl G. J., "Über ein
leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch
aufzulösen." *Journal fur die reine und angewandte Mathematik* 30 (1846):
51–94.

[3] Forsythe, George E. and Peter
Henrici. "The Cyclic Jacobi Method for Computing the Principal Values of a Complex Matrix."
*Transactions of the American Mathematical Society* 94, no. 1 (January
1960): 1-23.

[4] Shiri, Aidin and Ghader
Khosroshahi. "An FPGA Implementation of Singular Value Decomposition", *ICEE 2019:
27th Iranian Conference on Electrical Engineering, Yazd, Iran, April 30–May 2,
2019*, 416-22, IEEE.

[5] Golub, Gene H. and Charles F. Van
Loan. *Matrix Computations*, 4th ed. Baltimore, MD: Johns Hopkins
University Press, 2013.

[6] Athi, Mrudula V., Seyed R.
Zekavat, and Alan A. Struthers. "Real-Time Signal Processing of Massive Sensor Arrays via a
Parallel Fast Converging SVD Algorithm: Latency, Throughput, and Resource Analysis."
*IEEE Sensors Journal* 16, no. 18 (January 2016): 2519-26. https://doi.org/10.1109/JSEN.2016.2517040.

[7] Brent, Richard P., Franklin T.
Luk, and Charles Van Loan. "Computation of the Singular Value Decomposition Using
Mesh-Connected Processors." *Journal of VLSI and Computer Systems* 1, 3
(1985): 242–70.

[8] Hemkumar, Nariankadu D.
*A Systolic VLSI Architecture for Complex SVD*.
Master’s thesis, Rice University, 1991.

[9] Duryea, R. A. *Finite
Precision Arithmetic in Singular Value Decomposition Architectures*. Ph.D. thesis,
Cornell University, 1987.

[10] Cavallaro, Joseph R. and Franklin
T. Luk. 1987. "CORDIC Arithmetic for an SVD Processor." *1987 IEEE 8th Symposium on
Computer Arithmetic (ARITH), Como, Italy, May 18-21, 1987*, 113-20. IEEE. https://doi.org/10.1109/ARITH.1987.6158686.

## Extended Capabilities

## Version History

**Introduced in R2023a**