# SVD_TRUNCATED The Truncated Singular Value Decomposition

SVD_TRUNCATED, a C++ code which demonstrates the computation of the reduced or truncated Singular Value Decomposition (SVD) of an M by N rectangular matrix, in cases where M < N or N < M.

The singular value decomposition of an M by N rectangular matrix A has the form

```        A(mxn) = U(mxm) * S(mxn) * V'(nxn)
```
where
• U is an orthogonal matrix, whose columns are the left singular vectors;
• S is a diagonal matrix, whose min(m,n) diagonal entries are the singular values;
• V is an orthogonal matrix, whose columns are the right singular vectors;
Note that the transpose of V is used in the decomposition, and that the diagonal matrix S is typically stored as a vector.

It is often the case that the matrix A has one dimension much bigger than the other. For instance, M = 3 and N = 10,000 might be such a case. For such examples, much of the computation and memory required for the standard SVD may not actually be needed. Instead, a truncated, or reduced version is appropriate. It will be computed faster, and require less memory to store the data.

If M < N, we have the "truncated V" SVD:

```        A(mxn) = U(mxm) * Sm(mxm) * Vm'(nxm)
```
Notice that, for our example, we will have to compute and store a Vm of size 30,000 instead of a V of size 1,000,000 entries.

If N < M, we have the "truncated U" SVD:

```        A(mxn) = Un(mxn) * Sn(nxn) * V'(nxn)
```
Similarly, in this case, the computation and storage of Un can be much reduced from that of U.

The LINPACK routine DSVDC can compute the "truncated U" version of the SVD. In order to compute the "truncated V" version with DSVDC, it's actually necessary to transpose the matrix, compute the truncated U version, and then transpose everything back...very carefully.

### Languages:

SVD_TRUNCATED is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

LINPACK_D, a C++ library which solves linear systems using double precision real arithmetic;

SVD_BASIS, a C++ program which computes a reduced basis for a collection of data vectors using the SVD.

SVD_SNOWFALL, a C++ library which reads a file containing historical snowfall data and analyzes the data with the Singular Value Decomposition (SVD), and plots created by GNUPLOT.

### Reference:

1. Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford, James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum, Sven Hammarling, Alan McKenney, Danny Sorensen,
LAPACK User's Guide,
Third Edition,
SIAM, 1999,
ISBN: 0898714478,
LC: QA76.73.F25L36
2. Gene Golub, Charles VanLoan,
Matrix Computations, Third Edition,
Johns Hopkins, 1996,
ISBN: 0-8018-4513-X,
LC: QA188.G65.
3. David Kahaner, Cleve Moler, Steven Nash,
Numerical Methods and Software,
Prentice Hall, 1989,
ISBN: 0-13-627258-4,
LC: TA345.K34.
4. Lloyd Trefethen, David Bau,
Numerical Linear Algebra,
SIAM, 1997,
ISBN: 0-89871-361-7,
LC: QA184.T74.

### Source Code:

Last revised on 20 April 2020.