# svd

## Description

`[___] = svd(`

produces an
economy-size decomposition of `A`

,"econ")`A`

. If `A`

is an
*m*-by-*n* matrix, then:

*m*>*n*— Only the first*n*columns of`U`

are computed and`S`

is*n*-by-*n*.*m*=*n*—`svd(A,"econ")`

is equivalent to`svd(A)`

.*m*<*n*— Only the first*m*columns of`V`

are computed, and`S`

is*m*-by-*m*.

`[___] = svd(`

produces a
different economy-size decomposition of `A`

,0)`A`

. If `A`

is an
*m*-by-*n* matrix, then:

*m*>*n*—`svd(A,0)`

is equivalent to`svd(A,"econ")`

.*m*<=*n*—`svd(A,0)`

is equivalent to`svd(A)`

.

Syntax is not recommended. Use the `"econ"`

option instead.

`[___] = svd(___,`

optionally specifies the output format for the singular values. You can use this option with
any of the previous input or output combinations. Specify `sigmaForm`

)`"vector"`

to
return the singular values as a column vector. Specify `"matrix"`

to return
the singular values in a diagonal matrix.

## Examples

### Singular Values of Fixed-Point Matrix

Compute the singular values of a full rank fixed-point matrix.

A = fi([1 0 1; -1 -2 0; 0 1 -1])

A = 1 0 1 -1 -2 0 0 1 -1 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 16 FractionLength: 14

Compute the singular values.

s = svd(A)

s = 2.4605 1.6996 0.2392 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 16

The singular values are returned in a column vector in decreasing order.

### Fixed-Point Singular Value Decomposition

Find the singular value decomposition of the rectangular fixed-point matrix `A`

.

Define the rectangular matrix `A`

.

```
m = 4;
n = 2;
rng('default');
A = fi(10*randn(m,n))
```

A = 5.3770 3.1875 18.3389 -13.0771 -22.5889 -4.3359 8.6221 3.4258 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 16 FractionLength: 10

Find the singular value decomposition of the fixed-point matrix `A`

.

[U,S,V] = svd(A)

U = 0.1591 0.2717 -0.9387 -0.1403 0.6397 -0.7548 -0.1219 0.0790 -0.7049 -0.5057 -0.3224 0.3786 0.2619 0.3174 0 0.9114 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

S = 31.0148 0 0 14.1290 0 0 0 0 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 16

V = 0.9920 0.1259 -0.1259 0.9920 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

Confirm the relation `A = U*S*V'`

.

U*S*V'

ans = 5.3770 3.1873 18.3390 -13.0773 -22.5890 -4.3360 8.6221 3.4257 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 99 FractionLength: 76

### Economy-Size Decomposition

Calculate the complete and economy-size decomposition of a rectangular fixed-point matrix.

Define the fixed-point matrix `A`

.

```
m = 5;
n = 3;
rng('default');
A = fi(10*randn(m,n))
```

A = 5.3770 -13.0762 -13.4980 18.3379 -4.3359 30.3496 -22.5879 3.4258 7.2539 8.6211 35.7832 -0.6309 3.1875 27.6953 7.1465 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 16 FractionLength: 9

Compute the complete decomposition.

[U,S,V] = svd(A)

U = 0.3081 -0.0950 0.4507 0.7929 0.2534 -0.1437 0.9533 -0.0877 0.2415 -0.0675 -0.0224 -0.2106 -0.8423 0.4887 -0.0831 -0.7299 -0.1909 0.2773 0.2722 -0.5290 -0.5926 -0.0375 -0.0541 0 0.8028 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

S = 48.4483 0 0 0 36.6720 0 0 0 26.9112 0 0 0 0 0 0 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 16

V = -0.1786 0.5444 0.8196 -0.9497 -0.3131 0.0009 -0.2571 0.7783 -0.5729 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

Compute the economy-size decomposition.

`[U,S,V] = svd(A,"econ")`

U = 0.3081 -0.0950 0.4507 -0.1437 0.9533 -0.0878 -0.0224 -0.2106 -0.8423 -0.7299 -0.1909 0.2773 -0.5926 -0.0374 -0.0541 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

S = 48.4485 0 0 0 36.6720 0 0 0 26.9112 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 16

V = -0.1786 0.5444 0.8196 -0.9497 -0.3131 0.0010 -0.2571 0.7783 -0.5729 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

Use the expected result `A = U*S*V'`

to determine the relative error of the calculation.

relativeError = norm(double(U*S*V'-A))/norm(double(A))

relativeError = 1.0359e-05

### Control Singular Value Output Format

Create a 3-by-3 magic square matrix and calculate the singular value decomposition. By default, the `svd`

function returns the singular values in a diagonal matrix when you specify multiple outputs.

Define the matrix `A`

.

m = 3; n = m; A = fi(magic(m))

A = 8 1 6 3 5 7 4 9 2 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 16 FractionLength: 11

Compute the singular value decomposition.

[U,S,V] = svd(A)

U = 0.5774 -0.7071 -0.4083 0.5773 -0.0000 0.8165 0.5773 0.7071 -0.4082 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

S = 15.0000 0 0 0 6.9283 0 0 0 3.4642 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 16

V = 0.5774 -0.4082 -0.7071 0.5773 0.8165 0.0000 0.5773 -0.4082 0.7071 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

Specify the `"vector"`

option to return the singular values in a column vector.

`[U,S,V] = svd(A,"vector")`

U = 0.5774 -0.7071 -0.4083 0.5773 -0.0000 0.8165 0.5773 0.7071 -0.4082 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

S = 15.0000 6.9283 3.4642 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 16

V = 0.5774 -0.4082 -0.7071 0.5773 0.8165 0.0000 0.5773 -0.4082 0.7071 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

If you specify one output argument, such as `S = svd(A)`

, then `svd`

switches behavior to return the singular values in a column vector by default. In that case, you can specify the `"matrix"`

option to return the singular values as a diagonal matrix.

## Input Arguments

`A`

— Input matrix

matrix

Input matrix, specified as a matrix. `A`

can be a fixed-point or
scaled double `fi`

data type.

**Data Types: **`fi`

**Complex Number Support: **Yes

`sigmaForm`

— Output format of singular values

`"vector"`

| `"matrix"`

Output format of singular values, specified as one of these values:

`"vector"`

—`S`

is a column vector. This behavior is the default when you specify one output,`S = svd(A)`

.`"matrix"`

—`S`

is a diagonal matrix. This behavior is the default when you specify multiple outputs,`[U,S,V] = svd(A)`

.

**Example: **`[U,S,V] = svd(X,"vector")`

returns `S`

as
a column vector instead of a diagonal matrix.

**Example: **`S = svd(X,"matrix")`

returns `S`

as a
diagonal matrix instead of a column vector.

**Data Types: **`char`

| `string`

## Output Arguments

`U`

— Left singular vectors

matrix

Left singular vectors, returned as the columns of a matrix.

The fixed-point data type is adjusted to avoid overflow and increase precision. For more information, see Algorithms.

`S`

— Singular values

diagonal matrix | column vector

Singular values, returned as a diagonal matrix or column vector. The singular values are nonnegative and returned in decreasing order.

The fixed-point data type is adjusted to avoid overflow and increase precision. For more information, see Algorithms.

`V`

— Right singular vectors

matrix

Right singular vectors, returned as the columns of a matrix.

The fixed-point data type is adjusted to avoid overflow and increase precision. For more information, see Algorithms.

## Tips

To have full control over the fixed-point types, use the `fixed.svd`

function.

## Algorithms

### Data Type Propagation

The `svd`

function adjusts the data type of a fixed-point input to
avoid overflow and increase precision. The fraction length of the singular vectors
`S`

is adjusted to a minimum of `16`

, and the word
length is increased to avoid overflow with a minimum of `32`

. The word
length of the left and right singular vectors `U`

and
`V`

are the same as the word length of `S`

. The
fraction length of `U`

and `V`

is two less than the
word length.

### Golub-Kahan-Reinsch

The Golub-Kahan-Reinsch algorithm is a sequential method that performs well on serial
computers. For parallel computing, as in FPGA and ASIC applications, use the `fixed.jacobiSVD`

function.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

`svd`

generates efficient, purely integer C code.

## Version History

**Introduced in R2022b**

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