# fixed.svd

## Syntax

## Description

returns
the singular values of matrix
`S`

= fixed.svd(`A`

)`A`

in descending order.

`[___] = fixed.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*—`fixed.svd(A,"econ")`

is equivalent to`fixed.svd(A)`

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

are computed, and`S`

is*m*-by-*m*.

`[___] = fixed.svd(`

produces a
different economy-size decomposition of `A`

,0)`A`

. If `A`

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

*m*>*n*—`fixed.svd(A,0)`

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

.*m*<=*n*—`fixed.svd(A,0)`

is equivalent to`fixed.svd(A)`

.

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

option instead.

`[___] = fixed.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 scaled-double matrix.

A = [1 0 1; -1 -2 0; 0 1 -1];

Define fixed-point types that will never 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 and another additional bit for intermediate CORDIC growth. Compute the fraction length based on the integer length and the desired word length.

svdUpperBound = fixed.singularValueUpperBound(3,3,max(abs(A(:)))); integerLength = ceil(log2(svdUpperBound)) + 2; wordLength = 16; fractionLength = wordLength - integerLength;

Cast the matrix `A`

to the scaled-double type.

T.A = fi([],1,wordLength,fractionLength,'DataType','ScaledDouble'); A = cast(A,'like',T.A)

A = 1 0 1 -1 -2 0 0 1 -1 DataTypeMode: Scaled double: binary point scaling Signedness: Signed WordLength: 16 FractionLength: 11

Compute the singular values.

s = fixed.svd(A)

s = 2.4605 1.6996 0.2392 DataTypeMode: Scaled double: binary point scaling Signedness: Signed WordLength: 16 FractionLength: 11

The singular values are returned in a column vector in decreasing order, and have the same data type as `A`

.

### 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 = 10*randn(m,n);
```

Define fixed-point types that will never overflow. First, use the `fixed.singularValueUpperBoun`

d 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 and another additional bit for intermediate CORDIC growth. Compute the fraction length based on the integer length and the desired word length.

svdUpperBound = fixed.singularValueUpperBound(m,n,max(abs(A(:)))); integerLength = ceil(log2(svdUpperBound)) + 2; wordLength = 32; fractionLength = wordLength - integerLength;

Cast the matrix `A`

to the signed fixed-point type.

T.A = fi([],1,wordLength,fractionLength,'DataType','Fixed'); A = cast(A,'like',T.A)

A = 5.3767 3.1877 18.3389 -13.0769 -22.5885 -4.3359 8.6217 3.4262 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 24

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

.

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

U = 0.1590 0.2717 -0.9387 -0.1403 0.6397 -0.7548 -0.1219 0.0790 -0.7049 -0.5057 -0.3224 0.3787 0.2619 0.3174 0 0.9114 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

S = 31.0141 0 0 14.1289 0 0 0 0 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 24

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 = 5.3767 3.1877 18.3389 -13.0769 -22.5885 -4.3359 8.6217 3.4262 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 99 FractionLength: 84

### Economy-Size Decomposition

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

Define the matrix `A`

.

A = [1 2; 3 4; 5 6; 7 8];

Define fixed-point types that will never 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 and another additional bit for intermediate CORDIC growth. Compute the fraction length based on the integer length and the desired word length.

svdUpperBound = fixed.singularValueUpperBound(4,2,max(abs(A(:)))); integerLength = ceil(log2(svdUpperBound)) + 2; wordLength = 32; fractionLength = wordLength - integerLength;

Cast the matrix `A`

to the signed fixed-point type.

T.A = fi([],1,wordLength,fractionLength,'DataType','Fixed'); A = cast(A,'like',T.A)

A = 1 2 3 4 5 6 7 8 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 25

Compute the complete decomposition.

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

U = -0.1525 0.8226 -0.4082 0.3651 -0.3499 0.4214 0.8165 -0.1826 -0.5474 0.0201 -0.4082 -0.7303 -0.7448 -0.3812 0 0.5477 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

S = 14.2691 0 0 0.6268 0 0 0 0 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 25

V = -0.6414 -0.7672 -0.7672 0.6414 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

Compute the economy-size decomposition.

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

U = -0.1525 0.8226 -0.3499 0.4214 -0.5474 0.0201 -0.7448 -0.3812 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

S = 14.2691 0 0 0.6268 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 25

V = -0.6414 -0.7672 -0.7672 0.6414 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

Since `A`

is 4-by-2,` fixed.svd(A,"econ")`

returns fewer columns in `U`

and fewer rows in `S`

compared to a complete decomposition. Extra rows of zeros in `S`

are excluded, along with the corresponding columns in `U`

that would multiply with those zeros in the expression` A = U*S*V'`

.

### Control Singular Value Output Format

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

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

Define the matrix `A`

.

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

Define fixed-point types that will never 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 and another additional bit for intermediate CORDIC growth. Compute the fraction length based on the integer length and the desired word length.

svdUpperBound = fixed.singularValueUpperBound(m,n,max(abs(A(:)))); integerLength = ceil(log2(svdUpperBound)) + 2; wordLength = 32; fractionLength = wordLength - integerLength;

Cast the matrix `A`

to the signed fixed-point type.

T.A = fi([],1,wordLength,fractionLength,'DataType','Fixed'); A = cast(A,'like',T.A)

A = 8 1 6 3 5 7 4 9 2 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 25

Compute the singular value decomposition.

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

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

S = 15.0000 0 0 0 6.9282 0 0 0 3.4641 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 25

V = 0.5774 -0.4082 -0.7071 0.5774 0.8165 -0.0000 0.5774 -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] = fixed.svd(A,"vector")`

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

S = 15.0000 6.9282 3.4641 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 25

V = 0.5774 -0.4082 -0.7071 0.5774 0.8165 -0.0000 0.5774 -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 = fixed.svd(A)`

, then `fixed.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.

### Compute Fixed-Point Singular Value Decomposition and Generate Code

Compute the fixed-point singular value decomposition, verify the results, and generate purely integer C code.

Define the input matrix `A`

.

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

The `fixed.svd`

function also accepts complex inputs.

A = 10*complex(rand(m,n),rand(m,n));

Define fixed-point types that will never overflow. Use the `fixed.singularValueUpperBound`

function to determine the upper bound on the singular values. Define the integer length based on the value of the upper bound, with one additional bit for the sign and another additional bit for intermediate CORDIC growth. Compute the fraction length based on the integer length and the desired word length.

svdUpperBound = fixed.singularValueUpperBound(m,n,max(abs(A(:)))); integerLength = ceil(log2(svdUpperBound)) + 2; wordLength = 32; fractionLength = wordLength - integerLength;

Specify the desired data type for the input matrix `A`

.

dataType = 'Fixed'; T.A = fi([],1,wordLength,fractionLength,'DataType',dataType); disp(T.A)

[] DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 23

Cast the matrix `A`

to the signed fixed-point type.

`A = cast(A,'like',T.A);`

Generate a MATLAB® executable (MEX) file for execution speed. Use the `"econ"`

flag to compute the economy-size singular-value decomposition. Use the `"vector"`

flag to return the singular values as a vector, `s`

. The flags must be constant for code generation. Use the `-nargout 3`

flag to indicate to the `codegen`

function that it is to generate code for the three-output syntax.

codegen fixed.svd -o fixedSVD_mex -args {A,coder.Constant("econ"),coder.Constant("vector")} -nargout 3

Code generation successful.

Run the MEX file.

[U,s,V] = fixedSVD_mex(A,"econ","vector")

U = 0.2509 + 0.1236i 0.1980 + 0.1578i 0.1745 + 0.0268i -0.0755 + 0.2443i 0.1601 + 0.2073i -0.3227 - 0.1684i 0.1420 - 0.3385i -0.3686 + 0.1899i 0.2937 + 0.1868i 0.0574 - 0.2108i 0.0884 - 0.0633i 0.1079 - 0.0866i 0.2071 + 0.3019i 0.0678 + 0.1740i -0.3790 + 0.1518i 0.3808 - 0.1483i 0.2262 + 0.2405i 0.5884 - 0.1889i -0.0693 - 0.3624i -0.0547 + 0.1581i 0.2435 + 0.2111i -0.0568 + 0.3536i -0.1593 + 0.1445i -0.3714 + 0.0026i 0.2195 + 0.2411i -0.0713 - 0.2517i -0.2848 + 0.2641i -0.4055 + 0.0481i 0.1461 + 0.3184i -0.1657 + 0.0477i 0.0478 + 0.2812i 0.2972 + 0.0396i 0.2204 + 0.1354i -0.2868 - 0.0745i 0.3628 - 0.2391i 0.0565 - 0.1251i 0.1863 + 0.2334i -0.1495 + 0.0491i 0.1422 - 0.1789i 0.3594 - 0.0775i DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

s = 45.9444 16.1174 10.7897 8.3153 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 23

V = 0.5637 + 0.0000i -0.4038 + 0.0000i -0.2061 + 0.0000i 0.6904 + 0.0000i 0.4261 + 0.0228i -0.3782 - 0.4565i 0.0076 + 0.3326i -0.5667 - 0.1863i 0.4980 + 0.0276i 0.3223 + 0.4193i -0.5081 - 0.2300i -0.3698 + 0.1540i 0.5014 + 0.0041i 0.4531 - 0.0243i 0.7280 - 0.0758i 0.0729 - 0.0402i DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 30

Verify the singular values. Since singular values are unique, you can use the svd function to verify that `fixed.svd`

gives a comparable result within the precision of the selected fixed-point type.

sExpected = svd(double(A))

`sExpected = `*4×1*
45.9444
16.1174
10.7897
8.3153

singularValueRelativeError = norm(double(s)-double(sExpected))/norm(double(sExpected))

singularValueRelativeError = 4.8247e-07

Singular vectors are not unique. You can verify the singular vectors by confirming that `A ≈ U*S*V'`

and that the singular vector matrices are orthonormal.

First, expand the singular value vector `s`

into matrix `S`

.

S = zeros(size(U,2),size(V,2),'like',s); for i = 1:min(m,n) S(i,i) = s(i); end S

S = 45.9444 0 0 0 0 16.1174 0 0 0 0 10.7897 0 0 0 0 8.3153 DataTypeMode: Fixed-point: binary point scaling Signedness: Signed WordLength: 32 FractionLength: 23

Verify that `U*S*V'`

is approximately equal to `A`

.

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

decompositionRelativeError = 4.5856e-07

`U`

and `V`

are orthonormal. Verify that `U'U`

and `V'V`

are approximately equal to the identity matrix.

UtransposeU = double(U'*U)

`UtransposeU = `*4×4 complex*
1.0000 + 0.0000i -0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 - 0.0000i
-0.0000 - 0.0000i 1.0000 + 0.0000i -0.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 - 0.0000i -0.0000 - 0.0000i 1.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 - 0.0000i 0.0000 - 0.0000i 1.0000 + 0.0000i

VtransposeV = double(V'*V)

`VtransposeV = `*4×4 complex*
1.0000 + 0.0000i -0.0000 - 0.0000i -0.0000 - 0.0000i -0.0000 - 0.0000i
-0.0000 + 0.0000i 1.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 - 0.0000i
-0.0000 + 0.0000i 0.0000 - 0.0000i 1.0000 + 0.0000i -0.0000 - 0.0000i
-0.0000 + 0.0000i 0.0000 + 0.0000i -0.0000 + 0.0000i 1.0000 + 0.0000i

Generate C code. If the input is fixed point, you can verify that the generated C code consists only of integer types.

cfg = coder.config('lib'); if isfi(A) && isfixed(A) cfg.PurelyIntegerCode = true; end codegen fixed.svd -args {A, coder.Constant("econ"), coder.Constant("vector")} -config cfg -nargout 3 -launchreport

Code generation successful: To view the report, open('codegen/lib/fixed_svd/html/report.mldatx')

The MATLAB code for `fixed.svd`

does not appear in the code generation report because `fixed.svd`

is a MATLAB toolbox function.

## Input Arguments

`A`

— Input matrix

matrix

Input matrix, specified as a matrix. `A`

can be a signed
fixed-point `fi`

, a signed scaled double `fi`

,
`double`

, or `single`

data type.

**Data Types: **`single`

| `double`

| `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 = fixed.svd(A)`

.`"matrix"`

—`S`

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

.

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

returns
`S`

as a column vector instead of a diagonal matrix.

**Example: **`S = fixed.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.

For fixed-point and scaled-double inputs, `U`

is returned as a
signed fixed-point or scaled-double `fi`

with the same word length as
`A`

and fraction length equal to two less than the word length. One
of these integer bits is used for the sign. The other integer bit allows
`+1`

to be represented exactly.

For floating-point input, `U`

has the same data type as
`A`

.

`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 singular values
`S`

have the same data type as `A`

.

`V`

— Right singular vectors

matrix

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

For fixed-point input and scaled-double inputs, `V`

is returned
as a signed fixed-point or scaled-double `fi`

with the same word length
as `A`

and fraction length equal to two less than the word length.
One of these integer bits is used for the sign. The other integer bit allows
`+1`

to be represented exactly.

For floating-point input, `V`

has the same data type as
`A`

. One of these integer bits is used for the sign, and the other
integer bit allows `+1`

to be represented exactly.

## Tips

The `fixed.svd`

function allows full control over the fixed-point
types. `fixed.svd`

computes in-place in the same data type as the input,
which may overflow but will produce efficient code. The `svd`

function adjusts the data type of a fixed-point input
to avoid overflow and increase precision.

## Algorithms

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™.

`fixed.svd`

generates efficient, purely integer C code.

## Version History

**Introduced in R2022b**

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