# min

Minimum elements of symbolic input

## Description

example

M = min(A) returns the minimum elements of a symbolic input.

• If A is a vector, then min(A) returns the minimum of A.

• If A is a matrix, then min(A) is a row vector containing the minimum value of each column.

For an input A that contains symbolic expression, the symbolic min function returns an unevaluated expression that is reduced by eliminating arguments that do not represent minimum values. The output may have another argument that represents the condition for the symbolic variable. For example, syms x; min([1 x]) returns the output min([1, x], [], 2, 'Omitnan', ~in(x, 'real')) in the Command Window since x is complex.

example

M = min(A,[],nanflag) specifies whether to include or omit NaN values in the calculation. For example, min(A,[],'includenan') includes all NaN values in A while min(A,[],'omitnan') ignores them.

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M = min(A,[],dim) returns the minimum element along dimension dim. For example, if A is a matrix, then min(A,[],2) is a column vector containing the minimum value of each row.

M = min(A,[],dim,nanflag) also specifies the dimension to operate along when using the nanflag option.

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M = min(A,[],'all') returns the minimum over all elements of A.

M = min(A,[],'all',nanflag) computes the minimum over all elements of A when using the nanflag option.

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C = min(A,B) returns an array with the smallest elements taken from A or B.

C = min(A,B,nanflag) also specifies how to treat NaN values.

## Examples

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Create a symbolic vector of real elements. Find the smallest real element using the symbolic min function.

syms x real
A = [23 42 37 18 x];
M = min(A)
M = $\mathrm{min}\left(\left[18,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)$

The symbolic min function returns an unevaluated expression. The expression is reduced by eliminating arguments that do not represent minimum values.

Create a symbolic vector and compute its minimum, excluding NaN values.

syms x positive
A = [1.75 x 3.25 -2.5 NaN 0.5 NaN 0.2 -4*x];
M = min(A,[],'omitnan')
M =

$\mathrm{min}\left(\left[-\frac{5}{2},-4 x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)$

min(A) will also produce this result since 'omitnan' is the default option.

Use the 'includenan' flag to return NaN.

M = min(A,[],'includenan')
M = $\mathrm{NaN}$

Create a symbolic matrix and find the smallest element in each column.

syms x real
A = [1 x -0.5; -2 1 x]
A =

$\left(\begin{array}{ccc}1& x& -\frac{1}{2}\\ -2& 1& x\end{array}\right)$

M = min(A)
M =

$\left(\begin{array}{ccc}-2& \mathrm{min}\left(\left[1,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)& \mathrm{min}\left(\left[-\frac{1}{2},x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)\end{array}\right)$

Create a symbolic matrix and find the smallest element in each row.

syms x real
A = [1 x -0.5; -2 1 x]
A =

$\left(\begin{array}{ccc}1& x& -\frac{1}{2}\\ -2& 1& x\end{array}\right)$

M = min(A,[],2)
M =

$\left(\begin{array}{c}\mathrm{min}\left(\left[-\frac{1}{2},x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)\\ \mathrm{min}\left(\left[-2,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)\end{array}\right)$

Create a symbolic matrix.

syms x real
A = [1 x -0.5; -2 1 x]
A =

$\left(\begin{array}{ccc}1& x& -\frac{1}{2}\\ -2& 1& x\end{array}\right)$

To find the minimum over all dimensions of a matrix, use the 'all' option.

M = min(A,[],'all')
M = $\mathrm{min}\left(\left[-2,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)$

Create two symbolic matrices with complex elements. Find the smallest elements taken from the two matrices, which are complex values with the smallest magnitude.

syms x y
A = [x 2+1i; 3 4i; -5 y]
A =

$\left(\begin{array}{cc}x& 2+\mathrm{i}\\ 3& 4 \mathrm{i}\\ -5& y\end{array}\right)$

B = [1 y; 2i 1+1i; -3 x]
B =

$\left(\begin{array}{cc}1& y\\ 2 \mathrm{i}& 1+\mathrm{i}\\ -3& x\end{array}\right)$

C = min(A,B)
C =

$\left(\begin{array}{cc}\mathrm{min}\left(\left[1,x\right],\left[\right],2,\mathrm{"omitnan"},x\notin \mathbb{R}\right)& \mathrm{min}\left(\left[2+\mathrm{i},y\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symtrue}\right)\\ 2 \mathrm{i}& 1+\mathrm{i}\\ -3& \mathrm{min}\left(\left[x,y\right],\left[\right],2,\mathrm{"omitnan"},x\notin \mathbb{R}\vee y\notin \mathbb{R}\right)\end{array}\right)$

Define the following expression by using the symbolic min function. Assume that the variable $x$ is real.

$\mathit{f}\left(\mathit{x}\right)=\left\{\begin{array}{ll}0& \mathit{x}>1\\ \sqrt{1-\mathit{x}}& \mathit{x}<1\end{array}$

syms x real
f(x) = sqrt(1 - min(x,1))
f(x) =

$\sqrt{1-\mathrm{min}\left(\left[1,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)}$

Plot the expression by using fplot.

fplot(f,[-5 5])

## Input Arguments

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Input array, specified as a symbolic expression, vector, or matrix of symbolic expressions.

• If A is complex, then min(A) returns the complex number with the smallest magnitude. If magnitudes are equal, then min(A) returns the value with the smallest magnitude and the smallest phase angle.

• If A is a scalar, then min(A) returns A.

• If A is a 0-by-0 empty array, then min(A) is an empty array as well.

Data Types: sym | single | double
Complex Number Support: Yes

NaN condition, specified as one of these values:

• 'omitnan' — Ignore all NaN values in the input. If all elements are NaN, then min returns the first one.

• 'includenan' — Include the NaN values in the input for the calculation.

Data Types: char

Dimension to operate along, specified as a positive integer scalar. If no value is specified, then the default is the first array dimension whose size does not equal 1.

Dimension dim indicates the dimension whose length reduces to 1. The size(M,dim) is 1, while the sizes of all other dimensions remain the same, unless size(A,dim) is 0. If size(A,dim) is 0, then min(A,dim) returns an empty array with the same size as A.

Consider a two-dimensional input array, A:

• If dim = 1, then min(A,[],1) returns a row vector containing the smallest element in each column.

• If dim = 2, then min(A,[],2) returns a column vector containing the smallest element in each row.

min returns A if dim is greater than ndims(A).

Additional input array, specified as a symbolic expression, vector, or matrix of symbolic expressions. Inputs A and B must either be the same size or have sizes that are compatible (for example, A is an M-by-N matrix and B is a scalar or 1-by-N row vector). For more information, see Compatible Array Sizes for Basic Operations.

Data Types: sym | single | double
Complex Number Support: Yes

## Output Arguments

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Minimum values, returned as a symbolic expression, vector, or matrix of symbolic expressions. size(M,dim) is 1, while the sizes of all other dimensions match the size of the corresponding dimension in A, unless size(A,dim) is 0. If size(A,dim) is 0, then M is an empty array with the same size as A.

Minimum elements from A or B, returned as a symbolic expression, vector, or matrix of symbolic expressions. The size of C is determined by implicit expansion of the dimensions of A and B. For more information, see Compatible Array Sizes for Basic Operations.

## Version History

Introduced in R2021a

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