max

Maximum elements of symbolic input

Syntax

M = max(A)

M = max(A,[],"all")

M = max(A,[],dim)

M = max(A,[],___,nanflag)

[M,I] = max(___)

C = max(A,B)

C = max(A,B,nanflag)

Description

M = max(A) returns the maximum elements of a symbolic input.

• If A is a vector, then max(A) returns the maximum of A.

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

For an input A that contains a symbolic expression, the symbolic max function returns an unevaluated expression that excludes elements in A that do not represent maximum values. The output might have another argument that represents the condition for the symbolic variable. For example, syms x; max([1 x]) returns the output max([1, x], [], 2, 'omitnan', ~in(x, 'real')) in the Command Window because x is complex.

M = max(A,[],"all") returns the maximum over all elements of A.

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

M = max(A,[],___,nanflag) specifies whether to include or omit NaN values in A for any of the previous syntaxes. For example, max(A,[],"includenan") includes all NaN values when computing the maximum, while max(A,[],"omitnan") ignores all NaN values.

[M,I] = max(___) also returns the index into the operating dimension that corresponds to the first occurrence of the maximum value of A.

C = max(A,B) returns an array containing the largest elements taken from A or B.

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

Examples

Largest Real Element

Create a symbolic vector of real elements. Find the largest real element using the symbolic max function.

syms x real
A = [23 42 37 18 x];
M = max(A)

M = $$\max \left(\left[42,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)$$

The symbolic max function returns an unevaluated expression. The expression excludes elements in A that do not represent maximum values.

Largest Element Involving NaN

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

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

M = 
$$\max \left(\left[\frac{13}{4},x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)$$

max(A) also produces this result because "omitnan" is the default option.

Use the "includenan" option to return NaN.

M = max(A,[],"includenan")

M = $$\mathrm{NaN}$$

Largest Element in Each Matrix Column

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

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

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

M = max(A)

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

Largest Element in Each Matrix Row

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

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

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

M = max(A,[],2)

M = 
$$\left(\begin{array}{c}\max \left(\left[1,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)\\ \max \left(\left[2,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)\end{array}\right)$$

Largest Element Indices

Create a symbolic matrix A. Find the largest elements in each row of A and return the column indices corresponding to the first occurrence of the maximum value in each row. Because A contains a symbolic variable x with an undetermined value, max cannot explicitly evaluate the maximum value in each row and returns the indices as -1.

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

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

[M,I] = max(A,[],2)

M = 
$$\left(\begin{array}{c}\max \left(\left[1,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)\\ \max \left(\left[2,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)\end{array}\right)$$

I = 2×1

    -1
    -1

Substitute a symbolic number for the variable x in the symbolic matrix A. Find the largest elements in each row again. Here, max is able to evaluate the maximum values and returns the column indices corresponding to the first occurrence of the maximum value in each row.

A = subs(A,x,sqrt(2))

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

[M,I] = max(A,[],2)

M = 
$$\left(\begin{array}{c}\sqrt{2}\\ 2\end{array}\right)$$

I = 2×1

     2
     2

Maximum of Matrix

Create a symbolic matrix. Find the maximum over all dimensions of the matrix by using the max function with the "all" option.

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

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

M = max(A,[],"all")

M = $$\max \left(\left[2,x\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symfalse}\right)$$

Largest Element Comparison

Create two symbolic matrices with complex elements. Find the largest elements taken from the two matrices. The largest elements are the complex values with the largest magnitude.

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

A = 
$$\left(\begin{array}{cc}x& 2+\mathrm{i}\\ 3& 4\hspace{0.17em}\mathrm{i}\\ -5& y\end{array}\right)$$

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

B = 
$$\left(\begin{array}{cc}1& y\\ 2\hspace{0.17em}\mathrm{i}& 1+\mathrm{i}\\ -3& x\end{array}\right)$$

C = max(A,B)

C = 
$$\left(\begin{array}{cc}\max \left(\left[1,x\right],\left[\right],2,\mathrm{"omitnan"},x\notin \mathbb{R}\right)& \max \left(\left[2+\mathrm{i},y\right],\left[\right],2,\mathrm{"omitnan"},\mathrm{symtrue}\right)\\ 3& 4\hspace{0.17em}\mathrm{i}\\ -5& \max \left(\left[x,y\right],\left[\right],2,\mathrm{"omitnan"},x\notin \mathbb{R}\vee y\notin \mathbb{R}\right)\end{array}\right)$$

Plot Expression with Maximum Function

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

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

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

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

Plot the expression by using fplot.

fplot(f,[-5 5])

Input Arguments

A — Input array

symbolic expression | vector of symbolic expressions | matrix of symbolic expressions

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

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

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

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

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

dim — Dimension to operate along

positive integer scalar

Dimension to operate along, specified as a positive integer scalar. If you do not specify a value, then the default is the first array dimension whose size does not equal 1.

Dimension dim indicates the dimension whose length reduces to 1. So, 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 max(A,dim) returns an empty array with the same size as A.

Consider a two-dimensional input array, A:

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

  

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

  

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

nanflag — NaN condition

"omitnan" (default) | "includenan"

NaN condition, specified as one of these values:

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

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

B — Additional input array

symbolic expression | vector of symbolic expressions | matrix of symbolic expressions

Additional input array, specified as a symbolic expression, vector of symbolic expressions, 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

M — Maximum values

symbolic expression | vector of symbolic expressions | matrix of symbolic expressions

Maximum values, returned as a symbolic expression, vector of symbolic expressions, 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.

I — Index

scalar | vector | matrix

Index, returned as a scalar, vector, or matrix. I is the same size as the output M.

If the largest element occurs more than once, then I contains the index to the first occurrence of the value.

Data Types: double

C — Maximum elements from A or B

symbolic expression | vector of symbolic expressions | matrix of symbolic expressions

Maximum 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

R2021a: Find maximum elements of symbolic expressions

The max function can now operate on symbolic expressions involving symbolic variables. In previous releases, such symbolic expressions resulted in an error.