power, .^

Element-wise power

Syntax

C = A.^B

C = power(A,B)

Description

C = A.^B raises each element of A to the corresponding powers in B. The sizes of A and B must be the same or be compatible.

If the sizes of A and B are compatible, then the two arrays implicitly expand to match each other. For example, if one of A or B is a scalar, then the scalar is combined with each element of the other array. Also, vectors with different orientations (one row vector and one column vector) implicitly expand to form a matrix.

C = power(A,B) is an alternate way to execute A.^B, but is rarely used. It enables operator overloading for classes.

Examples

Square Each Element of Vector

Create a vector, A, and square each element.

A = 1:5;
C = A.^2

C = 1×5

     1     4     9    16    25

Find Inverse of Each Matrix Element

Create a matrix, A, and take the inverse of each element.

A = [1 2 3; 4 5 6; 7 8 9];
C = A.^-1

C = 3×3

    1.0000    0.5000    0.3333
    0.2500    0.2000    0.1667
    0.1429    0.1250    0.1111

An inversion of the elements is not equal to the inverse of the matrix, which is instead written A^-1 or inv(A).

Row Vector to Power of Column Vector

Create a 1-by-2 row vector and a 3-by-1 column vector and raise the row vector to the power of the column vector.

a = [2 3];
b = (1:3)';
a.^b

ans = 3×2

     2     3
     4     9
     8    27

The result is a 3-by-2 matrix, where each (i,j) element in the matrix is equal to a(j) .^ b(i):

$$a=\left[a_1{a}_2\right],b=\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right],a.ˆb=\left[\begin{array}{cc}{a_1}^{b_1}& {a_2}^{b_1}\\ {a_1}^{b_2}& {a_2}^{b_2}\\ {a_1}^{b_3}& {a_2}^{b_3}\end{array}\right].$$

Find Roots of Number

Calculate the roots of -1 to the 1/3 power.

A = -1;
B = 1/3;
C = A.^B

C = 
0.5000 + 0.8660i

For negative base A and noninteger B, the power function returns complex results.

Use the nthroot function to obtain the real roots.

C = nthroot(A,3)

C = 
-1

Raise Table to Power of Another Table

Since R2023a

Create two tables and raise the first table to the power of the second. The row names (if present in both) and variable names must be the same, but do not need to be in the same orders. Rows and variables of the output are in the same orders as the first input.

A = table([1;2],[3;4],VariableNames=["V1","V2"],RowNames=["R1","R2"])

A=2×2 table
          V1    V2
          __    __

    R1    1     3 
    R2    2     4 

B = table([4;2],[3;1],VariableNames=["V2","V1"],RowNames=["R2","R1"])

B=2×2 table
          V2    V1
          __    __

    R2    4     3 
    R1    2     1 

C = A .^ B

C=2×2 table
          V1    V2 
          __    ___

    R1    1       9
    R2    8     256

Input Arguments

A, B — Operands

scalars | vectors | matrices | multidimensional arrays | tables | timetables

Operands, specified as scalars, vectors, matrices, multidimensional arrays, tables, or timetables. 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.

• Operands with an integer data type cannot be complex.

Inputs that are tables or timetables must meet the following conditions: (since R2023a)

• If an input is a table or timetable, then all its variables must have data types that support the operation.

• If only one input is a table or timetable, then the other input must be a numeric or logical array.

• If both inputs are tables or timetables, then:

  • Both inputs must have the same size, or one of them must be a one-row table.

  • Both inputs must have variables with the same names. However, the variables in each input can be in a different order.

  • If both inputs are tables and they both have row names, then their row names must be the same. However, the row names in each input can be in a different order.

  • If both inputs are timetables, then their row times must be the same. However, the row times in each input can be in a different order.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | char | table | timetable
Complex Number Support: Yes

More About

IEEE Compliance

For real inputs, power has a few behaviors that differ from those recommended in the IEEE^®-754 Standard.

	MATLAB®	IEEE
power(1,NaN)	NaN	1
power(NaN,0)	NaN	1

Tips

• Using times or .* for repeated multiplications can be faster than using power or .^ with integer exponents. However, round-off errors are greater with repeated multiplications because the errors accumulate. This is particularly important for the double data type. For data types where round-off errors are not an issue, such as int64, you can use times. For example, if you define x = int64(randi([1, 10], 10000, 1));, then the operation y = x.*x.*x; is faster than y = x.^3;.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The power function fully supports tall arrays. For more information, see Tall Arrays.

C/C++ Code Generation

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

Usage notes and limitations:

• When both X and Y are real, but power(X,Y) or X .^ Y is complex, simulation produces an error and generated code returns NaN. To obtain the correct complex result, convert value X complex by using complex(X). For example, power(complex(X),Y) or complex(X).^Y.

• Code generation does not support sparse matrix inputs for this function.

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

• When both X and Y are real, but power(X,Y) or X .^ Y is complex, simulation produces an error and generated code returns NaN. To obtain the correct complex result, convert value X complex by using complex(X). For example, power(complex(X),Y) or complex(X).^Y.

• Code generation does not support sparse matrix inputs for this function.

HDL Code Generation

Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.

Both inputs must be scalar, and the exponent input, k, must be an integer.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

The power function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The power function supports GPU array input with these usage notes and limitations:

• If base A or exponent B are sparse, B must be a scalar.

• If the output of the function running on the GPU can be complex, then you must explicitly specify its input arguments as complex. For more information, see Work with Complex Numbers on a GPU (Parallel Computing Toolbox).

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

The power function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2026a: Improved performance with integer exponents

The power function (.^) shows improved performance when computing element-wise powers with integer exponents. For example, this code raises every element in a 5000-by-5000 array to the power of 3. The code is about 3.8x faster than in the previous release.

function t = timingPower
x = rand(5000);
y = @() x.^3;
t = timeit(y);
end

The approximate execution times are:

R2025b: 0.80 s

R2026a: 0.21 s

The code was timed on a Windows^® 11, AMD EPYC™ 74F3 24-Core Processor @ 3.19 GHz test system by calling the timingPower function.

Compatibility Considerations

This performance improvement arises from code changes that also result in slightly different round-off behavior in double precision, leading to more accurate results. For example, this code now returns a result that is more accurate to the 15th digit after the decimal point in long scientific notation.

format longE
y = 0.4543.^3

y =
     9.376229100700000e-02

Previously, the same code returned this result.

format longE
y = 0.4543.^3

y =
     9.376229100699998e-02

Note that MATLAB converts a decimal number input, like 0.4543, to the nearest representable double-precision binary value, which might not be exactly equal to the original decimal input. For this reason, although the .^ operator is now more accurate in double precision, the result might differ from a calculation performed using the exact representation of the operands.

R2026a: Combining exponentiation operators with unary operators or logical negations without parentheses will not be supported in a future release

When you specify a sequence of exponentiation operators combined with unary operators or logical negations in the exponents, you will have to use parentheses to explicitly specify the order of operations. Omitting parentheses in operations that include ^-, .^-, ^+, .^+, ^~, or .^~ when chained with other exponentiation operators will result in an error in a future release.

For example, the result of an operation such as y = 4.^-3.^-2 depends on the order in which exponentiation and negation are performed. For this reason, use parentheses to explicitly specify the intended order of operations. For example:

y = (4.^(-3)).^(-2)

y =
    4096

y = 4.^(-(3.^(-2)))

y =
    0.8572

y = 4.^((-3).^(-2))

y =
    1.1665

R2023a: Perform operations directly on tables and timetables

The power operator supports operations directly on tables and timetables without indexing to access their variables. All variables must have data types that support the operation. For more information, see Direct Calculations on Tables and Timetables.

R2016b: Implicit expansion change affects arguments for operators

Starting in R2016b with the addition of implicit expansion, some combinations of arguments for basic operations that previously returned errors now produce results. For example, you previously could not add a row and a column vector, but those operands are now valid for addition. In other words, an expression like [1 2] + [1; 2] previously returned a size mismatch error, but now it executes.

If your code uses element-wise operators and relies on the errors that MATLAB previously returned for mismatched sizes, particularly within a try/catch block, then your code might no longer catch those errors.

For more information on the required input sizes for basic array operations, see Compatible Array Sizes for Basic Operations.