mpower, ^

Matrix power

Syntax

C = A^B

C = mpower(A,B)

Description

C = A^B computes A to the B power and returns the result in C.

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

Examples

Square a Matrix

Create a 2-by-2 matrix and square it.

A = [1 2; 3 4];
C = A^2

C = 2×2

     7    10
    15    22

The syntax A^2 is equivalent to A*A.

Matrix Exponents

Create a 2-by-2 matrix and use it as the exponent for a scalar.

B = [0 1; 1 0];
C = 2^B

C = 2×2

    1.2500    0.7500
    0.7500    1.2500

Compute C by first finding the eigenvalues D and eigenvectors V of the matrix B.

[V,D] = eig(B)

V = 2×2

   -0.7071    0.7071
    0.7071    0.7071

D = 2×2

    -1     0
     0     1

Next, use the formula 2^B = V*2^D/V to compute the power.

C = V*2^D/V

C = 2×2

    1.2500    0.7500
    0.7500    1.2500

Input Arguments

A, B — Operands

scalar | matrix

Operands, specified as scalars or matrices. Inputs A and B must be one of the following combinations:

• Base A and exponent B are both scalars, in which case A^B is equivalent to A.^B.

• Base A is a square matrix and exponent B is a scalar. If B is a positive integer, the power is computed by repeated squaring. For other values of B the calculation uses an eigenvalue decomposition (for most matrices) or a Schur decomposition (for defective matrices).

• Base A is a scalar and exponent B is a square matrix. The calculation uses an eigenvalue decomposition.

Operands with an integer data type cannot be complex.

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

Tips

• MATLAB^® computes X^(-1) and inv(X) in the same manner, and both are subject to the same limitations. For more information, see inv.

References

[1] Higham, Nicholas J., and Lijing Lin. “An Improved Schur--Padé Algorithm for Fractional Powers of a Matrix and Their Fréchet Derivatives.” SIAM Journal on Matrix Analysis and Applications 34, no. 3 (January 2013): 1341–1360. https://doi.org/10.1137/130906118.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• If A is a 2-by-2 or larger matrix and B is Inf or -Inf, then A^B returns a matrix of NaN values.

• For A^b, if b is a noninteger scalar, then at least one of A or b must be complex.

• 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:

• If A is a 2-by-2 or larger matrix and B is Inf or -Inf, then A^B returns a matrix of NaN values.

• For A^b, if b is a noninteger scalar, then at least one of A or b must be complex.

• 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.

This 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 mpower function supports GPU array input with these usage notes and limitations:

• If base A is a sparse matrix, exponent B must be a nonnegative real integer.

• If A is a sparse scalar, B must be a scalar.

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™.

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

Version History

Introduced before R2006a

R2021b: Improved algorithm for defective matrices

The algorithm for defective matrices raised to a real power was improved. In previous releases, mpower used an algorithm based on eigenvalue decomposition for these inputs that can return incorrect results for defective matrices. The new algorithm for defective matrices is instead based on the Schur decomposition.