mpower - Matrix power - MATLAB (original) (raw)

Syntax

Description

C = [A](#btx%5F%5F27-1-A)^[B](#btx%5F%5F27-1-A)computes A to the B power and returns the result in C.

example

C = mpower([A](#btx%5F%5F27-1-A),[B](#btx%5F%5F27-1-A)) is an alternate way to execute A^B, but is rarely used. It enables operator overloading for classes.

Examples

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

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

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

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 = 2×2

-0.7071 0.7071 0.7071 0.7071

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

C = 2×2

1.2500    0.7500
0.7500    1.2500

Input Arguments

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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 exponentB 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 exponentB 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

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Usage notes and limitations:

• If A is a 2-by-2 or larger matrix and B isInf or -Inf, thenA^B returns a matrix of NaN values.
• For A^b, if b is a noninteger scalar, then at least one of A orb must be complex.
• Code generation does not support sparse matrix inputs for this function.

Usage notes and limitations:

• If A is a 2-by-2 or larger matrix andB 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 orb must be complex.
• Code generation does not support sparse matrix inputs for this function.

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

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

• If base A is a sparse matrix, exponentB 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).

Version History

Introduced before R2006a

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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.