# Question:Why is the performance of LinearAlgebra[MatrixPower] (or MTM[mpower]) so terrible?

## Question:Why is the performance of LinearAlgebra[MatrixPower] (or MTM[mpower]) so terrible?

Maple 2023

In practice, I need calculate the (principal) squareroot of some suitable large matrix exactly (so the desired result should not involve floating-point numbers, otherwise the decomposition will be of no theoretical value as a certificate …). But I find it difficult to do so in Maple. (What about the Efficient Computations - Maple Help (maplesoft.com)?)
Below are some matrices: (They are not contrived academic examples.)

```M__4 := <1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0|0,3,0,-6,0,3,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,-3,0,0,0,0,0,0,0,0,6,0,0,0,0,-3,0|-2,0,7,0,-8,0,3,0,0,0,0,0,0,0,0,1,0,-7,0,2,0,0,0,0,0,0,4,0,3,0,0,0,0,-3,0,0|0,-6,0,13,0,-8,0,1,0,0,0,0,0,0,0,0,6,0,-4,0,4,0,0,0,0,0,0,-1,0,-11,0,0,0,0,6,0|1,0,-8,0,13,0,-6,0,0,0,0,0,0,0,0,4,0,8,0,-1,0,0,0,0,0,0,-11,0,-6,0,0,0,0,6,0,0|0,3,0,-8,0,7,0,-2,0,0,0,0,0,0,0,0,-3,0,-1,0,1,0,0,0,0,0,0,2,0,4,0,0,0,0,-3,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,0,0,1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,4,0,0,0,-2,0,0,0,0,0,0,0,0,-6,0,6,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,-2,0,-1,0,0,0,0,2,0,2,0,0,-1|0,0,0,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,-6,0,-6,0,0,0,0,0,0,6,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,-2,0,-6,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,0,0,-3,0,3,0,0,0,0,0,0,6,0,3,0,-6,0,0,0,0,-3,0,3,0,0,0|-2,0,1,0,4,0,-3,0,0,0,0,0,0,0,0,7,0,-1,0,2,0,0,0,0,0,0,-8,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|2,0,-7,0,8,0,-3,0,0,0,0,0,0,0,0,-1,0,7,0,-2,0,0,0,0,0,0,-4,0,-3,0,0,0,0,3,0,0|0,3,0,-4,0,-1,0,2,0,0,0,0,0,0,0,0,-3,0,7,0,-7,0,0,0,0,0,0,-2,0,8,0,0,0,0,-3,0|-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0|0,-3,0,4,0,1,0,-2,0,0,0,0,0,0,0,0,3,0,-7,0,7,0,0,0,0,0,0,2,0,-8,0,0,0,0,3,0|0,0,0,0,0,0,0,0,-6,0,-1,0,-6,0,6,0,0,0,0,0,0,13,0,8,0,-11,0,0,0,0,-8,0,4,0,0,1|0,0,0,0,0,0,0,0,0,-6,0,-6,0,6,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,-2,0,-3,0,3,0,0,0,0,0,0,8,0,7,0,-4,0,0,0,0,-7,0,-1,0,0,2|0,0,0,0,0,0,0,0,0,6,0,-6,0,0,0,0,0,0,0,0,0,0,-6,0,12,0,0,0,0,0,0,-6,0,0,0,0|0,0,0,0,0,0,0,0,6,0,-1,0,6,0,-6,0,0,0,0,0,0,-11,0,-4,0,13,0,0,0,0,4,0,-8,0,0,1|1,0,4,0,-11,0,6,0,0,0,0,0,0,0,0,-8,0,-4,0,-1,0,0,0,0,0,0,13,0,6,0,0,0,0,-6,0,0|0,0,0,-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,6,0,-11,0,4,0,1,0,0,0,0,0,0,0,0,-6,0,8,0,-8,0,0,0,0,0,0,-1,0,13,0,0,0,0,-6,0|0,0,0,0,0,0,0,0,3,0,2,0,3,0,-3,0,0,0,0,0,0,-8,0,-7,0,4,0,0,0,0,7,0,1,0,0,-2|0,0,0,0,0,0,0,0,0,-2,0,6,0,-2,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,4,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,2,0,-3,0,3,0,0,0,0,0,0,4,0,-1,0,-8,0,0,0,0,1,0,7,0,0,-2|0,0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,-6,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,2,0,1,0,0,0,0,-2,0,-2,0,0,1>:
M__3 := 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```

Unexpectedly, `LinearAlgebra[MatrixPower]` does not work well. For instance:

 > restart;
 >
 (1)
 > M__4 := <1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0|0,3,0,-6,0,3,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,-3,0,0,0,0,0,0,0,0,6,0,0,0,0,-3,0|-2,0,7,0,-8,0,3,0,0,0,0,0,0,0,0,1,0,-7,0,2,0,0,0,0,0,0,4,0,3,0,0,0,0,-3,0,0|0,-6,0,13,0,-8,0,1,0,0,0,0,0,0,0,0,6,0,-4,0,4,0,0,0,0,0,0,-1,0,-11,0,0,0,0,6,0|1,0,-8,0,13,0,-6,0,0,0,0,0,0,0,0,4,0,8,0,-1,0,0,0,0,0,0,-11,0,-6,0,0,0,0,6,0,0|0,3,0,-8,0,7,0,-2,0,0,0,0,0,0,0,0,-3,0,-1,0,1,0,0,0,0,0,0,2,0,4,0,0,0,0,-3,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,0,0,1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,4,0,0,0,-2,0,0,0,0,0,0,0,0,-6,0,6,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,-2,0,-1,0,0,0,0,2,0,2,0,0,-1|0,0,0,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,-6,0,-6,0,0,0,0,0,0,6,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,-2,0,-6,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,0,0,-3,0,3,0,0,0,0,0,0,6,0,3,0,-6,0,0,0,0,-3,0,3,0,0,0|-2,0,1,0,4,0,-3,0,0,0,0,0,0,0,0,7,0,-1,0,2,0,0,0,0,0,0,-8,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|2,0,-7,0,8,0,-3,0,0,0,0,0,0,0,0,-1,0,7,0,-2,0,0,0,0,0,0,-4,0,-3,0,0,0,0,3,0,0|0,3,0,-4,0,-1,0,2,0,0,0,0,0,0,0,0,-3,0,7,0,-7,0,0,0,0,0,0,-2,0,8,0,0,0,0,-3,0|-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0|0,-3,0,4,0,1,0,-2,0,0,0,0,0,0,0,0,3,0,-7,0,7,0,0,0,0,0,0,2,0,-8,0,0,0,0,3,0|0,0,0,0,0,0,0,0,-6,0,-1,0,-6,0,6,0,0,0,0,0,0,13,0,8,0,-11,0,0,0,0,-8,0,4,0,0,1|0,0,0,0,0,0,0,0,0,-6,0,-6,0,6,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,-2,0,-3,0,3,0,0,0,0,0,0,8,0,7,0,-4,0,0,0,0,-7,0,-1,0,0,2|0,0,0,0,0,0,0,0,0,6,0,-6,0,0,0,0,0,0,0,0,0,0,-6,0,12,0,0,0,0,0,0,-6,0,0,0,0|0,0,0,0,0,0,0,0,6,0,-1,0,6,0,-6,0,0,0,0,0,0,-11,0,-4,0,13,0,0,0,0,4,0,-8,0,0,1|1,0,4,0,-11,0,6,0,0,0,0,0,0,0,0,-8,0,-4,0,-1,0,0,0,0,0,0,13,0,6,0,0,0,0,-6,0,0|0,0,0,-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,6,0,-11,0,4,0,1,0,0,0,0,0,0,0,0,-6,0,8,0,-8,0,0,0,0,0,0,-1,0,13,0,0,0,0,-6,0|0,0,0,0,0,0,0,0,3,0,2,0,3,0,-3,0,0,0,0,0,0,-8,0,-7,0,4,0,0,0,0,7,0,1,0,0,-2|0,0,0,0,0,0,0,0,0,-2,0,6,0,-2,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,4,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,2,0,-3,0,3,0,0,0,0,0,0,4,0,-1,0,-8,0,0,0,0,1,0,7,0,0,-2|0,0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,-6,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,2,0,1,0,0,0,0,-2,0,-2,0,0,1>:
 > M__3 := 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 > M__2 := 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 > M__1 := 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 >
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 10 x 10 matrix
 IntegerCharacteristicPolynomial: Used total of  2  prime(s)
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 10 x 10 matrix
 IntegerCharacteristicPolynomial: Used total of  2  prime(s)
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 10 x 10 matrix
 IntegerCharacteristicPolynomial: Used total of  2  prime(s)
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 6 x 6 matrix
 IntegerCharacteristicPolynomial: Used total of  1  prime(s)
 >
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 13 x 13 matrix
 IntegerCharacteristicPolynomial: Used total of  4  prime(s)
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix
 IntegerCharacteristicPolynomial: Used total of  1  prime(s)
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix
 IntegerCharacteristicPolynomial: Used total of  1  prime(s)
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix
 IntegerCharacteristicPolynomial: Used total of  1  prime(s)
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix
 IntegerCharacteristicPolynomial: Used total of  1  prime(s)
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix
 IntegerCharacteristicPolynomial: Used total of  1  prime(s)
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix
 IntegerCharacteristicPolynomial: Used total of  1  prime(s)
 memory used=2.84GiB, alloc change=8.00MiB, cpu time=3.05m, real time=3.01m, gc time=5.84s
 (2)
 >
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 6 x 6 matrix
 IntegerCharacteristicPolynomial: Used total of  1  prime(s)
 IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 6 x 6 matrix
 IntegerCharacteristicPolynomial: Used total of  1  prime(s)
 memory used=2.40GiB, alloc change=0 bytes, cpu time=2.85m, real time=2.82m, gc time=4.08s
 (3)
 >
 CharacteristicPolynomial: working on determinant of minor 2
 CharacteristicPolynomial: working on determinant of minor 3
 CharacteristicPolynomial: working on determinant of minor 4
 CharacteristicPolynomial: working on determinant of minor 5
 CharacteristicPolynomial: working on determinant of minor 6
 CharacteristicPolynomial: working on determinant of minor 2
 CharacteristicPolynomial: working on determinant of minor 3
 CharacteristicPolynomial: working on determinant of minor 4
 CharacteristicPolynomial: working on determinant of minor 5
 CharacteristicPolynomial: working on determinant of minor 6
 CharacteristicPolynomial: working on determinant of minor 2
 CharacteristicPolynomial: working on determinant of minor 3
 CharacteristicPolynomial: working on determinant of minor 4
 CharacteristicPolynomial: working on determinant of minor 5
 CharacteristicPolynomial: working on determinant of minor 6
 CharacteristicPolynomial: working on determinant of minor 2
 CharacteristicPolynomial: working on determinant of minor 3
 CharacteristicPolynomial: working on determinant of minor 4
 CharacteristicPolynomial: working on determinant of minor 5
 CharacteristicPolynomial: working on determinant of minor 6
 CharacteristicPolynomial: working on determinant of minor 2
 CharacteristicPolynomial: working on determinant of minor 3
 CharacteristicPolynomial: working on determinant of minor 4
 CharacteristicPolynomial: working on determinant of minor 5
 CharacteristicPolynomial: working on determinant of minor 6
 >
 Eigenvalues: calling external function
 Eigenvalues: CLAPACK hw_dgeevx_
 memory used=468.66MiB, alloc change=0 bytes, cpu time=18.48s, real time=19.71s, gc time=1.09s
 (4)
 >
 Eigenvalues: calling external function
 Eigenvalues: CLAPACK hw_dgeevx_
 Multiply: copying first Matrix to enable external call
 Multiply: calling external function
 Multiply: NAG hw_f06yaf
 unknown: NAG hw_f06yaf
 unknown: NAG hw_f06yaf
 unknown: NAG hw_f06yaf
 memory used=13.43GiB, alloc change=0 bytes, cpu time=8.66m, real time=8.46m, gc time=26.41s
 (5)
 >
 Eigenvalues: calling external function
 Eigenvalues: CLAPACK hw_dgeevx_
 Multiply: copying first Matrix to enable external call
 Multiply: calling external function
 Multiply: NAG hw_f06yaf
 unknown: NAG hw_f06yaf
 memory used=13.45GiB, alloc change=0 bytes, cpu time=8.39m, real time=8.23m, gc time=26.12s
 (6)
 >
 Eigenvalues: calling external function
 Eigenvalues: CLAPACK hw_dgeevx_
 Multiply: copying first Matrix to enable external call
 Multiply: calling external function
 Multiply: NAG hw_f06yaf
 unknown: NAG hw_f06yaf
 >
 Eigenvalues: calling external function
 Eigenvalues: copying first Matrix, to enable external call
 Eigenvalues: CLAPACK hw_dgeevx_
 memory used=418.17MiB, alloc change=0 bytes, cpu time=16.33s, real time=16.16s, gc time=796.88ms
 (7)
 >
 Eigenvalues: calling external function
 Eigenvalues: copying first Matrix, to enable external call
 Eigenvalues: CLAPACK sw_dgeevx_
 Multiply: copying first Matrix to enable external call
 Multiply: copying second Matrix to enable external call
 Multiply: calling external function
 Multiply: NAG sw_f06yaf
 unknown: copying second Matrix to enable external call
 unknown: calling external function
 unknown: NAG sw_f06yaf

=

Download performance_of_`sqrtm`.mw

The last six numerical experiments all fail to compute the square root. (Accordingly, here it is impossible to convert each numeric element to one of the "simplest" algebraic numbers that approximates it well.) As you can see, if I execute them directly (i.e., without converting each entry to the nearest floating-point value), the elapsed time required to run the procedure is still unacceptable! (Note that they can be evaluated symbolically in fact.) Is this a bug of Maple? And how do I get the desired results in Maple efficiently?

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