Conjugate transpose: Difference between revisions

=={{header|Wren}}==

{{libheader|Wren-fmt}}

{{libheader|Wren-complex}}

Although the third example is in fact a unitary matrix, the ''isUnitary'' method of the above module returns false.

This is because the methods in the module work as accurately as they can within the confines of 64-bit floating point arithmetic and don't therefore allow for the small rounding error that occurs due to the use of the irrational number, sqrt(2).

However, if we use the ''almostEquals'' method with the default tolerance of 1.0e-14, then we do get a ''true'' result.

<lang ecmascript>import "/complex" for Complex, CMatrix

import "/fmt" for Fmt

var cm1 = CMatrix.new(

[

[Complex.new(3), Complex.new(2, 1)],

[Complex.new(2, -1), Complex.one ]

]

)

var cm2 = CMatrix.fromReals([ [1, 1, 0], [0, 1, 1], [1, 0, 1] ])

var x = 2.sqrt/2

var cm3 = CMatrix.new(

[

[Complex.new(x), Complex.new(x), Complex.zero],

[Complex.new(0, -x), Complex.new(0, x), Complex.zero],

[Complex.zero, Complex.zero, Complex.imagOne]

]

)

for (cm in [cm1, cm2, cm3]) {

System.print("Matrix:")

Fmt.mprint(cm, 5, 3)

System.print("\nConjugate transpose:")

Fmt.mprint(cm.conjTranspose, 5, 3)

System.print("\nHermitian : %(cm.isHermitian)")

System.print("Normal : %(cm.isNormal)")

System.print("Unitary : %(cm.isUnitary)")

System.print()

}

System.print("For the final example if we use a tolerance of 1e-14:")

var cm4 = cm3 * cm3.conjTranspose

var id = CMatrix.identity(3)

System.print("Unitary : %(cm4.almostEquals(id))")</lang>

{{out}}

<pre>

Matrix:

|3.000 + 0.000i 2.000 + 1.000i|

|2.000 - 1.000i 1.000 + 0.000i|

Conjugate transpose:

|3.000 + 0.000i 2.000 + 1.000i|

|2.000 - 1.000i 1.000 + 0.000i|

Hermitian : true

Normal : true

Unitary : false

Matrix:

|1.000 + 0.000i 1.000 + 0.000i 0.000 + 0.000i|

|0.000 + 0.000i 1.000 + 0.000i 1.000 + 0.000i|

|1.000 + 0.000i 0.000 + 0.000i 1.000 + 0.000i|

Conjugate transpose:

|1.000 + 0.000i 0.000 + 0.000i 1.000 + 0.000i|

|1.000 + 0.000i 1.000 + 0.000i 0.000 + 0.000i|

|0.000 + 0.000i 1.000 + 0.000i 1.000 + 0.000i|

Hermitian : false

Normal : true

Unitary : false

Matrix:

|0.707 + 0.000i 0.707 + 0.000i 0.000 + 0.000i|

|0.000 - 0.707i 0.000 + 0.707i 0.000 + 0.000i|

|0.000 + 0.000i 0.000 + 0.000i 0.000 + 1.000i|

Conjugate transpose:

|0.707 + 0.000i 0.000 + 0.707i 0.000 + 0.000i|

|0.707 + 0.000i 0.000 - 0.707i 0.000 + 0.000i|

|0.000 + 0.000i 0.000 + 0.000i 0.000 - 1.000i|

Hermitian : false

Normal : true

Unitary : false

For the final example if we use a tolerance of 1e-14:

Unitary : true

</pre>