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Harry Garst

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These are questions asked by Harry Garst

Adjoint Matrix formulated as permuted H...

Harry Garst 234 Maple 2020
linearalgebra linear-algebra adjoint
December 08 2023
1 0

For a project I need to construct a large symbolic adjoint matrix, hoping it can be factored afterward into nice expressions.
In the worksheet, I present an adjoint matrix using permuted Hadamard products. What puzzles me is that only for even dimensions, I need to multiply it (elementwise) with a parity matrix. Okay, not a specific Maple question, but maybe someone can help me out.

Download Adjoint.mw

seq slower than proc ...

Harry Garst 234 Maple 2020
performance programming linear-algebra
May 22 2023
1 4

I learned about Dodgson calculation of the determinant only recently (https://en.m.wikipedia.org/wiki/Dodgson_condensation).
I am only interested in symbolic expressions of the determinant.
Furthermore, I compared several methods. Not surprisingly, the build in method is the fastest. But why is the seq method slower than the proc method for the Dodgson method? Is there anything I could do to program it more efficiently?
 

> 

restart; with(LinearAlgebra)
> 

with(combinat); with(GroupTheory)
> 

DetDef := proc (A) local i, n, sigma; description "Jeremy Johnson. Downloaded from https://www.cs.drexel.edu/~jjohnson/2016-17/winter/cs300/lectures/determinant.mw"; n := RowDimension(A); add(PermParity(Perm(sigma))*mul(A[i, sigma[i]], i = 1 .. n), `in`(sigma, permute([`$`(1 .. n)]))) end proc
> 

InnerMatrix := proc (M::Matrix) SubMatrix(M, 2 .. RowDimension(M)-1, 2 .. ColumnDimension(M)-1) end proc
> 

MatrixDet := proc (M::Matrix) local C, n, i, j; n := RowDimension(M)-1; C := Matrix(n, n); seq(seq(assign('C[i, j]', Determinant(M([i, i+1], [j, j+1]))), j = 1 .. n), i = 1 .. n); return C end proc
> 

Dodgson := proc(M::Matrix)
 MatrixDet(M);
InnerMatrix(M) ^~ (-1) *~ MatrixDet(MatrixDet(M));
do if 1 < RowDimension(%) then InnerMatrix(`%%`) ^~ (-1) *~ MatrixDet(%);
end if;
until RowDimension(%) = 1;
Trace(%):
end proc:
> 

Dodgsonseq := proc (E::Matrix) local w, dim, Z; dim := RowDimension(E); Z[dim] := E; Z[dim-1] := MatrixDet(E); Z[dim-2] := `~`[`*`](`~`[`^`](InnerMatrix(E), -1), MatrixDet(MatrixDet(E))); seq(assign('Z[w-1]', `~`[`*`](`~`[`^`](InnerMatrix(Z[w+1]), -1), MatrixDet(Z[w]))), w = dim-1 .. 1, -1); Trace(Z[1]) end proc
> 

LaPlace := proc (M::Matrix) local c; add((-1)^(c+1)*M[1, c]*Minor(M, 1, c), c = 1 .. ColumnDimension(M)) end proc
> 

dim := 7; A := Matrix(dim, dim, shape = symmetric, symbol = a)

7

 (1)
> 

start_time := time(); st := time[real](); Det1 := abs(A); CPUtime_used_Build_in_Determinant := time()-start_time; REALtime_used_Build_in_Determinant := time[real]()-st; start_time := time(); st := time[real](); Det2 := DetDef(A); CPUtime_used_Jeremy_Johnson_Determinant := time()-start_time; REALtime_used_Jeremy_Johnson_Determinant := time[real]()-st; start_time := time(); st := time[real](); Det3 := Dodgsonseq(A); CPUtime_usedDodgsonseq := time()-start_time; REALCPUtime_usedDodgsonseq := time[real]()-st; start_time := time(); st := time[real](); Det4 := Dodgson(A); CPUtime_usedDodgson := time()-start_time; REALtime_usedDodgson := time[real]()-st; start_time := time(); st := time[real](); Det5 := LaPlace(A); CPUtime_usedLaPlace := time()-start_time; REALtime_usedLaPlace := time[real]()-st; simplify(Det1-Det2); simplify(Det1-Det3); simplify(Det1-Det4); simplify(Det1-Det5)
``

0.32e-1

  

0.34e-1

  

0.93e-1

  

.108

  

47.094

  

41.295

  

40.766

  

38.158

  

0.31e-1

  

0.50e-1

  

0

  

0

  

0

  

0

 (2)

Download test_Determinants_symbolic.mw

multipliying nonconformable matrices sho...

Harry Garst 234 Maple 2021
multiplication matrix linearalgebra
October 26 2022
1 0

I made an error by trying to multiply two nonconformable matrices. I think, I should receive an error message. But in this example this did not occur.

Strange.mw

> 

restart; with(LinearAlgebra); alias(`&bigotimes;` = LinearAlgebra:-KroneckerProduct); interface(rtablesize = 16); kernelopts(version); interface(version)

`&bigotimes;`

[10, 10]

`Maple 2021.2, X86 64 LINUX, Nov 23 2021, Build ID 1576349`

`Standard Worksheet Interface, Maple 2021.2, Linux, November 23 2021 Build ID 1576349`

Matrices nonconformable, Maple should give error message:

> 

`&bigotimes;`(Matrix(1, 4, 1), IdentityMatrix(4)); Dimension(%); M := Matrix(4, 4, shape = symmetric, symbol = m); Dimension(%)

Matrix([[1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1]])

4, 16

M := Matrix(4, 4, {(1, 1) = m[1, 1], (1, 2) = m[1, 2], (1, 3) = m[1, 3], (1, 4) = m[1, 4], (2, 2) = m[2, 2], (2, 3) = m[2, 3], (2, 4) = m[2, 4], (3, 3) = m[3, 3], (3, 4) = m[3, 4], (4, 4) = m[4, 4]}, storage = triangular[upper], shape = [symmetric])

4, 4

> 

MatrixMatrixMultiply(`&bigotimes;`(Matrix(1, 4, 1), IdentityMatrix(4)), DiagonalMatrix(Diagonal(M)))

Matrix([[m[1, 1], 0, 0, 0], [0, m[2, 2], 0, 0], [0, 0, m[3, 3], 0], [0, 0, 0, m[4, 4]]])

> 

`&bigotimes;`(Matrix(1, 4, 1), IdentityMatrix(4)).DiagonalMatrix(Diagonal(M))

Matrix([[m[1, 1], 0, 0, 0], [0, m[2, 2], 0, 0], [0, 0, m[3, 3], 0], [0, 0, 0, m[4, 4]]])

> 

KroneckerProduct(Matrix(1, 4, 1), IdentityMatrix(4)).DiagonalMatrix(Diagonal(M))

Matrix([[m[1, 1], 0, 0, 0], [0, m[2, 2], 0, 0], [0, 0, m[3, 3], 0], [0, 0, 0, m[4, 4]]])

> 

(Matrix(4, 16, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (1, 5) = 1, (1, 6) = 0, (1, 7) = 0, (1, 8) = 0, (1, 9) = 1, (1, 10) = 0, (1, 11) = 0, (1, 12) = 0, (1, 13) = 1, (1, 14) = 0, (1, 15) = 0, (1, 16) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (2, 4) = 0, (2, 5) = 0, (2, 6) = 1, (2, 7) = 0, (2, 8) = 0, (2, 9) = 0, (2, 10) = 1, (2, 11) = 0, (2, 12) = 0, (2, 13) = 0, (2, 14) = 1, (2, 15) = 0, (2, 16) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1, (3, 4) = 0, (3, 5) = 0, (3, 6) = 0, (3, 7) = 1, (3, 8) = 0, (3, 9) = 0, (3, 10) = 0, (3, 11) = 1, (3, 12) = 0, (3, 13) = 0, (3, 14) = 0, (3, 15) = 1, (3, 16) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1, (4, 5) = 0, (4, 6) = 0, (4, 7) = 0, (4, 8) = 1, (4, 9) = 0, (4, 10) = 0, (4, 11) = 0, (4, 12) = 1, (4, 13) = 0, (4, 14) = 0, (4, 15) = 0, (4, 16) = 1})).DiagonalMatrix(Diagonal(M))

Matrix([[m[1, 1], 0, 0, 0], [0, m[2, 2], 0, 0], [0, 0, m[3, 3], 0], [0, 0, 0, m[4, 4]]])

> 

`&bigotimes;`(Matrix(1, 4, 1), IdentityMatrix(4)).Matrix(4, 4, shape = diagonal, symbol = m)

Matrix([[m[1, 1], 0, 0, 0], [0, m[2, 2], 0, 0], [0, 0, m[3, 3], 0], [0, 0, 0, m[4, 4]]])

> 

whattype(`&bigotimes;`(Matrix(1, 4, 1), IdentityMatrix(4))); Dimension(`&bigotimes;`(Matrix(1, 4, 1), IdentityMatrix(4)))

Matrix

4, 16

> 

whattype(DiagonalMatrix(Diagonal(M))); Dimension(M)

Matrix

4, 4

hereafter results are correct: Matrices

nonconformable, therefore error messages appear.

> 

`&bigotimes;`(Matrix(1, 4, 1), IdentityMatrix(4)).M

Error, (in LinearAlgebra:-Multiply) first matrix column dimension (16) <> second matrix row dimension (4)

> 

`&bigotimes;`(Matrix(1, 4, 1), IdentityMatrix(4)).Matrix(4, 4, shape = symmetric, symbol = m)

Error, (in LinearAlgebra:-Multiply) first matrix column dimension (16) <> second matrix row dimension (4)

Download Strange.mw

indexed versus symbol...

Harry Garst 234 Maple
indexed
February 24 2022
2 3

I don't know how to fix this. What's wrong? Is there an alternative way to include underscores in a cat command?

``

> 

restart; with(LinearAlgebra)
> 

komb := proc (n::integer, m::integer) local L, a, j; L := [seq(cat(a, "__", j, ",", j), j = 1 .. n)]; add(mul(k), `in`(k, combinat:-choose(L, m))); return % end proc

proc (n::integer, m::integer) local L, a, j, k; L := [seq(cat(a, "__", j, ",", j), j = 1 .. n)]; add(mul(k), `in`(k, combinat:-choose(L, m))); return % end proc

 (1)
> 

komb(5, 5)

`a__1,1`*`a__2,2`*`a__3,3`*`a__4,4`*`a__5,5`

 (2)
> 

whattype(`a__1,1`)

symbol

 (3)
> 

A := Matrix(5, 5, shape = diagonal, symbol = a)

Matrix(%id = 36893489895174306860)

 (4)
> 

abs(A)

a[1, 1]*a[2, 2]*a[3, 3]*a[4, 4]*a[5, 5]

 (5)
> 

whattype(a[1, 1])

indexed

 (6)
> 

simplify(komb(5, 5)-abs(A))

`a__1,1`*`a__2,2`*`a__3,3`*`a__4,4`*`a__5,5`-a[1, 1]*a[2, 2]*a[3, 3]*a[4, 4]*a[5, 5]

 (7)

``

Download indexed.mw

input for Hessian in package VectorCalcu...

Harry Garst 234 Maple 2020
vectorcalculus duplicate-question hessian
July 09 2021
1 1

Using the VectorCalculus package in Maple 2020.2, the Jacobian does accept my target function, whereas the Hessian does not. Both the Jacobian and the Hessian need an algebraic expression for its input.

What can I do to make it work?

kind regards, Harry

 

vectorcalculus_question.mw

 

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