laporte bernard

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9 years, 298 days

MaplePrimes Activity


These are replies submitted by laporte bernard

@Markiyan Hirnyk 


"PROOF 1, using the definition of big-O":

"By the triangle inequality":

abs(2*x^3+x^2-3*x+2) <= abs(2*x^3)+abs(x^2)+abs(3*x)+2 and abs(2*x^3)+abs(x^2)+abs(3*x)+2 = 2*abs(x^3)+abs(x^2)+3*abs(x)+2;

abs(2*x^3+x^2-3*x+2) <= 2*abs(x)^3+abs(x)^2+3*abs(x)+2

(1)

"Now, if x&ge;2, then x^2&le;x^3, x&le;x^3 and 2&le;x^3":

"Thus":

abs(2*x^3)+abs(x^2)+abs(3*x)+2 <= 2*abs(x^3)+abs(x^3)+3*abs(x^3)+abs(x^3) and 2*abs(x^3)+abs(x^3)+3*abs(x^3)+abs(x^3) = 7*abs(x^3);

2*abs(x)^3+abs(x)^2+3*abs(x)+2 <= 7*abs(x)^3

(2)

"Using these inequalities , C=7 and k=2, we see that F is O(x^3)":

>There are other proofs also at infinity; can you find them ?

Source : www.math.fsu.edu 

``


Download Big-O_Maple.mw

@Markiyan Hirnyk 

Hi Markiyan Hirnyk

I thought it was obvious that we were at infinity ... Ok, I forgot to write down the little "x" !

Anyhow I see you are a smart guy and you saw quickly what was wrong !!!

Best wishes

@Carl Love 

Hi Carl Love,

No, so far I do not have any question

Best wishes

@tomleslie 

``

 

 

``

(1)

``

 

 

NULL

Gerschgorin := proc (A::Matrix) local Delta, m, n, AA, R, C, i, c, eig, P, Plt; Delta := proc (i, j) if i = j then 0 else 1 end if end proc; m, n := LinearAlgebra[Dimension](A); AA := Matrix(m, n, proc (i, j) options operator, arrow; Delta(i, j)*abs(A[i, j]) end proc); R := evalm(`&*`(AA, Vector(m, 1))); C := {seq(('plottools[circle]')([Re(A[i, i]), Im(A[i, i])], R[i], color = violet), i = 1 .. m)}; c := {seq(('plottools[point]')([Re(A[i, i]), Im(A[i, i])], color = blue, symbol = diamond), i = 1 .. m)}; eig := evalf(LinearAlgebra[Eigenvalues](A)); P := {seq(('plottools[point]')([Re(eig[i]), Im(eig[i])], color = red, symbol = box), i = 1 .. m)}; Plt := `union`(`union`(C, c), P); plots[display](eval(Plt), scaling = constrained) end proc

A := Matrix([[5, 8, 4, -3], [8, -9, 7, 5], [0, 4, 4, 2], [5, -5, 9, -9]]); evalf(LinearAlgebra[Eigenvalues](A), 3); Gerschgorin(A)

A := Matrix(4, 4, {(1, 1) = 5, (1, 2) = 8, (1, 3) = 4, (1, 4) = -3, (2, 1) = 8, (2, 2) = -9, (2, 3) = 7, (2, 4) = 5, (3, 1) = 0, (3, 2) = 4, (3, 3) = 4, (3, 4) = 2, (4, 1) = 5, (4, 2) = -5, (4, 3) = 9, (4, 4) = -9})

 

Vector[column]([[11.0], [3.71], [-11.8+6.01*I], [-11.8-6.01*I]])

 

 

A matrix with real entries and one with complex entries; both have complex eigenvalues.

F := Matrix([[2, -1/2, -1/3, 0], [0, 6, 1, 0], [1/3, -1/3, 5, 1/3], [-1/2, 1/4, -1/4, 4]]); evalf(LinearAlgebra[Eigenvalues](F)); Gerschgorin(F)

F := Matrix(4, 4, {(1, 1) = 2, (1, 2) = -1/2, (1, 3) = -1/3, (1, 4) = 0, (2, 1) = 0, (2, 2) = 6, (2, 3) = 1, (2, 4) = 0, (3, 1) = 1/3, (3, 2) = -1/3, (3, 3) = 5, (3, 4) = 1/3, (4, 1) = -1/2, (4, 2) = 1/4, (4, 3) = -1/4, (4, 4) = 4})

 

Vector[column]([[5.427845847+.1338767333*I], [5.427845847-.1338767333*I], [4.116425921], [2.027882385]])

 

 

de.mw

NULL


Download gerschorin_maple_15_11_2014.mw

Hi Tomeslie here is a response !

 

@Rouben Rostamian  


 

 

``

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NULL

Gerschgorin := proc (A::Matrix) local Delta, m, n, AA, R, C, i, c, eig, P, Plt; Delta := proc (i, j) if i = j then 0 else 1 end if end proc; m, n := LinearAlgebra[Dimension](A); AA := Matrix(m, n, proc (i, j) options operator, arrow; Delta(i, j)*abs(A[i, j]) end proc); R := evalm(`&*`(AA, Vector(m, 1))); C := {seq(('plottools[circle]')([Re(A[i, i]), Im(A[i, i])], R[i], color = violet), i = 1 .. m)}; c := {seq(('plottools[point]')([Re(A[i, i]), Im(A[i, i])], color = blue, symbol = diamond), i = 1 .. m)}; eig := evalf(LinearAlgebra[Eigenvalues](A)); P := {seq(('plottools[point]')([Re(eig[i]), Im(eig[i])], color = red, symbol = box), i = 1 .. m)}; Plt := `union`(`union`(C, c), P); plots[display](eval(Plt), scaling = constrained) end proc

A := Matrix([[5, 8, 4, -3], [8, -9, 7, 5], [0, 4, 4, 2], [5, -5, 9, -9]]); evalf(LinearAlgebra[Eigenvalues](A), 3); Gerschgorin(A)

A := Matrix(4, 4, {(1, 1) = 5, (1, 2) = 8, (1, 3) = 4, (1, 4) = -3, (2, 1) = 8, (2, 2) = -9, (2, 3) = 7, (2, 4) = 5, (3, 1) = 0, (3, 2) = 4, (3, 3) = 4, (3, 4) = 2, (4, 1) = 5, (4, 2) = -5, (4, 3) = 9, (4, 4) = -9})

 

Vector[column]([[11.0], [3.71], [-11.8+6.01*I], [-11.8-6.01*I]])

 

 

A matrix with real enties and one with complex entries; both have complex eigenvalues.

F := Matrix([[2, -1/2, -1/3, 0], [0, 6, 1, 0], [1/3, -1/3, 5, 1/3], [-1/2, 1/4, -1/4, 4]]); evalf(LinearAlgebra[Eigenvalues](F)); Gerschgorin(F)

F := Matrix(4, 4, {(1, 1) = 2, (1, 2) = -1/2, (1, 3) = -1/3, (1, 4) = 0, (2, 1) = 0, (2, 2) = 6, (2, 3) = 1, (2, 4) = 0, (3, 1) = 1/3, (3, 2) = -1/3, (3, 3) = 5, (3, 4) = 1/3, (4, 1) = -1/2, (4, 2) = 1/4, (4, 3) = -1/4, (4, 4) = 4})

 

Vector[column]([[5.427845847+.1338767333*I], [5.427845847-.1338767333*I], [4.116425921], [2.027882385]])

 

 

``


Download gerschorin_maple_15_11_2014.mw

 

 

``

(1)

 

 

 

 

 

 

 

NULL

Gerschgorin := proc (A::Matrix) local Delta, m, n, AA, R, C, i, c, eig, P, Plt; Delta := proc (i, j) if i = j then 0 else 1 end if end proc; m, n := LinearAlgebra[Dimension](A); AA := Matrix(m, n, proc (i, j) options operator, arrow; Delta(i, j)*abs(A[i, j]) end proc); R := evalm(`&*`(AA, Vector(m, 1))); C := {seq(('plottools[circle]')([Re(A[i, i]), Im(A[i, i])], R[i], color = violet), i = 1 .. m)}; c := {seq(('plottools[point]')([Re(A[i, i]), Im(A[i, i])], color = blue, symbol = diamond), i = 1 .. m)}; eig := evalf(LinearAlgebra[Eigenvalues](A)); P := {seq(('plottools[point]')([Re(eig[i]), Im(eig[i])], color = red, symbol = box), i = 1 .. m)}; Plt := `union`(`union`(C, c), P); plots[display](eval(Plt), scaling = constrained) end proc

A := Matrix([[5, 8, 4, -3], [8, -9, 7, 5], [0, 4, 4, 2], [5, -5, 9, -9]]); evalf(LinearAlgebra[Eigenvalues](A), 3); Gerschgorin(A)

A := Matrix(4, 4, {(1, 1) = 5, (1, 2) = 8, (1, 3) = 4, (1, 4) = -3, (2, 1) = 8, (2, 2) = -9, (2, 3) = 7, (2, 4) = 5, (3, 1) = 0, (3, 2) = 4, (3, 3) = 4, (3, 4) = 2, (4, 1) = 5, (4, 2) = -5, (4, 3) = 9, (4, 4) = -9})

 

Vector[column]([[11.0], [3.71], [-11.8+6.01*I], [-11.8-6.01*I]])

 

 

A matrix with real enties and one with complex entries; both have complex eigenvalues.

F := Matrix([[2, -1/2, -1/3, 0], [0, 6, 1, 0], [1/3, -1/3, 5, 1/3], [-1/2, 1/4, -1/4, 4]]); evalf(LinearAlgebra[Eigenvalues](F)); Gerschgorin(F)

F := Matrix(4, 4, {(1, 1) = 2, (1, 2) = -1/2, (1, 3) = -1/3, (1, 4) = 0, (2, 1) = 0, (2, 2) = 6, (2, 3) = 1, (2, 4) = 0, (3, 1) = 1/3, (3, 2) = -1/3, (3, 3) = 5, (3, 4) = 1/3, (4, 1) = -1/2, (4, 2) = 1/4, (4, 3) = -1/4, (4, 4) = 4})

 

Vector[column]([[5.427845847+.1338767333*I], [5.427845847-.1338767333*I], [4.116425921], [2.027882385]])

 

 

``


Download gerschorin_maple_15_11_2014.mw

@Rouben Rostamian  

Hi Rouben Rostamian,

I just want to know the procedure to copy paste an entire worksheet to mapleprimes !

I hope it's clear enough...

@Markiyan Hirnyk                                                                                                                                        

 

Thanks for your help!

Anyhow, the centers of the circles are all lying on the Real x-axis.

Y=Im z, x= Re z.

 

Here are the equations of the circles:

  1. (x-3)^2 + y^2 =25/36
  2. (x-4)^2 +y^2 =1
  3. (x-5)^2 +y^2 =1
  4. (x-6)^2 +y^2 =1                                                                                                                                                            I tried to plot them, without success...                                                                                                     

@Kitonum                                                                                                                                              

 

hi,

Indeed it's shorter !

@Carl Love                                                                                                                                               

hello,

nice job !

@Carl Love                                                                                                                                               

 

hi,

many thanks about that quick and nice response !

@Kitonum                                                                                                                                             

 

Very nice way congratulations !!

@Kitonum                                                                                                                                                     

Hello, I tried a formula, based on the number of surjections :                                                                             

> sum((-1)^j*binomial(3, j)*2^j*factorial(2*3-j), j = 0 .. 3);
                                     240
 Manys thanks for your replies !

@Carl Love                                                                                                                                                     

smart answer indeed !

@Kitonum 

I see you are right !!

How did you find the y extent ?

Regards

Laporte Bernard 20

 

 

 

 

@Kitonum 

 

Hi, well done and thank you so much :

LAPORTE BERNARD 20

 

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