The calculator

[Gunther]

Dave,

Gunther....
sin(45) is a constant   :P

yes, that's true. But the value is sqrt(2)/2 which is irrational. 

Gunther

[dedndave]

constants can be irrational - take my wife, for example   :lol:

[Gunther]

Dave,

constants can be irrational - take my wife, for example   :lol:

I hope she didn't read your posts, otherwise ...  :lol:

Gunther

[RuiLoureiro]

No, that wouldn't be a linear equation system, because the sine is bent. You're talking about a non-linear equation system. That's not so easy to solve and most of these systems can only be solved approximately. That's another point.
Gunther

Gunther,

        If the constants are real (like, 1, 1.2, pi, sqrt(5),...)
        the system is a linear equation system.

        The system is non-linear in x if we use sin(x) instead of x
        or x^2, e^x, ...

        For instance:  2 sin(3t) x+  y=5
                               2 x           - 3y=10       for instance: t={1,2,...}
--------------------------------------------------------------------
        For each t, we get a real number A=2*sin(3t),
        we have a linear equation system,
        we may solve it, we may have or not a solution,
        etc. etc.
---------------------------------------------------------------------
        But this:     2 sin(3x)+  y =5
                            2 x        - 3y=10
        is a non-linear equation system
        and what you said is correct.

[RuiLoureiro]

constants can be irrational - take my wife, for example   :lol:

(...)

[RuiLoureiro]

 
Gunther,
          you tried to show us an example of
          a linear equation system A
          that you say "it has one property"
          and another linear equation system B
          that you say "it has another property".
          Now, i show that they have the same:
          the question is the starting point.

This is what you wrote
-----------------------------

My question has that background:
Given is the following linear equation system A:
x -  2y = 1
x + 1y = 4

The solution is: x = 3 and y = 1
If we change the coefficients a bit, we've the system A':
x -     2y = 1
x + 1.2y = 4

The solution is now: x = 2.875 and y = 0.9375
But the relationship between the two solutions is still recognizable.

    Well,
    Do this:
                (1) replace +1y  by +ay where "a" is a real number
    we get:
                    x - 2y = 1
                    x + ay = 4
           
                (2) study this system
                    (if you know how to do that.
                    (explanations: $150 per hour. very cheap!).
    we get:
                    if a = -2 the system has no solution
                    if a<>-2 the system has a unique solution

                Using "The calculator", we get:

                x-2y=1; x-   2y=4; The System of linear equations has no solution                 
                x-2y=1; x-1.9y=4; X=  61.0   Y=  30.0   Determinant:  0.1
                x-2y=1; x-2.1y=4; X= -59.0   Y= -30.0   Determinant: -0.1               

Well, if we want to be far from a=-2, we may choose a=1 or a=200

          x-2y=1; x+200y=4; X=1.0297029702970297  Y=0.01485148514851485
          x-2y=1; x+201y=4; X=1.0295566502463054  Y=0.0147783251231527
          ...

Following what you call "an ill conditioned linear equation system",
i would say: that's a simple example of "an ill conditioned linear equation system",and this is your linear equation system A.

Well, you may say: the starting point is the first equation system.
And my question is: why to change "1" and no "-2" or "4" or other ?
-----------------------------------------------------------------------------------
This is what you wrote
----------------------------

Lets have a look at the following system B:
x   +     2y  = 3
2x +  4.1y = 4

Doing the same
           
    we get:
                    x +  2y = 3
                   2x + ay = 4

    studying the system:
   
                    if a = 4  the system has no solution
                    if a<>4  the system has a unique solution

                Using "The calculator", we get:

                    x+2y=3; 2x+   4y=4;  The System of linear equations has no solution
                    x+2y=3; 2x+3.9y=4;  X= -37.0   Y=  20.0   Determinant: -0.1
                    x+2y=3; 2x+4.1y=4;  X=  43.0   Y= -20.0   Determinant:  0.1

---------------------------------------------------------------------------
my conclusion is this: 

          The calculator should solve the linear equation system with
          real constant coefficients;
          We may study the system, if and when we want;
          You can call the name you want, for me, no problems.
---------------------------------------------------------------------------
I want to remember that many of us are here
doing this things to do something useful or
to pass the time, no to get more problems
for our lives

I will post the NEXT version v3.10.2 (calcula60.exe)
soon as possible

[Gunther]

Hi Rui,

        But this:     2 sin(3x)+  y =5
                            2 x        - 3y=10
        is a non-linear equation system
        and what you said is correct.

no doubt about that.

I want to remember that many of us are here
doing this things to do something useful or
to pass the time, no to get more problems
for our lives

That's clear. It wasn't my matter of concern to make you problems.

          Now, i show that they have the same:

That's not my point of view. I want you cause no further problems; therefore I won't answer in detail. Only that: The condition of a linear equation system is a measure for the behavior of the system. If we call M the matrix of the coefficients, it's defined as:

cond (M) = ||M|| ||M^-1||

or formulated in a sentence: The condition of a linear equation system is the norm of the matrix M multiplied by the norm of the inverse matrix of M. The system A has a much better condition than system B. That's the difference.

But again, I don't want that you've further problems in your spare time. You may ignore this point. Anyway, your Calculator is a solid work. Thank you for that.

Gunther 

[RuiLoureiro]

Hi Gunther,

That's clear. It wasn't my matter of concern to make you problems.

I wanted to say the same to you

That's not my point of view.
...
The system A has a much better condition than system B.
That's the difference.

It's clear: now, i understand your point of view.
         Well, nothing to say: i showed my point of view.         
         Thank you for that, also.  :icon14:
         RuiLoureiro

Anyway, your Calculator is a solid work.

:t

[RuiLoureiro]

Hi all,                               
-----------------------------------------------------------------------
                              LASTEST VERSION
-----------------------------------------------------------------------
        ****-- The (powerful) Calculator v3.10.2 --****
                                 (calcula60.exe)
                       
        This is the lastest version and this is the powerful calculator
        v3.10.2

            ---------------------------------------------------------------
            COMPLEX NUMBERS and COMPLEX FUNCTIONS
            ---------------------------------------------------------------

        Now, we may solve any complex expression in the same way we do for
        real expressions.

        We may use any function:

                                z= a + i b          (a,b are reals, i^2=-1)
                                  = |z| e^(i.angleR)

                conj(a+ib)      = a-ib
                inv(z)             = 1/z

                abs(a+ib)       = sqr(a^2+b^2)+i0
                sqr(a+ib)       = (a+ib)^(1/2)
                rnd(a+ib)       = rnd(a)+i rnd(b)
                rndi(a+ib)      = rndi(a)+i rndi(b)
       
                e^(a+ib)        = e^a * e^ib = e^a * ( cos(b)+i sin(b))

                (a+ib)^(c+id)   = e^( (c+id)*ln(a+ib) )
               
                ln(a+ib)        = ln (|a+ib|* e^i angleR )
                log(a+ib)       = (ln (|a+ib|* e^i angleR )) / ln(10)
               
                sin(z)          = (e^iz - e^-iz)/2i
                cos(z)          = (e^iz + e^-iz)/2
                tan(z)          = (e^2iz - 1) / i (e^2iz + 1)
                sec(z)          = 1 / cos(z)
                csc(z)          = 1 / sin(z)
                cot(z)          = 1 / tan(z)
               
                arcsin(z)       = -i  * ln(i*z + sqrt(1-z^2))   
                arccos(z)       = -i  * ln(z +i sqrt(1-z^2))
                arctan(z)       = i/2 * ln((i+z) / (i-z))
                arcsec(z)       = arccos(1/z)
                arccsc(z)       = arcsin(1/z)
                arccot(z)       = arctan(1/z)

                sind(z)
                cosd(z)
                tand(z)
                secd(z)
                cscd(z)
                cotd(z)

                arcsind(z)
                arccosd(z)
                arctand(z)
                arcsecd(z)
                arccscd(z)
                arccotd(z)

                sinh(z)         = (e^z - e^-z) / 2
                cosh(z)         = (e^z + e^-z) / 2
                tanh(z)         = (e^2z - 1) / (e^2z + 1)
                sech(z)         = 1 / cosh(z)
                csch(z)         = 1 / sinh(z)
                coth(z)         = 1 / tanh(z)

                arcsinh(z)      = ln(z + sqrt(z^2+1))
                arccosh(z)      = ln(z + sqrt(z^2-1))
                arctanh(z)      = 1/2 * ln((1+z) / (1-z))
                arcsech(z)      = arccosh(1/z)
                arccsch(z)      = arcsinh(1/z)
                arccoth(z)      = arctanh(1/z)
               
                round(a+ib, n)  - round 'a' and 'b' to n decimal places

        The operation rules are the same for real numbers

        The result may be a complex number, INFINITY or indeterminate form.
        We may use the division by 0 to generate th infinity.

        We need to use brackets when we have powers of powers

        (1-i2)^(1-i)^(i2) gives "Complex power too complex- use brackets"

        We need to do ((1-i2)^(1-i))^(i2) or (1-i2)^((1-i)^(i2)).

        Any complex number is a number with a real part 'a'
        and an imaginary part 'b'. For instance, z1=-2+i3 or z1=i3-2.

        We may define it typing z1=(-2,3); or z1=-2+i3 for example.

        To build a complex expression
        we may define a set of real/complex constants z1, z2, z3, z4, ...
        and then we write the expression

        For instance,

                (1-i2)*(2-i3)  or (a-ib)*(c-id) where a=12;b=-4;c=1;d=-3;
                                                are real constants

                z1+z2*z3-z4   or  z5=z1+z2*z3-z4  or z1=z1+z2*(z3/z4)+3-i2

        If the expression uses '^', we may use brackets or constants
        previously defined. See the following example:

            z1=(e^i*(2-i3)+ e^-i*(2-i3) ) /2 =  1.0806046117362794-i1.6209069176044192

        is not equal to
       
            z2=(e^(i*(2-i3))+ e^(-i*(2-i3)) ) /2 = -4.1896256909688072+i9.1092278937553366

            cos(2-i3)=-4.1896256909688072+i9.1092278937553366

        but is equal to

            x1=i*(2-i3)
            x2=-x1

            z3=(e^x1+e^x2)/2 = -4.1896256909688072+i9.1092278937553366

        After defining any constant/matrix we may type the constant name
        to see its value. We may use also list c or list r or list l or
        list.

        To delete all defined variables, type: del a (delete all)

        If you know any bug or something else, please post it       
        If you have any suggestion, please send me a personal message or ...

        I hope i have not did many mistakes!

        What will coming next ?
       
        Maybe the function norm(a), where a=matrix
        and something about complex numbers.
        ---------------------------------------------------------
        Nov. 2013

        Good luck !
        Rui Loureiro

        note:   RulesV3_10_2I   - rules in English
                   RulesV3_10_2P   - regras em Português

[Gunther]

Rui,

impressive. You are a hard working man.  :t Thank you for providing.

Gunther

[RuiLoureiro]

Rui,
impressive. You are a hard working man.  :t Thank you for providing.
Gunther

Thank you, Gunther   :t

Hi all,                               
-----------------------------------------------------------------------
                              LASTEST VERSION
-----------------------------------------------------------------------
        ****-- The (powerful) Calculator v3.11.1 --****
                               (calcula61.exe)
                       
        This is the lastest version and this is the powerful calculator
        v3.11.1

            ---------------------------------------------------------------------
            LINEAR EQUATION SYSTEMS WITH COMPLEX VARIABLES
                                  NORM of a MATRIX           
            ---------------------------------------------------------------------
        Now, we may solve linear equation systems with complex variables,
        and we may compute the norm of a matrix.

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
                           Systems of 2 complex variables x,y
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

-------------
example 1
-------------
type the following lines (for each one, press COMPUTE)
or copy and paste:

z1=1-i2
z2=2+i3

x1=3;x2=5;x3=-7;

x1 X+ x2 Y=z1; x2 X- x3 Y=z2;

X=  0.75+i7.25
Y= -0.25-i4.75
Determinant: -4.0+i0
----------------------------------------
PROOF:

    type the following lines ( press COMPUTE):

X1=  0.75+i7.25
Y1= -0.25-i4.75

w1= x1*X1+ x2*Y1
  = 1.0-i2.0     = z1
 
w2= x2*X1- x3*Y1
  = 2.0+i3.0     = z2
---------------------------------------
------------
example 2
------------
    type the following lines ( press COMPUTE):

z1 X+ x2 Y=x1; x2 X- z2 Y=x2;
X=  0.9302752293577981+i0.3009174311926605 
Y=  0.2935779816513761+i0.3119266055045871
Determinant: -33.0+i1.0


++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
                       Systems of 3 complex variables x,y,z
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

------------
example 3
------------
type the following lines (for each one, press COMPUTE):

z1=1-i2
z2=2+i3

x1=3;x2=5;x3=-7;
x+y-x3 z=z1; x2 x-iy+2z=20; (x1+ix2)x+2y-z=8;

result:
X=  5.4763271162123386-i2.6248206599713056
Y= -10.3565279770444763-i9.0616929698708752
Z=  0.8400286944045911+i1.3837876614060258
Determinant:  42.0+i32.0
-------------------------------------------------
------------   
example 4
------------
type the following line ( press COMPUTE):

x+y-x3 z=z1; x2 x-iy+2z=20; z2 x+2y-z=x1+ix2;

result:
X=  4.5773011617515639-i1.0393208221626452
Y= -4.5656836461126005-i3.1689008042895442
Z=  0.1411974977658623+i0.3154602323503127
Determinant:  54.0+i21.0

PROOF:

    type the following lines ( press COMPUTE):

    X1=  4.5773011617515639-i1.0393208221626452
    Y1= -4.5656836461126005-i3.1689008042895442
    Z1=  0.1411974977658623+i0.3154602323503127


w1=   X1+   Y1-x3* Z1
  =0.9999999999999995-i2.0000000000000005 = z1=1-i2

w2=x2*X1-i* Y1+2*  Z1
  =20.0-i9.9746599868666408e-17 = 20

w3=z2*X1+2* Y1-    Z1
  =3.0000000000000001+i5.0000000000000002 = x1+ix2 = 3+i5

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
                      Systems of 4 complex variables x,y,z,t
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

--------------------------------
example 5: real variables
--------------------------------
    type the following line ( press COMPUTE):

x+y+z+t=10; -x-y+2z+2t=20; 3x+2y-z-3t=8; -2x+y+z-5t=18;

X=  7.6666666666666667
Y= -7.6666666666666667
Z=  15.1666666666666667
T= -5.1666666666666667
Determinant: -36.0

    type the following line ( press COMPUTE):

matA=[1,1,1,1;  -1,-1,2,2;  3,2,-1,-3; -2,1,1,-5];
det(matA)=-36.0
The triangular matrix is correct
--------------------------------------------------------
------------
example 6- same as example 5
------------
    type the following line ( press COMPUTE):

x+y+z+t=10; -x-y+2z+2t=20; 3x+2y-z-3t=8; -2x+y+z-5t=18+i0;

X=  7.6666666666666667+i0
Y= -7.6666666666666667+i0
Z=  15.1666666666666667+i0
T= -5.1666666666666667+i0
Determinant: -36.0+i0
----------------------------------------------------------
------------
example 7
------------
    type the following line ( press COMPUTE):

x+iy+z+t=10; -x-y+2z+2t=20; 3x+2y-z-3t=8; -2x+y+z-5t=18+i0;

X=  5.1111111111111111-i2.5555555555555556
Y=  0+i7.6666666666666667
Z=  15.1666666666666667+i0
T= -2.6111111111111111+i2.5555555555555556
Determinant:  0-i36.0
----------------------------------------------
PROOF:

    type the following lines ( press COMPUTE):

X1=5.1111111111111111-i2.5555555555555556
Y1=0+i7.6666666666666667
Z1=15.1666666666666667+i0
T1=-2.6111111111111111+i2.5555555555555556

W1=X1+i*Y1+Z1+T1
  =10.0+i0

W2=-X1-Y1+2*Z1+2*T1
  =20.0+i0.0000000000000001

W3=3*X1+2*Y1-Z1-3*T1
  =7.9999999999999999-i1.9949319973733282e-16

W4=-2*X1+Y1+Z1-5*T1
  =18.0-i0.0000000000000001
----------------------------------------------
------------
example 8
------------
    type the following lines ( press COMPUTE):

z1=1-i2
z2=2+i3

x1=3;x2=5;x3=-7;

z1 x+iy+z+t=x1-ix2; -x-z2 y+2z+2t=20; 3x+2y-z1 z-3t=8; -2x+y+z-5t=18+i2;

X=  5.2599406528189911-i7.4712166172106825
Y=  5.7750741839762611+i2.2890207715133531
Z=  16.267062314540059+i3.4747774480712166
T= -1.2955489614243323+i3.7412462908011869

Determinant: -46.0-i68.0
----------------------------------------------
PROOF:

    type the following lines ( press COMPUTE):

X1=  5.2599406528189911-i7.4712166172106825
Y1=  5.7750741839762611+i2.2890207715133531
Z1=  16.267062314540059+i3.4747774480712166
T1= -1.2955489614243323+i3.7412462908011869

w1= z1*X1+ i* Y1+   Z1+  T1
  = 2.9999999999999997-i5.0000000000000001 = x1-ix2 =3-i5
 
w2=   -X1-z2 *Y1+2* Z1+2*T1
  = 19.999999999999999+i0 = 20
 
w3= 3* X1+2*  Y1-z1*Z1-3*T1
  = 8.0000000000000002-i6.0108168442596366e-16 = 8
 
w4=-2* X1+    Y1+   Z1-5*T1
  = 17.999999999999999+i2.0000000000000002 = 18+i2
;«««««««««««««««««««««««««««««««««««««««««««««««««««
                             NORM of a MATRIX
;«««««««««««««««««««««««««««««««««««««««««««««««««««
    type the following lines ( press COMPUTE):
matA=[1,-2; -3, 4];
norm(matA) 

we get:
norm -max. of the sum of each line (matA)= 7.0
norm -max. of the sum of each column (matA)= 6.0

[Gunther]

Hi Rui,

only one thing: Wow!  :t

Gunther

[RuiLoureiro]

 
Hi Gunther
                       :t

Hi all,
        This is only to say that "The powerful calculator" (calcula62)
        is near to use matrices with COMPLEX NUMBERS.
        I need only to do some tests.       
        RuiLoureiro

Hi qWord,
        When i post calcula62, could you test it with matlab ?
        Ok, thank you  :t
       
See this:

Do:
        matX=[1+i2,9-i3; 3-i5.4, 4+i1.2];
We get:
        The complex matrix is defined

Do:
        matY=matX^-1;
We get:       
        matY=[0.0095362094447181-i0.061193609684003, 0.0622844675909634+i0.1261524385952565;
              0.0854036174255753+i0.0331480047856992,0.027359420085861 -i0.0187381237244];
        The complex matrix is defined
        Errors in the identity matrix are below 1e-18
        The inverted matrix is correct
-------------------------------------------------
Do:

Code Select Expand


matA=[1, 2, 3, 2, 0, 2,1,-2, 3, 2,0,1  ; 1, 4,5, 6, 0, 3,-5, 0, 1, 2, 7,0 ; -3, 0, 7,-2,-3,5,1 ,-1, 2,9,-2,0;
      9, 1, 1, 3, 4, 6,2, 5, 6, 8,2,3  ; 1,-1,1,-1, 8, 0, 4, 1,-1, 0, 5,1 ;  2,-1,-5, 2, 0,5,7 ,-9, 3,5,-3,1;
      2, 3, 1, 3, 5, 6,4, 5,-6,-8,7,0  ;-1,-3,1,-7,-8, 0, 0, 2,-5, 3, 4,1 ; 12,-4, 5, 7, 1,0,2 , 0,-3,0,-4,-1;
      1,-1, 1,-1,-9,-3,5,-7, 2, 2,1,-1 ; 0,-5,3,-2,-1,-7, 6,-1,-2,-5,-2,8 ;  0, 0, 1,-3, 0,1,-5,-2, 0,1, 5,1+i0];   


We get:
The complex matrix is defined

Do:
    matB=matA^-1;

We get:

Code Select Expand


matB=[0.3489475139105589+i0,-0.077717764135831+i0,-0.1305663289999746+i0,
-0.01189321415923582+i0, 0.030658417448943+i0,-0.01544063546736273+i0,
-0.032359956048502+i0, 0.12026503888873099+i0, 0.0563587763184989+i0,
-0.0469873206919989+i0,-0.0552819588513436+i0, 0.0028764341340867+i0;
1.2038253746424853+i0,-0.0737862490216782+i0,-0.3707567189349224+i0,
-0.1747472582414871+i0, 0.1591538921209745+i0,-0.0028790830112823+i0,
-0.1494282660592322+i0, 0.4412261789225444+i0, 0.01548276728050134+i0,
-0.2686901806756077+i0,-0.1567441075089298+i0,-0.2763391412739283+i0;
0.0187343364849522+i0,-0.0207830852282367+i0, 0.0772439172493358+i0,
-0.01210186072460702+i0, 0.0069412698217746+i0,-0.055170392364791+i0,
0.0258755788715113+i0,-0.0499767559328416+i0, 0.0257864718665081+i0,
0.0349311858466001+i0, 0.0159887049951585+i0, 0.0485851419165665+i0;
-0.3314381496330013+i0, 0.149112029548901+i0, 0.0633701813335249+i0,
0.0279434023893145+i0,-0.043087392918502+i0, 0.0414049652003719+i0,
0.01215734823711377+i0,-0.0944356492076842+i0,-0.0062092805019976+i0,
0.036984525440662+i0, 0.0541658475877536+i0,-0.058825516372493+i0;
0.1306324231186913+i0,-0.0212040294890572+i0,-0.0315844618695761+i0,
-0.0359397656450297+i0, 0.0930859455477016+i0, 0.006451085834335+i0,
-0.0357521157353953+i0, 0.0202766124971702+i0, 0.01438058210203112+i0,
-0.0674981469058792+i0,-0.0234276869891709+i0,-0.0083228389532894+i0;
-0.3692504255738974+i0,-0.021892117367119+i0, 0.1657167584891193+i0,
0.0705876315716985+i0,-0.1292871530040815+i0, 0.026429924473778+i0,
0.1266082036440829+i0,-0.164592953418614+i0,-0.01335398125911003+i0,
0.0718698527221532+i0, 0.0405050165976217+i0, 0.1594134514897885+i0;
0.0490849707981281+i0,-0.0171928398021129+i0,-0.00158264398711094+i0,
0.0022457425854745+i0, 0.0498990230369092+i0, 0.0029350115469339+i0,
0.0180160649764132+i0, 0.0216114527675041+i0,-0.01354256886325167+i0,
0.0354693568163014+i0,-0.0026549908694263+i0,-0.087100970997439+i0;
-0.2044568349995381+i0, 0.01297154108852978+i0, 0.0532421387390865+i0,
0.0737142372186105+i0,-0.0384820735367924+i0,-0.047213045317213+i0,
0.0295741565155512+i0,-0.0456359070616521+i0,-0.026393327801645+i0,
0.0326274528589774+i0, 0.0201915799013743+i0,-0.0406533648942984+i0;
-0.5932972720613709+i0,-0.0209918266600847+i0, 0.2026367180091205+i0,
0.1572055516251538+i0,-0.1259187657287346+i0,-0.0612563733893335+i0,
0.10269198782892028+i0,-0.3301642954853802+i0,-0.0588064472272272+i0,
0.2259098962133411+i0, 0.0763407165237221+i0, 0.1953977685856945+i0;
0.3182471547714092+i0, 0.0517233171449673+i0,-0.10861579851959029+i0,
-0.055216064571107+i0, 0.11886820932720037+i0, 0.0414594815056652+i0,
-0.1230670565368351+i0, 0.2047330641547552+i0, 0.01459943016238838+i0,
-0.12498644339395018+i0,-0.0577935633223318+i0,-0.1656982226986158+i0;
-0.2799018148394809+i0, 0.068480479218752+i0, 0.0509548448741404+i0,
0.0509176474488098+i0, 0.01172613482006036+i0,-0.0045568810249207+i0,
0.0402069822428226+i0,-0.0780196840570518+i0,-0.0297156907710767+i0,
0.10156931114386624+i0, 0.0259680263379561+i0, 0.062108712424104+i0;
0.2739747436075013+i0, 0.0358644611698261+i0,-0.10043750004637351+i0,
-0.01537054898749617+i0,-0.01188421088973993+i0, 0.0545292542113109+i0,
-0.0447509545683625+i0, 0.1334135675731232+i0,-0.0175808083399441+i0,
-0.1361058847726861+i0, 0.0610654582586491+i0,-0.0461320667215304+i0];


The complex matrix is defined
Errors in the identity matrix are below 1e-18
The inverted matrix is correct
-----------------------------------------------------------------
Do:
matR=[1, 2, 3, 2, 0, 2,1,-2, 3, 2,0,1  ; 1, 4,5, 6, 0, 3,-5, 0, 1, 2, 7,0 ; -3, 0, 7,-2,-3,5,1 ,-1, 2,9,-2,0;
      9, 1, 1, 3, 4, 6,2, 5, 6, 8,2,3  ; 1,-1,1,-1, 8, 0, 4, 1,-1, 0, 5,1 ;  2,-1,-5, 2, 0,5,7 ,-9, 3,5,-3,1;
      2, 3, 1, 3, 5, 6,4, 5,-6,-8,7,0  ;-1,-3,1,-7,-8, 0, 0, 2,-5, 3, 4,1 ; 12,-4, 5, 7, 1,0,2 , 0,-3,0,-4,-1;
      1,-1, 1,-1,-9,-3,5,-7, 2, 2,1,-1 ; 0,-5,3,-2,-1,-7, 6,-1,-2,-5,-2,8 ;  0, 0, 1,-3, 0,1,-5,-2, 0,1, 5,1];   

The real matrix is defined

NOTE: this matR is equal matA
      ***********************
Do:
   matS=matR^-1;
The real matrix is defined
The identity matrix is correct

Code Select Expand


matS=[0.3489475139105589,-0.077717764135831,-0.1305663289999746,-0.01189321415923582,
0.030658417448943,-0.01544063546736273,-0.032359956048502, 0.12026503888873098,
0.0563587763184989,-0.0469873206919989,-0.0552819588513436, 0.0028764341340867;
1.2038253746424853,-0.0737862490216782,-0.3707567189349224,-0.1747472582414871,
0.1591538921209745,-0.0028790830112823,-0.1494282660592322, 0.4412261789225444,
0.01548276728050134,-0.2686901806756077,-0.1567441075089298,-0.2763391412739283;
0.0187343364849522,-0.0207830852282367, 0.0772439172493358,-0.01210186072460701,
0.0069412698217746,-0.055170392364791, 0.0258755788715113,-0.0499767559328416,
0.0257864718665081, 0.0349311858466001, 0.0159887049951585, 0.0485851419165665;
-0.3314381496330014, 0.149112029548901, 0.0633701813335249, 0.0279434023893145,
-0.043087392918502, 0.0414049652003719, 0.01215734823711377,-0.0944356492076842,
-0.0062092805019976, 0.036984525440662, 0.0541658475877536,-0.058825516372493;
0.1306324231186913,-0.0212040294890573,-0.0315844618695761,-0.0359397656450297,
0.0930859455477016, 0.006451085834335,-0.0357521157353953, 0.0202766124971702,
0.01438058210203112,-0.0674981469058792,-0.0234276869891709,-0.0083228389532894;
-0.3692504255738975,-0.021892117367119, 0.1657167584891193, 0.0705876315716985,
-0.1292871530040815, 0.026429924473778, 0.1266082036440829,-0.164592953418614,
-0.01335398125911003, 0.0718698527221532, 0.0405050165976217, 0.1594134514897885;
0.0490849707981281,-0.0171928398021129,-0.00158264398711094, 0.0022457425854745,
0.0498990230369092, 0.0029350115469339, 0.0180160649764132, 0.0216114527675041,
-0.01354256886325167, 0.0354693568163014,-0.0026549908694263,-0.087100970997439;
-0.2044568349995381, 0.01297154108852978, 0.0532421387390865, 0.0737142372186105,
-0.0384820735367924,-0.047213045317213, 0.0295741565155512,-0.0456359070616521,
-0.026393327801645, 0.0326274528589774, 0.0201915799013743,-0.0406533648942984;
-0.5932972720613709,-0.0209918266600847, 0.2026367180091205, 0.1572055516251538,
-0.1259187657287346,-0.0612563733893335, 0.10269198782892028,-0.3301642954853802,
-0.0588064472272272, 0.2259098962133411, 0.0763407165237221, 0.1953977685856945;
0.3182471547714092, 0.0517233171449673,-0.1086157985195903,-0.055216064571107,
0.11886820932720038, 0.0414594815056652,-0.1230670565368351, 0.2047330641547552,
0.01459943016238838,-0.12498644339395018,-0.0577935633223318,-0.1656982226986158;
-0.2799018148394809, 0.068480479218752, 0.0509548448741404, 0.0509176474488098,
0.01172613482006036,-0.0045568810249207, 0.0402069822428226,-0.0780196840570518,
-0.0297156907710767, 0.10156931114386624, 0.0259680263379561, 0.062108712424104;
0.2739747436075013, 0.0358644611698261,-0.10043750004637351,-0.01537054898749618,
-0.01188421088973993, 0.0545292542113109,-0.0447509545683625, 0.1334135675731232,
-0.0175808083399441,-0.1361058847726861, 0.0610654582586491,-0.0461320667215304];


[RuiLoureiro]

 
Hi
        --**** The calculator v3.12.1 ****--
                              calcula62

    This is the latest version.

    Now, we may do almost all operations with complex numbers,
    the same way we do for real numbers.

    In the next version, the calculator will do more
    matrix operations with real and complex numbers,
    i hope. And, i hope to do a general revision too.

    Dez. 2013

    Good luck !   
    RuiLoureiro

   
1. matrix definition:

a11=2.453 ;a12=-5.236 ;a21=9.123  ;a22=-3.154;
z11=(2,3) ; z12=(3,-2); z21=(8,-3); z22=(0,-5);

matA=[a11+ia12, a12-ia21; 0, 2+ia22];
matB=[z11, z12; z21, z22];

2. Operations:

    2.1  - ROUND

            round(matA, 1) or
           
            matC=round(matB, 1)

    2.2  - CONJUGATE

            conj(matA) or

            matC=conj(matB)

    2.3  - CHANGE

            - lines:         change l(matA;1,2) or
                        matX=change l(matA;1,2)

            - columns:       change c(matA;1,2) or
                        matY=change c(matA;1,2)

    2.4  - ADD,SUB,MUL (x=+,-,*)
                matX=matA x matB;

    2.5  - SCALAR:
   
                matX=-matB;

                matX=-2.3*matB;

                matX=(-2.3+i3.4)*matB;
               
    2.6  - TRANSPOSE:
   
                matX=matA^t;

    2.7  - INVERSE:
   
                matX=matA^-1;

    2.8  - DETERMINANT:
   
                det(matA)        - Gauss

                delta(matA)     - Laplace (note: more than 12x12 use Gauss)

---------------------
Inverse example
20x20 matrix
---------------------

Code Select Expand


g=[1.23+i3,2,3,-4.45,5,6,7,-1.24,0,10,11,1.12,-13,14.11,15,-1.16,7,8,19,-4;
1,2.24,3,-4,5.23,6,7,8,9,-1.08,11,12,-13.23,14,15,16,-17,18.45,1,20;
1,0,3,4,5.23,-6,7,8,9,10,-2.11,0,-13.34,14,0,-16,17,18.58,19,-20;
1,2,1,0,0,6,7.25,8,9,10.12,0,12,13.45,14,15,16,17.21,18.29,-19.23,2;
-1,2.45,0,4,5,6,7.45,8,9,0,11,12,13,4,15,0,17.22,18.21,19,1;
1,2,3,4,5.24,6,0.27,8,9.89,10,11,2,-13,14,15,16,17,3,19,20;
20,9,18,17,16,5,14,13,12,11.38,10,9,8,7,6.34,5,4,3.25,2,-1;
1,2.23,-3,4,5,6,7,-1,9,10,11.45,12,13,14.89,15,-16,17,18,19,20;
1,0,3,-4,5,6,7.34,8.34,9,1,1,12.29,-13.59,14,5,16,17,18.34,19,20;
1,2,0,4,5,0,1,8,9,10,-11,12.24,13,14.22,15,16,17.45,18,19,20;
20,19.34,1.84,17,16,15,14.35,13,12.68,11,10,9,8.34,7,6.34,5,4,3,2,1;
1,2,3,4,5.24,6,7,8.24,9,10,11.23,12,13.34,14,15,-1,-17.23,1,19,1;
1,-2.81,1,4.23,5,6.21,7,8.89,9,10,11,0,13,14,15,16,17,18,19.23,20;
-1,2,3,4,5,-6.23,-7,8,9,10,1,12,13,-14,15.33,6,17,18,19,20;
1,0,0,4,5,6.34,7,8,9,10,-11.89,1.23,13,14.34,15.21,16.34,0,18,19,20;
1.34,2,3,4,5,6.23,7,0,9,0,11,0,1.35,14,15,16.62,17,18,1.92,0;
1,-2,3,4,5.45,6,1,-8,9,10,1.12,12,13,14,15,-16,17.43,2,19,0;
1,2,1,0,-1,6,7,8,9,10,11,-12.23,13,14,15,16,17,18,0,20.23;
-1.45,1,3,4.56,0,6,7,0,9,10,11,0,13.34,14,-15,16,7,18,9.34,20;
1,2,3,-4,5,6,7,8,9.45,10,0,12.48,13,14,-1.54,16.21,-17,0,19,-2];

det(g)=-7.4473766747107554e+24+i4.3514436660822637e+23

f=g^-1;

Code Select Expand


f=[-0.0194101390259696-i0.00113411916989698,-0.1370614197121183-i0.0080083910445378,
0.0066192461046662+i0.000386757348184,-0.0465182165804565-i0.0027180228386169,
-0.386511062558009-i0.0225835376468868,-0.368461737789722-i0.0215289297846751,
-0.1614524182309398-i0.009433537920411,-0.2385493532116904-i0.01393825124497746,
0.4595938754918398+i0.0268537089747356, 0.0388561956287034+i0.0022703369755786,
0.1837886974705918+i0.01073862916349639, 0.5117449650708355+i0.0299008561560835,
0.1620417483438316+i0.0094679720157843, 0.1277470356882338+i0.0074641589056987,
-0.1869878416007705-i0.01092555264098641, 0.1301116930261207+i0.0076023239756936,
0.1197250236072046+i0.0069954390438745, 0.0993442813080926+i0.005804608287473,
-0.0020717516763048-i0.00012105082236762,-0.2827179734166223-i0.0165189890137946;
0.0337604928916169+i0.0019725990690909, 0.0299769462062127+i0.01392917529187581,
-0.01306560097164575-i0.0006726957847483,-0.0192320231024803+i0.0047275184685492,
0.12136210891085013+i0.0392800567360801, 0.0866611744166858+i0.0374457534790036,
0.0386556888522107+i0.0164079654184201, 0.055693308951244+i0.0242431149745011,
-0.11648218123635306-i0.0467072621036945, 0.0604679439001707-i0.0039488483427679,
-0.0280615129773847-i0.01867794007322,-0.11348444865514418-i0.0520072339698397,
-0.1499236106659666-i0.0164678574176749,-0.0331475772238254-i0.01298258004956008,
0.0202305291773326+i0.0190030602964516,-0.0270031861935396-i0.01322289367416595,
-0.0440565862158477-i0.01216732501495661, 0.0358409776223077-i0.0100960862034869,
0.0154972153526858+i0.0002105464274417, 0.0667522074753097+i0.0287318504226455;
0.0406117941812262+i0.0023729152193714, 0.0818252740755585+i0.0167559402015824,
-0.01085170799121976-i0.0008092116085059,-0.01108316472790187+i0.0056869136256106,
0.1517248804988679+i0.0472514896246874, 0.0585950532450137+i0.0450449357517479,
0.0715022541055779+i0.0197377720948801,-0.0520828615153939+i0.0291629745696253,
-0.10913404205833358-i0.0561859603594483, 0.0430512462722827-i0.0047502214101024,
-0.0543945245926987-i0.0224684118332657,-0.12046089994367322-i0.0625615001741418,
-0.0724493563580234-i0.0198098184821957,-0.0432071231790731-i0.01561724441069838,
0.0189264569889868+i0.0228595114428722,-0.10750152552027906-i0.0159063269040366,
0.0513258689097457-i0.01463654280255493, 0.0355983503043648-i0.0121449700467418,
0.043286961062462+i0.0002532743880342, 0.0170873264137525+i0.0345626469245062;
-0.0159470029981016-i0.000931770853282,-0.0579278258913802-i0.0065795425692905,
-0.0002637164077883+i0.0003177525201018, 0.01317218340486517-i0.0022330761412033,
-0.0542926443389835-i0.0185542072666661,-0.0126062849965109-i0.0176877613994824,
-0.017181424227656-i0.0077504162797713,-0.0209795548031127-i0.01145140352135348,
0.0591087304758857+i0.022062499241108,-0.0074086132114467+i0.00186526590602018,
0.0163702885251149+i0.0088226545537184, 0.1342346535807686+i0.0245659777155073,
0.0000232256387358+i0.0077787066810621, 0.01368776224649405+i0.0061324117404943,
0.0219876624186343-i0.0089762273463687, 0.0539338060356428+i0.0062459255480201,
-0.0231183686896271+i0.0057473203698566,-0.0451954066975247+i0.0047689563500441,
0.029589433178144-i0.0000994530654642,-0.0979559737147994-i0.01357168884654252;
0.0270098173828801+i0.00157816240411039, 0.2822294053188393+i0.01114392737497949,
0.0135642886383116-i0.0005381849832173, 0.0039312280452027+i0.0037822140488184,
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det(f)=-1.3381861640951734e-25-i7.8189165959126229e-27

[RuiLoureiro]

Hi
        --**** The calculator v2014.01 ****--
                            (calcula63)
   
        This is the latest powerful version.

        In this new version, we may define up to
        30 constants or matrices and matrices
        20x21. Each matrix may be real or complex.

        We may compute the determinant of any
        defined matrix N by (N+1) - the same
        as N by N.

        We may do linear transformations also:
        add lines, sub lines, etc.

        See the examples in the files:
       
            .  Examples_real_matrix.txt
            .  Examples_complex_matrix.txt
            .  Examples_equations.txt

        Finally, we may transfer all work done
        to the file TheCalculator.txt. To do
        this, press ToFile.
        To start a new page, press Del File.
       

        To transfer to the file ThisWork.txt
        use
                file(ThisWork)
       
        Last note: now the calculator has a
        unique table of erros with 190 messages.

           
    Jan. 2014

    Good luck !   
    RuiLoureiro