@database Mathan.guide @master Develop:mathan/doc/Mathan.guide @node "Main" "Mathan V1.0" ----------------------- @{b}@{fg shine} Mathan V1.2 @{ub}@{fg text} ----------------------- Mathematical software for the Amiga. @{b}Wellcome to Mathan.@{ub} @{"I.      Brief Introduction " link "Introduction" 0} @{"II.     Distribution " link "Distribution" 0} 1996 Enrique Jimenez @{"III.    Disclaimer " link "Disclaimer" 0} @{"IV.     System requirements " link "Requirements" 0} @{"V.      Starting " link "Starting" 0} Enrique Jimenez @{"VI.     Functions " link "Functions" 0} Romero Robledo 20, 3A @{"VII.    Graphics " link "Graphics" 0} 28008 MADRID @{"VIII.   Objects " link "Objects" 0} SPAIN @{"IX.     Configuration " link "Configuration" 0} @{"X.      DisplayM " link "External" 0} E-Mail: @{"XI.     Bugs " link "Bugs" 0} @{"XII.    Application examples " link "Applications" 0} n109701@eicaiii.uc3m.es @{"XIII.   Future " link "Future" 0} @{"XIV.    History " link "History" 0} @{"XV.     Author " link "Author" 0} @endnode @node "Introduction" "Introduction" INTRODUCTION (Sorry for my bad english) Did you ever need inverting a 5x5 matrix, a calculator with complex numbers, or solving (x+3)^4+2x+3=0?... The Amiga has plenty of killer applications related to music, graphics, word proccesors, databases, etc..., but it lacks of engineering software (CAD/CAM, etc... ) and math programs like Mathemathica or MatLab for the IBM PC and Macintosh. Mathan isn't close to these programs (yet), but it has a lot of advanced features: You can plot a 3D surface and then save it as an IFF ILBM picture , solve a linear simultaneous equation system or find the eigenvalues of a square matrix. If you hate reading docs and want to see a fast demonstration of Mathan try the next command from the shell: >mathan examples/demo.mth (Use the cursor keys to rotate and +/- to magnify the image) @endnode @node "Disclaimer" "Disclaimer" DISCLAIMER This software and information is provided "as is". You use it at your own risk. No representations or warranties are made with respect to the accuracy, reliability, performance, currentness or operation of this software and information. The autor will not be liable for any damages resulting from the use of this software. @endnode @node "Distribution" "Distribution" DISTRIBUTION This software and information is freely distributable, with the following limitations: - The package must remain complete and unchanged. - The package may be included on CD-ROMs, BBS, magazines or commercial distributions of software packages, but I ask you to inform the @{"Author" link "Author" 0}. If you are using this program I ask you to send me a postcard or a message by E-Mail, so I know how many people is using my program and where my program is used. I have no intention of stopping development, but I would like to know whether the time I am investing is worth it. @endnode @node "Requirements" "Requirements" REQUIREMENTS Mathan requires the following system to run: * Workbench 3.0 or above. * 68020 or better. @endnode @node "Starting" "Starting" STARTING THE PROGRAM. There are two ways to start Mathan: from @{b}CLI@{ub} or from @{b}Workbench@{ub}. When started from the @{b}Shell@{ub} without parameters, Mathan shows a prompt where you can enter the operations. If the parameter is a file name, then Mathan reads the operations from the file. To quit Mathan you may write the command 'Exit' or 'Quit' (NOTE the Upper case). When started from the @{b}Workbench@{ub}, or when the parameter from the @{b}Shell@{ub} is -rexx, Mathan runs in the background, and accepts commands through an ARexx port called 'Mathan'. Mathan terminates when you send the command 'Exit' or 'Quit'. Mathan may be used with any text editor or word processor with ARexx support. You can find some excelent shareware text editors in the public domain, like @{b}Blacks Editor@{ub} by Marco Negri, or you can use the AmigaDos text editor, @{b}Ed@{ub}. There are supplied ARexx scripts for @{b}Blacks Editor@{ub}, @{b}ProWrite@{ub} and @{b}Ed@{ub}. If you use one of these programs, then you can try opening the file examples/demo.mth (for example), put the cursor in the line you want to evaluate and execute the ARexx macro for your editor (with @{b}Blacks@{ub} @{b}Editor@{ub} and @{b}ProWrite@{ub} you can assign the ARexx macro to a keyboard short cut, but with @{b}Ed@{ub} you can use @{b}FKey@{ub}). Otherwise you need to write an ARexx script for your text editor, then assign the ARexx script to any menu item or keystroke, like CTRL-RETURN, to invoke Mathan to insert a result. SEE THE SUPPLIED SCRIPTS. The configuration can be a bit complicated for a novice user, so I hope to include more ARexx scripts in further versions. If you have any trouble, don't hesitate to contact me. @endnode @node "Functions" "Functions" FUNCTIONS Mathan supports standard built-in functions and @{"user functions" link "user_f" 0}. Built-in functions: Trigonometric (angular measurement unit = radians) Cos()  cosine Sin()  sine Tan()  tangent ArcCos()  arc cosine ArcSin()  arc sine ArcTan()  arc tangent Cosh()  hyperbolic cosine Sinh()  hyperbolic sine Tanh()  hyperbolic tangent Sec() 1/Cos() Cosec() 1/Sec() Sinc() Sin(x)/x Cosc() Cos(x)/x Logarithmic and exponential Exp()  e to the exponent Ln()  base e logarithm Log()  base 10 logarithm Other scientific calculations Abs()  absolute value Sgn() sign Int() interger Sqrt()  square root @{"Complex" link "Complex" 0} related functions Mod()  magnitude of a complex number Arg()  phase of a complex number Re()  real part of a complex number Im()  imaginay part of a complex number @{"Matrix" link "Matrix" 0} related functions @{"Inverse()" link "Inverse_f" 0}  inverse of a matrix @{"Transpose()" link "Transpose_f" 0}  transpose of a matrix @{"EigenValues()" link "EigenValues_f" 0}  eigenvalues of a matrix @{"Det()" link "Det_f" 0}  determinant of a matrix @{"Pol()" link "Pol_f" 0}  characteristic polynomial of a matrix @{"LU()" link "LU_f" 0}  LU decomposition of a matrix @{"QR()" link "QR_f" 0}  QR decomposition of a matrix @{"LES()" link "LES_f" 0}  linear equation solver @{"Polynomial" link "Polynomial" 0} related funtions @{"Roots()" link "Roots_f" 0}  polynomial roots @{"Special functions" link "Special" 0} Sum()  number addition Prod()  number multiplication NIntegrate() numerical integration u() unit step @{"Graphics" link "Graphics" 0} functions @{"Draw()" link "graphics" 0}  draw 2D/3D functions Special commands Exit quit Mathan Quit quit Mathan Predefined constants pi pi=3.14159... i i=Sqrt(-1) j j=Sqrt(-1) @endnode @node "user_f" "User functions" USER FUNCTIONS You can define functions of eight parameters. The format is: ()= * The function name can be up to 255 characters. * The expresion may be any object: number, matrix, etc... * Everything is case dependent. Some examples: /* Constants */ a=3 /* A function of three variables */ f(x,y,z)=Sin(x)*Cos(y)+z /* You can use long names */ abcdefghij(var)=Sqrt(var+1) /* A matrix function of two parameters */ W(q,r)={{1,Sin(r),Cos(q)},{Sin(r),r,6},{Cos(q),6,q}} @endnode @node "Inverse_f" "Inverse" Inverse() Inverse of a matrix. Example: >a={{1,2,3,4},{2,3,4,1},{3,4,1,2},{4,1,2,3}} 1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 >b=Inverse(a) -0.225 0.025 0.025 0.275 0.025 0.025 0.275 -0.225 0.025 0.275 -0.225 0.025 0.275 -0.225 0.025 0.025 You can verify the result calculating a*b. The result is the identity matrix, but due to the number precision there are some rounding errors, so you can get numbers like 1e-16. You can assume these numbers to be 0. @endnode @node "Transpose_f" "Transpose" Transpose() Transpose of a matrix. Example: >c={{1,8,4},{2,3,1}} 1 8 4 2 3 1 >Transpose(c) 1 2 8 3 4 1 @endnode @node "LU_f" "LU" LU() LU decomposition of a matrix. The format is: LU(,"

","","") The function returns three matrices, P is a permutation matrix, L a low triangular and U an upper triangular matrix, and A=P*L*U Example: >a={{1,2,3,4},{2,3,4,1},{3,4,1,2},{4,1,2,3}} 1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 >LU(a,"p","l","u") >p 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 >l 1 0 0 0 0.75 1 0 0 0.5 0.769231 1 0 0.25 0.538462 0.818182 1 >u 4 1 2 3 0 3.25 -0.5 -0.25 0 0 3.384615 -0.307692 0 0 0 3.636364 You can verify that a=p*l*u. >p*l*u 1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 @endnode @node "QR_f" "QR" QR() QR decomposition of a matrix. The format is: QR(,"","") A=Q*R Q is ortogonal and R triangular. Example: >a={{1,2,3,4},{2,3,4,1},{3,4,1,2},{4,1,2,3}} 1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 >QR(a,"q","r") >q 0.182574 0.365148 0.490098 0.770154 0.365148 0.426006 0.536774 -0.630126 0.547723 0.486864 -0.676802 0.070014 0.730297 -0.669439 0.11669 0.070014 >r 5.477226 4.38178 4.016632 4.38178 0 3.286335 1.947458 0.852013 0 0 3.173968 1.493632 0 0 0 2.80056 >q*r 1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 @endnode @node "Det_f" "Det" Det() Determinant of a matrix. Example: >a={{1,2,3,4},{2,3,4,1},{3,4,1,2},{4,1,2,3}} 1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 >Det(a) 160 @endnode @node "LES_f" "LES" LES() Linear equation system solver. Format: LES(,) The function returns the x solution of: A*x=b Example: Imagine the system: 2u + v + w = 1 4u + v = -2 -2u + 2v + w = 7 In the matrix notation: 2 1 1 u 1 4 1 0 * v = -2 -2 2 1 w 7 >A={{2,1,1},{4,1,0},{-2,2,1}} >b={1,-2,7} >LES(A,b) -1 2 1 Then the solution is u = -1 v = 2 w = 1 @endnode @node "Pol_f" "Pol" Pol() Characteristic polynomial of a matrix. Det(A-xI) Example: >mat={{1,0,3},{3,-1,2},{4,1,2}} >Pol(mat) (x)^3-2*(x)^2-15*x-17 @endnode @node "EigenValues_f" "EigenValues" EigenValues() Eigenvalues of a matrix. Det(A-xI)=0 Example: >mat={{1,0,3},{3,-1,2},{4,1,2}} >EigenValues(mat) 5.377185 0 -1.688687 -0.556536 -1.688687 0.556536 The first column is the real part, and the second the imaginary part, so the eigenvalues are: 5.377185 -1.688687 - 0.556536*j -1.688687 + 0.556536*j You can get the same result with >Roots(Pol(mat)) @endnode @node "Roots_f" "Roots" POLYNOMIAL ROOTS The Roots() function calculates the polynomial roots, but this will not work with multivariable polynomials. Example: >Roots(x^4+5x^3+7x^2-3x-10) -2 0 1 0 -2 -1 -2 1 The roots are -2,1,-2+i and -2-i. You can verify the result: >(x+2)(x-1)(x+2-i)(x+2+i) (x)^4+5*(x)^3+7*(x)^2-3*x-10+0*i NOTES: * Mathan can't calculate the roots of multivariable polynomials like x^2+2y-3z+1. * For polynomials like Sin(x)*Sin(x)-1, the roots are for the polynomial variable Sin(x), not x. Example: >f(x)=Sin(x)*Sin(x)-1 >Roots(f(x)) -1 0 1 0 Then the roots are in Sin(x)=-1, Sin(x)=1. The solutions are x=pi/2+k*pi (k=0,1,2,...) @endnode @node "Special" "Special" SPECIAL FUNCTIONS Sum() Format: Sum("",) ={"",,} Example: >Sum("x^2",{"x",0,5}) 55 (0^2+1^2+2^2+3^2+4^2+5^2 = 55) Prod() Format: Prod("",) ={"",,} Example: >Prod("x^2",{"x",1,5}) 14400 (1^2*2^2*3^2*4^2*5^2 = 14400) NIntegrate() Format: NIntegrate("",) ={"",,} Example: >NIntegrate("x*x",{"x",0,3}) 9 u() Unit step, or Heaviside function: u(x)=0 if x<0 u(x)=1 if x>=0 Example: >Draw("{x,1+(x^2-x)*u(x-1)}",{"x",-2,2,100}) @endnode @node "Graphics" "Graphics" GRAPHICS Drawing functions. Draw("",,[]) = {x,y,z,[color]} or {x,y} = {"name",start,stop,[count]} Drawing 2D functions. This example draws the Sin() function calculating 30 points between 0 and 2pi. The more points (count parameter), the more accurate plot. Note that the count parameter is not the step between points. The step is calculated (stop-start)/count. Draw("{x,Sin(x)}",{"x",0,2pi,30}) (@{"click here" link "sin.gif/main" 0} to see the result) You can print or save (IFF ILBM) the image in the Project menu, or set axis labels and ranges in the settings menu. Mathan can draw multiple functions if is a matrix, for example: Draw("{{x,Sinc(x)},{x,Exp(-x)}}",{"x",0,2pi,40}) (@{"click here" link "multi2.gif/main" 0}) There is a mode submenu in the settings menu to change the way points are connected (@{"click here" link "multi1.gif/main" 0}). Drawing 3D functions. Try the next line (draws a @{"cube" link "cube.gif/main" 0}): Draw("{Cos(x),Sin(x),y}",{"x",-pi,pi,4},{"y",0,Sqrt(2),1}) This example draws the function f(x,y)=x*x-y*y in the range -2