Mathomatic Command Reference
LHS is shorthand for the Left-Hand Side of an equation. Similarly, RHS used here always means the Right-Hand Side.
In this document, text enclosed by straight brackets [like this] means it is optional and may be omitted. The word "expression" always means a mathematical expression or formula.
At the Mathomatic main prompt, you may enter:
Mathomatic has about 40 commands that may be typed at the main prompt. They are simple English words and are described below, in alphabetical order. If the command is longer than 4 letters, you only need to type in the first 4 letters for Mathomatic to recognize the command. Option words and arguments for commands follow the command name and are separated by spaces. Commands are not executed until you press the Enter key, and any missing command line arguments that don't have a default are prompted for. For example, the Mathomatic command
help calculate
gives short help and usage information for the calculate command. "help" is the command, "calculate" is the argument (which is also a command, in this case).
Many commands have an optional equation number range argument, which specifies the equation spaces that the command is to operate on. An equation number range may be a single equation number, or a range of equation numbers separated by a dash (like "2-7", which means every equation space between and including equation numbers 2 and 7), or the word "all", which specifies all equation spaces. If omitted, the current expression or equation is assumed. If pluralized as "equation-number-ranges" in the command syntax, that means multiple equation numbers and ranges may be specified for that command, separated by spaces.
A greater-than character (>) may be appended to the end of any command line, followed by a file name. This will redirect the output of the command to that file. If the file already exists, it will be overwritten without asking. Note that any debugging output will be redirected, too. Two greater-than characters (>>) next to each other will append command output to a file, like the Unix shell does. For example, the Mathomatic command
list export all >filename
will output all stored expressions and equations to a file in exportable, single-line per equation format, so they can be read in by a different math program. "list" is the command, "export" is the option word, and "all" is the equation number range.
Command option words, such as "export" in the above list command line, always come immediately after the command name and before anything else on the command line. These words direct the command to do a different, but related, task.
If Mathomatic becomes unresponsive (a rare occurrence), pressing Control-C once will usually safely abort the current operation and return you to the main prompt. If not, pressing Control-C three times in a row will exit Mathomatic, with a warning displayed the second time.
Syntax: #["+" or "−"]equation-number
To change the current equation number at the main prompt, type a pound sign (#) followed by the equation number you wish to select. The equation number may be preceded by plus (+) or minus (−), to select an equation relative to the current equation. This syntax also works when prompted for an expression, the RHS or the expression at that equation number is substituted.
New feature: Selecting expressions is now possible by typing only the equation number at the main prompt. This is called the "autoselect" option.
Syntax: variable or "0"
Mathomatic can solve symbolic equations for any variable or for zero. Solving is accomplished internally by applying identical mathematical operations to both sides of the equation and simplifying, or by plugging the general coefficients of the solve variable into the quadratic formula. The Mathomatic solve algorithm is the best possible for general algebra, however the result is not verified by plugging the solutions into the original equation unless the "solve verify" command is used for solving.
To automatically solve the current equation for a variable, type the variable name at the main prompt. Mathomatic will proceed to manipulate the current equation until all of the solutions for the specified variable are determined. If successful, the current equation is replaced with the solutions and then displayed.
Automatic cubic (third degree), quintic (fifth degree), and higher degree polynomial equation solving is not supported. Cubic polynomial equations can be manually solved with the equations in file "tests/cubic.in". Quartic (fourth degree) polynomial equations can be automatically solved if they are biquadratic; that is, containing only degree four, degree two, and degree zero terms of the solve variable. Biquadratic polynomial equations of any degree can be generally solved by Mathomatic because they can be plugged into the quadratic formula.
Note that running the simplify command is a good idea after solving. The solve routine only unfactors the equation as needed to solve it and the result is not completely simplified.
To solve for zero, type in "0" at the main prompt. Zero solving is a special solve that will always be successful and will transform most divide operators into equivalent multiplications and subtractions, so that the result will more likely be a valid polynomial equation:
1—> a=b+1/b ; this is actually a quadratic equation
1
#1: a = b + -
b
1—> 0 ; solve for zero
#1: 0 = (b*(b - a)) + 1
1—> unfactor ; expand, showing that this is a quadratic polynomial equation in "b"
#1: 0 = (b^2) - (b*a) + 1
1—> b ; solve for "b"
Equation was quadratic.
1
(((((a^2) - 4)^-)*sign1) + a)
2
#1: b = -----------------------------
2
1—> a ; solve for "a", to check the answer
Raising both sides to the power of 2 and unfactoring...
((b^2) + 1)
#1: a = -----------
b
1—> simplify
1
#1: a = b + -
b
1—>
You can prefix the solve variable name with "=" to solve and swap equation sides, putting the solve variable on the right-hand side. Typing "=" by itself will only swap sides of the current equation and display.
To see all of the steps performed during a solve operation, type "set debug 2" before solving.
Syntax: approximate [equation-number-ranges]
This command operates on the current or specified equation spaces. It substitutes the special constants pi# and e# (which stand for the universal constants pi and e) with their respective floating point values and approximates all constants and roots. This allows them to combine with other constants and may help with simplification and comparisons.
Syntax: calculate ["factor"] [variable number-of-iterations]
This command is the formula calculator that prompts for the value of each normal variable in the current expression or the RHS of the current equation. It then temporarily substitutes any entered values, and simplifies, approximates, and substitutes all "sign" variables with all possible combinations of values (+1 and -1), displaying each solution as it does so. If all variables are supplied with constant values, then each solution will be a constant, otherwise the result will contain the variables you didn't enter values for. Nothing is modified by this command.
This command is used to temporarily plug in values and approximate expressions and expand "sign" variables. When an expression with only numbers is entered at the main prompt, this calculate command is automatically invoked on it, displaying the calculated result without storing.
If there are any simple numerical fractions in the result, that is displayed alongside the result.
To only simplify and expand "sign" variables in stored expressions without approximating, use the "simplify sign" command instead.
If a variable and number of iterations are specified on the calculate command line, you will be prompted for the initial value of the variable, and the calculation will be iterated, with the simplified result repeatedly plugged back into the variable. This will be done until convergence (the output equals the input) or when the specified number of iterations have been performed (if non-zero), whichever comes first. To see all of the intermediate values, type "set debug 1" before this.
"calculate factor" factors all integers and variables before display.
Examples of using the calculate command:
1—> y=x^2+x
#1: y = (x^2) + x
1—> x ; solve for x
Equation is a degree 2 polynomial in x.
Equation was quadratic.
1
-1·(1 + (((1 + (4·y))^—)·sign))
2
#1: x = ———————————————————————————————
2
1—> calculate
Enter y: 0
There are 2 solutions.
Solution number 1 with sign = 1:
x = -1
Solution number 2 with sign = -1:
x = 0
1—>
An example of iteration:
1—> x_new=(x_old+(y/x_old))/2 ; iterative formula for calculating the square root of y
y
(x_old + -----)
x_old
#2: x_new = ---------------
2
2—> calculate x_old 1000 ; iterate up to 1000 times to calculate the square root of 100
Enter y: 100
Enter initial x_old: 1
Convergence reached after 9 iterations.
x_new = 10
2—>
Syntax: clear [equation-number-ranges]
This command clears the specified equation spaces so that they are empty and can be reused. They are deleted from RAM only.
"clear all" quickly clears all equation spaces and restarts Mathomatic.
Syntax: code ["c" or "java" or "python" or "integer"] [equation-number-ranges]
This command outputs the current or specified equations as floating point or integer assignment statements in C, Java, or Python programming language code. The default is C double precision floating point code. The output from this command should compile correctly and emulate the equation from Mathomatic, if no warnings are given.
With "code integer", integer arithmetic is assumed, otherwise double precision floating point arithmetic is assumed. "code integer" is more generic and should work with any language.
To represent factorials, the user supplied function fact() is called,
since there is no equivalent function or operator in these languages.
fact() functions for several languages are supplied in the
directory "factorial" in the Mathomatic source distribution.
For the most efficient code, use the simplify and optimize commands on your equations before running this code command.
The C and Java languages require that all variables be defined before use. The variables command is provided for this. The output of the variables command should be prepended to the output of the code command before compiling.
Syntax: compare ["symbolic"] equation-number ["with" equation-number]
This command compares two equation spaces to see if they are mathematically the same (equal). If only one equation number is supplied, the comparison is between the current equation and the specified equation. The comparison will be faster and more accurate if both equations are previously solved for the same variable.
The simplify command is automatically used on both expressions if needed. If this compare command says the equations or expressions are identical, then they are definitely identical. If this command says the equations or expressions differ, then they might be identical if they are too hard for Mathomatic to simplify completely.
The "symbolic" option uses the "simplify symbolic" option when simplifying. This option sometimes simplifies more, but is not 100% mathematically correct.
Syntax: copy [equation-number-range]
This command simply duplicates the current or specified equation spaces. The new, duplicated expressions are stored in the next available equation spaces and displayed, along with their new equation numbers.
Syntax: derivative ["nosimplify"] [variable or "all"] [order]
Alternate command name: differentiate
This command computes the exact symbolic derivative of a function with respect to the specified variable, using the current expression or RHS of the current equation as the function. It does this by recursively applying the proper rule of differentiation for each operator encountered. The result is fully simplified with the simplify command, unless the "nosimplify" option is specified. If successful, the derivative is placed in the next available equation space, displayed, and becomes the current equation. The original equation is not modified.
Specifying "all" computes the derivative of the current expression with respect to all normal variables. It is equivalent to adding together the derivatives with respect to each variable.
Specifying the order allows you to repeatedly differentiate and simplify. The default is to differentiate once (order=1).
If differentiation fails, it is probably because symbolic logarithms are required. Symbolic logarithms are not implemented in Mathomatic. Also, factorial, modulus, and integral divide operators cannot be differentiated if they contain the specified variable. Because this command handles almost everything, a numerical differentiation command is not needed.
Some examples:
1—> x^3+x^2+x+1 #1: (x^3) + (x^2) + x + 1 1—> derivative ; no need to specify the variable if there is only one Differentiating with respect to (x) and simplifying... #2: (3*(x^2)) + (2*x) + 1 2—> a*x^n #3: a*(x^n) ; show a general rule of differentiation 3—> derivative x Differentiating with respect to (x) and simplifying... #4: a*n*(x^(n - 1)) 4—> integrate x ; undo the differentiation #5: a*(x^n) 5—>
Syntax: display ["factor"] [equation-number-ranges]
This command displays stored expressions in nice looking multi-line 2D (two dimensional) fraction format, where division is displayed as a numerator over a fractional line (made up of dashes) over a denominator. If the width (number of columns) required for this 2D display exceeds the screen width, the expression is displayed instead in single-line format by the list command. The screen width is set automatically on startup, or by the "set columns" option.
Non-integer constants are converted to reduced fractions, if they are exactly equal to a simple fraction and it would improve readability.
The "factor" option causes all integers, less than or equal to 15 decimal digits long, to be factored into their prime factors before display, including the numerator and denominator of fractions. To always factor integers like this before display, use the "set factor_integers" option.
Syntax: divide [variable]
This command is for doing and experimenting with polynomial and numerical division and Greatest Common Divisors (GCD). It simply prompts for two expressions and divides them, displaying the result and the GCD. Mathomatic has general symbolic polynomial division and GCD routines used by the simplify command which this divide command calls without any other processing if two polynomials are entered.
This command prompts for the dividend (the main polynomial) and then the divisor (what you want to divide the main polynomial by). The polynomial quotient, remainder, and GCD are displayed. The power of the highest power term in the dividend must be greater than or equal to the power of the highest power term in the divisor, otherwise the polynomial division will fail (as it should). In other words, the degree of the divisor polynomial must be less than or equal to the degree of the dividend polynomial. A variable may be specified on the command line as the base variable of the two polynomials, but is usually not necessary because a base variable is automatically selected.
If two numbers are entered instead of polynomials, the result of the numerical division, the GCD, and the Least Common Multiple (LCM) are displayed. The LCM of two numbers is the smallest positive number that can be evenly divided by both numbers separately, without remainder. The LCM is the same as the Lowest Common Denominator (LCD) of two fractions and is the two numbers multiplied together, divided by the GCD.
The Greatest Common Divisor of a and b is defined as the greatest positive number or polynomial that evenly divides both a and b without remainder. In Mathomatic, the GCD is not necessarily integer, unless both a and b are integers. The Euclidean GCD algorithm is used by Mathomatic to compute the GCD for numbers and polynomials.
The GCD is the best way to reduce any fraction to its simplest form. Just divide the numerator and denominator by their GCD, and replace them with the quotients (there will be no remainder), and the fraction is completely reduced. The GCD is also used when factoring polynomials and for simplifying.
The Euclidean GCD algorithm of successive divides is the best way to compute the GCD for numbers and polynomials. Multivariate polynomial GCD computation usually requires recursion of the GCD algorithm or other methods. Currently the polynomial GCD routine in Mathomatic is not recursive, making it univariate and simpler and faster. Because it is univariate, Mathomatic may be unable to find the GCD of polynomials with many variables.
The polynomial division algorithm in Mathomatic is generalized and able to handle any number of variables (multivariate), and division is always done with one selected base variable to be proper polynomial division. Being generalized, the coefficients of the polynomials may be any mathematical expression.
An example of polynomial division:
1—> divide Enter dividend: (x^4) - (7*(x^3)) + (18*(x^2)) - (22*x) + 12 Enter divisor: (x^2) - (2*x) + 2 Polynomial division successful using variable (x). The quotient is: 6 + (x^2) - (5*x) The remainder is: 0 Polynomial Greatest Common Divisor (iterations = 1): (x^2) - (2*x) + 2 1—>
The number of iterations displayed is the number of polynomial divides done to compute the GCD with the Euclidean algorithm.
Syntax: echo [text]
This command outputs a line of text, followed by a newline.
Syntax: edit [file-name]
This command invokes the ASCII text editor specified by the EDITOR environment variable. By default, all equation spaces are edited. Access to shell (/bin/sh) is required for this command to work.
Type "edit" at the Mathomatic prompt to edit all expressions and equations you have entered for the current session. When you are done editing Mathomatic expressions and commands, save and exit the editor to have them automatically read in by Mathomatic. If Mathomatic gets an error reading in its new input, observe where the error is and continue, to automatically re-enter the editor.
To edit an existing file and have it read in, specify the file name on the edit command line.
Syntax: eliminate variables or "all" ["using" equation-number]
This command is used to combine simultaneous equations, by automatically substituting variables in the current equation. It will scan the command line from left to right, replacing all occurrences of the specified variables in the current equation with the RHS of solved equations. The equation to solve can be specified with the "using" argument. If "using" is not specified, Mathomatic will search backwards, starting at the current equation minus one, for the first equation that contains the specified variable. The equation to solve is solved for the specified variable, then the RHS is inserted at every occurrence of the specified variable in the current equation. That effectively eliminates the specified variable from the current equation, resulting in one less unknown.
There is an advantage to eliminating multiple variables in one command: each equation will be used only once. If the same equation is solved and substituted into the current equation more than once, it will cancel out.
"eliminate all" is shorthand for specifying all normal variables on the command line. "repeat eliminate all" will eliminate all variables repeatedly until nothing more can be substituted, using each equation only once.
Here is a simple example of combining two equations:
1—> ; This arrives at the distance between two points in 3D space from the
1—> ; Pythagorean theorem (distance between two points on a 2D plane).
1—> ; The coordinate of point 1, 2D: (x1, y1), 3D: (x1, y1, z1).
1—> ; The coordinate of point 2, 2D: (x2, y2), 3D: (x2, y2, z2).
1—>
1—> L^2=(x1-x2)^2+(y1-y2)^2 ; Distance formula for a 2D Cartesian plane.
#1: L^2 = ((x1 − x2)^2) + ((y1 − y2)^2)
1—> distance^2=L^2+(z1-z2)^2 ; Add another leg.
#2: distance^2 = (L^2) + ((z1 − z2)^2)
2—> eliminate L ; Combine the two equations.
Solving equation #1 for (L) and substituting into the current equation...
#2: distance^2 = ((x1 − x2)^2) + ((y1 − y2)^2) + ((z1 − z2)^2)
2—> distance ; Solve to get the distance formula for 3D space.
1
#2: distance = ((((x1 − x2)^2) + ((y1 − y2)^2) + ((z1 − z2)^2))^—)·sign2
2
Finished reading file "pyth3d.in".
2—>
Syntax: extrema [variable] [order]
This command computes possible extrema (the minimums and maximums) of the current expression by default, or possible inflection points when order is 2. The result is placed in the next available equation space, displayed, and becomes the current equation. The original expression is not modified.
By default (order=1) this command computes stationary points. The stationary points of function f(x) are the values of x when the slope (derivative) equals zero. Stationary points are likely the local minimums and maximums of the function, unless the point is determined to be an inflection point.
For y=f(x), where f(x) is the RHS and x is the specified variable, this command gives the values of x that make the minimums and maximums of y. This is computed by taking the derivative of f(x), setting it equal to zero, and then solving for x.
The number of derivatives to take before solving can be specified by the order argument (default is 1). When order is 2, possible points of inflection are determined. A point of inflection is a point on a curve at which the second derivative changes sign from positive to negative or negative to positive.
1—> y=x^2 #1: y = x^2 1—> extrema x #2: x = 0 2—>
This function is a parabola, with the minimum at x=0.
Syntax: factor ["number" [integers]] or ["power"] [equation-number-range] [variables]
Alternate command name: collect
This command will factor manually entered integers when "numbers" is specified on the command line, otherwise this command will factor variables in expressions in the specified equation spaces. This command does not factor polynomials. To factor polynomials with repeated or symbolic factors, use the simplify command. To factor integers in equation spaces and display, use the "display factor" command.
"factor number" will prompt for an integer to factor, which may be up to 15 decimal digits. The plural "factor numbers" will repeatedly prompt for integers to factor. Multiple integers can be specified on the same line and should be separated with spaces.
Without the "number" option, this command will factor out repeated sub-expressions in equation spaces. When factoring expressions, this command does some basic simplification and factors out any common (equal) sub-expressions it can, unless variables are specified on the command line, in which case only common sub-expressions containing those variables are factored out. This collects together terms involving those variables.
For example, with the following expression:
(b*c) + (b*d)
variable b factors out and the result of this command is:
b*(c + d)
If no variables are specified on the command line, this command factors even more: the bases of common (equal) bases raised to any power are factored out. This is called Horner factoring or Horner's rule.
For example:
1—> (2+3x)^2*(x+y) #1: ((2 + (3·x))^2)·(x + y) 1—> unfactor ; expand #1: (4·x) + (12·(x^2)) + (9·(x^3)) + (4·y) + (12·x·y) + (9·(x^2)·y) 1—> factor x ; collect terms containing x #1: (x·(4 + (12·y))) + ((x^2)·(12 + (9·y))) + (9·(x^3)) + (4·y) 1—> x^3+2x^2+3x+4 ; enter another expression #2: (x^3) + (2·(x^2)) + (3·x) + 4 2—> factor ; Horner factoring #2: (x·((x·(x + 2)) + 3)) + 4 2—>
To undo any kind of factoring in selected equation spaces, use the unfactor command.
Syntax: fraction [equation-number-range]
Alternate command name: together
This command reduces and converts expressions with any algebraic fractions in them into a single simple algebraic fraction (usually the ratio of two polynomials), similar to what Maxima's rat() function does. It does this by combining all terms added together with like and unlike denominators to a single simple fraction with a like denominator. Unlike denominators are combined by converting the terms to what they would be over like (common) denominators. Fractions are reduced by cancelling out the Greatest Common Divisor (GCD) of the numerator and denominator. The result of this command is equivalent to the original expression. Note that algebraic fractions added together with like denominators are automatically combined by almost any Mathomatic command.
Example:
1—> 1/x+1/y+1/z
1 1 1
#1: — + — + —
x y z
1—> fraction
(((y + x)·z) + (x·y))
#1: —————————————————————
(x·y·z)
1—> unfactor
((y·z) + (x·z) + (x·y))
#1: ———————————————————————
(x·y·z)
1—> unfactor fraction
1 1 1
#1: — + — + —
x y z
1—>
If more simplification is needed, use the "simplify fraction" command instead.
Syntax: help [topic or command-names]
This command is provided as a quick reference while running Mathomatic. If the argument is a command name, a one line description and one line syntax of that command are displayed. Command names may be abbreviated.
Entering this command by itself will display a list of topics and commands. "help copyright" will display the copyright and license notice for the currently running version of Mathomatic.
To create a quick reference of all Mathomatic commands, type:
help all >quickref.txt
Syntax: imaginary [variable]
This command copies the imaginary part of a complex number expression to the next available equation space. If the current expression or RHS of the current equation is not complex, the warning message "Expression is not a mix" will be displayed. A complex number expression contains both imaginary and real number parts. To copy only the real number part, see the real command.
The separation variable may be specified on the command line, the default is the imaginary unit i#. i# is really a mathematical constant equal to the square root of -1, but it can often be specified where variables are required in Mathomatic.
If successful, the result may contain the imaginary unit i# or the specified separation variable.
1—> (a+b*i#)/(c+d*i#)
(a + (b·i#))
#1: ————————————
(c + (d·i#))
1—> imaginary
i#·((b·c) − (a·d))
#2: ——————————————————
((c^2) + (d^2))
2—>
Syntax: integrate ["constant" or "definite"] variable [order]
Alternate command name: integral
This command computes the exact symbolic integral of a polynomial function with respect to the specified variable, using the current expression or RHS of the current equation as the function. If successful, the simplified integral is placed in the next available equation space, displayed, and becomes the current equation.
The default is to compute and display the indefinite integral, also known as the antiderivative or primitive. The antiderivative is the inverse transformation of the derivative.
"integrate constant" simply adds a uniquely named constant of integration ("C_1", "C_2", etc.) to each integration result. The constants of integration here are actually variables that may be set to any constant.
"integrate definite" also integrates, but prompts you for the lower and upper bounds for definite integration. If g(x) is the indefinite integral (antiderivative) of f(x), the definite integral is:
g(upper_bound) − g(lower_bound)
Specifying the order allows you to repeatedly integrate. The default is to integrate once (order=1).
This command is only capable of integrating polynomials.
1—> x^3+x^2+x+1
#1: (x^3) + (x^2) + x + 1
1—> integrate x
(x^4) (x^3) (x^2)
#2: ----- + ----- + ----- + x
4 3 2
2—> derivative x ; check the result
Differentiating with respect to (x) and simplifying...
#3: (x^3) + (x^2) + x + 1
3—> compare 1
Comparing #1 with #3...
Expressions are identical.
3—>
Syntax: laplace ["inverse"] variable
This command computes the Laplace transform of a polynomial function of the specified variable, using the current expression or RHS of the current equation as the function. If successful, the transformed function is placed in the next available equation space, displayed, and becomes the current equation.
This command only works with polynomials.
A Laplace transform can be undone by applying the inverse Laplace transform. That is accomplished by specifying the "inverse" option to this command.
1—> y=1
#1: y = 1
1—> laplace x ; compute the Laplace transform of 1
1
#2: y = -
x
2—> a*x^n ; a general polynomial term
#3: a*(x^n)
3—> laplace x
a*(n!)
#4: -----------
(x^(n + 1))
4—> laplace inverse x
#5: a*(x^n)
5—>
Syntax: limit variable expression
This command takes the limit of the current expression as variable goes to the specified expression. The result is always an equation and placed in the next available equation space and displayed.
L'Hopital's rule for taking limits is not used by this command. Instead the limit is taken by simplifying, solving, and substituting. This command is experimental and does not know about negative infinity and occasionally gives a wrong answer when dealing with infinities.
1—> 2x/(x+1)
2·x
#1: ———————
(x + 1)
1—> limit x inf ; take the limit as x goes to infinity
Solving...
#2: answer = 2
1—>
Syntax: list ["export" or "maxima" or "gnuplot"] [equation-number-ranges]
This command displays stored expressions in single-line (one dimensional) format.
"list export" outputs expressions in general exportable, single-line format. You can cut-and-paste the expressions or redirect them to a file, so they can be read in with a different math program.
"list maxima" is for making output compatible with the free computer algebra system Maxima.
"list gnuplot" is for making expression output that is compatible with the free graphing utility Gnuplot.
The list command simply outputs expressions and equations as stored internally by Mathomatic. There is no simplification and nothing more is done.
To display equation spaces in better looking multi-line fraction format, use the display command.
Syntax: nintegrate ["trapezoid"] variable [partitions]
This is a numerical integrate command that will work with almost any expression and will not generally compute the exact symbolic integral except for the simplest of expressions. This command will prompt you for the lower and upper bounds to perform numerical definite integration on the current expression or the RHS of the current equation, with respect to the specified variable. These bounds may be any expression not containing infinity.
This command uses Simpson's rule to do the approximation. Accuracy varies widely, depending on the expression integrated, the interval between the lower and upper bounds, and the number of partitions. The default is to split the interval into 1000 partitions. Setting the number of partitions greater than 10000 seldom is helpful, because of accumulated floating point round-off error.
If "trapezoid" is specified on the command line, the trapezoid method is used instead, which is usually less accurate than Simpson's rule. The way the trapezoid method works is: the interval from the lower bound to the upper bound is divided into 1000 partitions to produce 1000 trapezoids, then the area of each trapezoid is added together to produce the result. This means that the accuracy usually decreases as the interval increases. Simpson's rule uses the same method, with quadratic curves bounding the top of each trapezoid, instead of straight lines, so that curves are approximated better.
If the integration fails, chances of success are greater if you reduce the number of variables involved in the integration.
If there are any singularities, such as division by zero, between the bounds of integration, the computed result will be wrong.
Here is an example of successful numerical integration:
1—> y=x^0.5/(1-x^3)
1
(x^—)
2
#1: y = ———————————
(1 − (x^3))
1—> nintegrate x
Warning: Singularity detected, result of numerical integration may be wrong.
Enter lower bound: 2
Enter upper bound: 4
Approximating the definite integral using Simpson's rule (1000 partitions)...
Numerical integration successful.
#2: y = -0.16256117185712
2—>
This example avoids the singularity at x=1 and is accurate to 12 digits.
Syntax: optimize [equation-number-range]
This command splits the specified equations into smaller, more efficient equations with no repeated expressions. Each repeated sub-expression becomes a new equation solved for a temporary variable (named "temp").
Note that the resulting assignment statements may be in the wrong order for inclusion in a computer program with the code command; the order and generated code should be visually checked before using.
To undo this command and substitute the split up equations into the original equation, use the eliminate command.
1—> y = (a+b+c+d)^(a+b+c+d) #1: y = (a + b + c + d)^(a + b + c + d) 1—> optimize #2: temp = a + b + c + d #1: y = temp^temp 1—> eliminate temp ; undo the optimization Solving equation #2 for (temp)... #1: y = (a + b + c + d)^(a + b + c + d) 1—>
Syntax: pause [text]
This command waits for the user to press the Enter key. It is useful in text files (scripts) that are read in to Mathomatic. Optionally, a one line text message may be displayed.
Typing "quit" or "exit" before pressing the Enter key will fail this command and abort the current script.
This command is ignored during test mode and when input is not a terminal.
Syntax: plot [expression]
This command automatically plots the given mathematical expression in 2D or 3D with the free graphing utility Gnuplot. If no expression is given, the current equation is plotted. The expression must contain the variable x to be successfully plotted. If it also contains the variable y, a 3D surface plot is performed in Cartesian space, with the x, y, and z axes projected on your 2D display.
Plotting of an equation space with the imaginary number i# in it is allowed, however plots are always plotted on real Cartesian space, showing only real number points. If there are no real number points, the plot will fail.
Gnuplot must be installed and accessible from shell by typing "gnuplot" for this command to work. All functions and operators of Gnuplot can be used in the plot expression.
Syntax: product variable start end [step-size]
This command performs a mathematical product (∏) of the current expression or the RHS of the current equation as the index variable goes from start to end in steps of step-size (default 1). The result is stored and displayed. The current equation is not changed.
1—> y=a*x #1: y = a*x 1—> product Enter variable: x x = 1 To: 10 #2: y = 3628800*(a^10) 1—> 10! Answer = 3628800 1—>
To see all of the intermediate results, type "set debug 1" before this.
Syntax: push [equation-number-range]
This command pushes the current or specified expressions into the readline history, for easy editing and re-entry by using the arrow keys. After this command, the pushed expressions are accessed by pressing the cursor UP (up arrow) key.
This command only exists if Mathomatic was compiled with readline support.
Syntax: quit [exit-value]
Alternate command name: exit
Type in this command to exit Mathomatic. All expressions in memory are discarded. To save all your expressions stored in equation spaces, use the save command before quitting.
The optional decimal exit value argument is the exit status returned to the operating system. The default is 0, meaning OK.
Another way to quickly exit Mathomatic is to enter your operating system's End-Of-File (EOF) character at the beginning of an input line. The EOF character for Unix-like operating systems is Control-D.
Syntax: read file-name
This command reads in a text file as if you typed the text of the file in at the main prompt. The text file (also known as a script) should contain Mathomatic expressions and commands. Read commands may be nested; that is, the file read in may contain further read commands. If any command or operation returns with an error, the read operation is aborted.
Expressions saved with the save command are restored using this read command.
This command is automatically executed when you start up Mathomatic with file names on the shell command line.
The default file name extension (suffix) for Mathomatic input files is ".in". A file name extension is not required.
Syntax: real [variable]
This command copies the real part of a complex number expression to the next available equation space. If the current expression or RHS of the current equation is not complex, the warning message "Expression is not a mix" will be displayed. A complex number expression contains both imaginary and real number parts. To copy only the imaginary number part, see the imaginary command.
The separation variable may be specified on the command line, the default is the imaginary unit i#.
There will be no imaginary numbers in the result.
1—> (a+b*i#)/(c+d*i#)
(a + (b·i#))
#1: ————————————
(c + (d·i#))
1—> real
((a·c) + (b·d))
#2: ———————————————
((c^2) + (d^2))
2—>
Syntax: repeat command-line
Any command may be preceded by "repeat", which sets the repeat flag for that command. Most commands ignore the repeat flag. Currently the calculate, eliminate, divide, roots, and simplify commands are repeatable.
Syntax: replace [variables ["with" expression]]
By default, this command prompts you for a replacement expression for each variable in the current expression or equation. If an empty line is entered for a variable, that variable remains unchanged. The result is placed in the current expression or equation and displayed.
This command is very useful for renaming or substituting variables. It is smart enough to do variable interchange.
If variables are specified on the command line, you will be prompted for those variables only and all other variables will be left unchanged.
If "with" is specified, you won't be prompted and all variables specified will be replaced with the expression that follows.
Syntax: roots root real-part imaginary-part
This command displays all complex number roots of a given positive integer root of a complex number. The number of the root equals the number of correct solutions. For example, "3" would give the 3 roots of the cubed root. This command will also convert rectangular coordinates to polar coordinates.
The floating point real part (X coordinate) and imaginary part (Y coordinate) of the complex number are prompted for. Just enter an empty line if the value is zero. The polar coordinates of the given complex number are displayed first, which consist of an amplitude (distance from the origin) and an angle (direction). Then each root is calculated and displayed, along with an "Inverse check" value, which should equal the original complex number. The "Inverse check" is calculated by repeated complex number multiplication of the root times itself.
Since double precision floating point is used, the results are only accurate from 10 to 12 decimal digits.
1—> roots Enter root (positive integer): 3 Enter real part (X): 8 Enter imaginary part (Y): The polar coordinates before root taking are: 8 amplitude and 0 radians (0 degrees). The 3 roots of (8)^(1/3) are: 2 Inverse check: 8 -1 +1.73205080757*i Inverse check: 8 -1 -1.73205080757*i Inverse check: 8 1—>
Syntax: save file-name
This command saves all expressions in all equation spaces into the specified text file. If the file exists, Mathomatic will ask you if you want to overwrite it. The saved expressions and equations can be reloaded (restored) at a later time by using the read command. You can edit the saved expressions with your favorite ASCII text editor.
Syntax: set [["no"] option] ...
This command sets various options listed below, for the current session. They remain in effect until you exit Mathomatic. Typing "set" without arguments shows all current option settings.
The specified option is turned on, unless it is preceded by "no", which turns it off. Some options can be followed by a number and some options can be followed by text, setting that option to the following value.
To permanently change the default settings of Mathomatic, set options can be put in the file ~/.mathomaticrc (for Microsoft Windows: mathomatic.rc in the same directory as the Mathomatic executable). It should be a text file with one set option per line, without the word "set". That file is loaded every time Mathomatic starts up, when not in test mode. The command "set save" conveniently saves all current session options in that file, making them permanent. "set no save" removes that file, so then Mathomatic will start up with all options set to the defaults.
"set precision" followed by an integer less than or equal to 14 sets the display precision in number of decimal digits for most numerical output. All arithmetic in Mathomatic is double precision floating point, so it is not useful to set this higher than 14 digits. Display output is rounded to the precision set by this option, though internally all constants are rounded to fit in a double precision float data type of approximately 15 decimal digits precision.
"set no autosolve" will disable solving by typing the solve variable at the main prompt, unless an equals sign (=) is included. This allows entry of single variable expressions into equation spaces. Solving is always allowed using the solve command.
"set no autocalc" will disable automatic approximation (calculation) of purely numerical input entered at the main prompt. Instead, numerical input will be entered into equation spaces, so it can be operated on by commands.
"set no autoselect" will disable selecting of equation spaces by only typing in the equation number. Selecting is still possible using the "#" operator.
"set auto" and "set no auto" turn on and off all three of the above options at once. If turned off, all expressions entered at the main prompt will be entered into equation spaces, so they can be operated on by Mathomatic commands.
"set debug" followed by an integer sets the debug level number. The initial debug level is 0, for no debugging. If the level number is 2 ("set debug=2"), Mathomatic will show you how it solves equations. Level 4 debugs the simplify command and its polynomial routines. Levels 5 and 6 show all intermediate expressions. Set the debug level to -1 for suppression of warnings and helpful messages.
"set case_sensitive" will set alphabetic case sensitive mode, while "set no case" will set case insensitive mode (all alphabetic characters will be converted to lower case). "set case" is the default.
"set color" enables color mode. When color mode is on, ANSI color escape sequences are output to make expressions easier to read. Requires a terminal emulator that supports ANSI color escape sequences. Put the line "no color" in your ~/.mathomaticrc file to always startup Mathomatic with color mode disabled, unless the -c or -b option is given.
"set bold_colors" enables highlighting in color mode. It makes all output brighter. Use this if any colors are difficult to see. This command can be shortened to "set bold". The -b option also sets this.
"set columns" followed by a positive integer sets the expected number of character columns (width) on a terminal screen with line wrap. When an expression is to be displayed in multi-line fraction format (two dimensional) and it is wider than this number of screen columns, single-line format is used instead, because otherwise the expression would not display properly due to wrap-around. "set no columns" or "set columns=0" does no checking for screen size and always displays in fraction format, which is useful for terminals that don't wrap lines. In most cases, this value is set automatically on startup. This value only affects 2D expression output.
"set wide" sets the number of screen columns (like "set columns=0" above does) and screen rows to 0, so that no checking for screen size is done, forcing 2D display of expressions that are too wide to display properly on a terminal with line wrap. Setting this option is useful if you are not using a terminal that wraps lines.
"set no display2d" will set the expression display mode to single-line format (one dimensional) using the list command, instead of the default fraction format (two dimensional) using the display command. Single-line format is useful when feeding Mathomatic output into another program.
"set no fractions_display" disables the automatic conversion of floating point fractions like .5 to 1/2 for display. This only affects numerical fractions.
"set no prompt" turns off Mathomatic prompt output, exactly like the -q (quiet mode) option does.
"set preserve" will set "preserve_surds" mode, which suppresses approximation of real roots of real rational numbers, if the result will be irrational. A surd is a quantity which can not be expressed by rational numbers. For example, 2^.5 (the square root of two, which is irrational and a surd) will remain 2^.5 unless explicitly approximated or "set no preserve" is entered. This option is turned on by default ("set preserve_surds"), allowing exact arithmetic and simplification of surds. Surds can always be manually approximated with the approximate and calculate commands.
"set rationalize" will set the "rationalize_denominators" option, which attempts to move radicals from the denominator of fractions to the numerator during simplification. This is the default.
"set modulus_mode" requires an integer from 0 to 2. When modulo arithmetic is performed with the modulus (%) operator, mode 0 returns a result that is the same sign as the dividend (same as C's % operator gives), mode 1 returns a result the same sign as the divisor (same as Python's % operator gives), mode 2 returns an always positive or zero result. Mode 2 is the default and is 100% mathematically correct and the type of modulo operation that can be generally simplified. Mode 0 is the remainder modulus used by the C and Java computer languages. This mode only affects modulo (%) operator numeric calculations. All modulus simplification rules are enabled, regardless of this mode.
"set finance" sets finance mode (fixed point display), which displays all constants with 2 digits after the decimal point (for example: "2.00") and negative numbers are always parenthesized (for example: "(-2.00)"). Displayed constants are rounded to the nearest cent, though internally there is no loss of accuracy. The number of digits to display after the decimal point may be specified with "set finance=number". This is not truly fixed point arithmetic, it is floating point displayed as fixed point. With double precision floating point, only the most significant 15 decimal digits will ever be correct. The default is no fixed point display (finance=0).
"set factor_integers" sets automatic factoring of integers for all displayed expressions. When set, all integers of up to 15 decimal digits are factored into their prime factors before the result of any command is displayed. This command can be shortened to "set factor".
"set right_associative_power" associates power operators from right to left in the absence of parentheses, so that x^a^b is interpreted as x^(a^b). Other math programs typically associate power operators from right to left. The default is "set no right", which associates power operators the same as all other operators, from left to right, resulting in (x^a)^b.
"set no negate_highest_precedence" changes the behavior of the mathematical expression parser to be the same as most other math programs, like Maxima. The default is "set negate", which gives the negate (-) operator the highest precedence of all operators, like programming languages and classic math do; for example, giving 25 for -5^2. "set no negate" gives the the negate operator the same precedence as the times and divide operators, giving -25 for -5^2, which seems to be standard among modern math programs.
"set special_variable_characters" followed by a string of 8-bit characters will allow Mathomatic to use those characters in variable names, in addition to the normal variable name characters, which are the alphanumeric characters and underline (_). For example, "set special $" will allow variable names like "$a" and "a$", and "set special []" will allow entry of array elements like "a[3]" for simulated array arithmetic. All non-alphanumeric characters in variable names are converted to underline characters (_) when exporting to a programming language or to a different program.
"set directory" followed by a directory name will change the current working directory to that directory. Not specifying a directory name defaults to your home directory. This command can be shortened to "set dir".
Syntax: simplify ["sign"] ["symbolic"] ["quick"] ["quickest"] ["fraction"] [equation-number-range]
This command fully simplifies expressions in selected equation spaces. The result is usually the smallest possible, easily readable expression that is equivalent to the original expression.
Use this command whenever you think an expression is not completely simplified or if you don't like the way an expression is factored. Sometimes simplifying more than once or using the "symbolic" option simplifies even more.
More than one option may be specified at a time.
"simplify sign" conveniently expands all "sign" variables by substituting them with all possible combinations of values (+1 and -1), storing the unique results into new equation spaces and fully simplifying. This will effectively create one simplified equation for each solution.
The "symbolic" option indicates (a^n)^m should always be reduced to a^(n*m). This often simplifies more and removes any absolute value operations: ((a^2)^.5 = a^(2*.5) = a^1 = a). Try this "symbolic" option if the simplify command doesn't simplify well, it often helps with powers raised to powers, though it is sometimes not 100% mathematically correct.
The "quick" option skips expanding sums raised to the power of 2 or higher, like (x+1)^5. Also, algebraic fractions will be simpler (less fractions within fractions) with this option.
The "quickest" option very basically simplifies without any unfactoring nor factoring. Running the simplify command with this option makes it complete almost instantaneously.
"simplify fraction" fully simplifies any expression with division in it down to the ratio of two polynomials or expressions, like Maxima's ratsimp() function does. The result will be a single simple algebraic fraction, like the fraction command produces, the difference here being it will be completely simplified. This is accomplished by full simplification without doing "unfactor fraction" and without doing polynomial or smart (algebraic) division on the divide operators.
This simplify command applies many algebraic transformations and their inverses (for example, unfactor and then factor) and then tries to combine and reduce algebraic fractions and rationalize their denominators. Complex fractions are converted to simple fractions by making the denominators of fractions added together the same, combining and simplifying. Polynomials with repeated or symbolic factors are factored next. Then smart (heuristic) and polynomial division are tried on any divides, possibly making complex fractions if it reduces the expression size. Lastly, the expressions are nicely factored and displayed.
Smart division is a symbolic division like polynomial division, but it tries every term in the dividend, instead of only the term with the base variable raised to the highest power, to make the expression smaller.
1—> (x+2^.5)^3 ; an irrational polynomial with repeated factors
1
#1: (x + (2^-))^3
2
1—> unfactor ; multiply it out
1 1
#1: (x^3) + (3*(x^2)*(2^-)) + (6*x) + (2*(2^-))
2 2
1—> simplify ; put it back together, since factored is its simplest form
1
#1: (x + (2^-))^3
2
1—>
"repeat simplify" repeatedly runs the simplify command until the result stabilizes to the smallest size expression.
Syntax: solve ["verify"] ["for"] variable or "0"
This command automatically solves the current equation for the specified variable or for zero. The current equation is replaced with the result. See Solving Equations for details.
The "verify" option checks the result by plugging all solutions into the original equation and then simplifying and comparing. If the resulting equation sides are identical, a "Solutions verified" message is displayed, meaning that all of the solutions are correct. Otherwise "Solution may be incorrect" is displayed, meaning at least one of the solutions is incorrect or unverifiable. The "verify" option only works when solving for a single variable.
The "for" option has no additional effect and is to make entering this solve command more natural.
Syntax: sum variable start end [step-size]
This command performs a mathematical summation (∑) of the current expression or the RHS of the current equation as the index variable goes from start to end in steps of step-size (default 1). The result is stored and displayed. The current equation is not changed.
1—> y=a*x #1: y = a*x 1—> sum Enter variable: x x = 1 To: 10 #2: y = 55*a 1—>
To see all of the intermediate results, type "set debug 1" before this.
Syntax: tally ["average"]
This command prompts for a value, adds it to a running (grand) total, simplifies and displays the running total and optional average, and repeats. The average is the arithmetic mean, that is the running total divided by the number of entries. No equation spaces are used.
It is a convenient way of adding, subtracting, and averaging many numbers and/or variables. Enter a minus sign (−) before each value you wish to subtract. Enter an empty line to end.
Syntax: taylor ["nosimplify"] variable order point
This command computes the Taylor series expansion of the current expression or RHS of the current equation, with respect to the specified variable. The Taylor series uses differentiation and is often used to approximate expressions near the specified point.
The Taylor series of f(x) at point a is the power series:
f(a) + f'(a)*(x-a)/1! + f''(a)*(x-a)^2/2! + f'''(a)*(x-a)^3/3! + ...
where f'(x) is the first derivative of f(x) with respect to x, f''(x) is the second derivative, etc.
This command prompts you for the point of expansion, which is usually a variable or 0, but may be any expression. Typically 0 is used to generate Maclaurin polynomials.
Then it prompts you for the order of the series, which is an integer indicating how many derivatives to take in the expansion. The default is a large number, stopping when the derivative reaches 0.
The result is simplified unless the "nosimplify" option is specified, and placed in the next available equation space, displayed, and becomes the current equation. The original expression is not modified.
1—> e^x
#1: e#^x
1—> taylor
Taylor series expansion around x = point.
Enter point: 0
Enter order (number of derivatives to take): 6
6 derivatives applied.
(x^2) (x^3) (x^4) (x^5) (x^6)
#2: 1 + x + ————— + ————— + ————— + ————— + —————
2 6 24 120 720
2—>
Syntax: unfactor ["fraction"] ["quick"] ["power"] [equation-number-range]
Alternate command name: expand
This command algebraically expands expressions in selected equation spaces by multiplying out products of sums and exponentiated sums and then simplifying a little. One or more options may be specified.
To illustrate what unfactoring does, suppose you have the following equations:
1—> a=b*(c+d) #1: a = b*(c + d) 1—> z=(x+y)^2 #2: z = (x + y)^2 2—> unfactor all #1: a = (b*c) + (b*d) #2: z = (x^2) + (2*x*y) + (y^2) 2—>
(x+y)^2 is called an exponentiated sum and is converted to (x+y)*(x+y) and then multiplied out, unless the "quick" option is given. Because this is a general but inefficient expansion method, exponentiated sums usually fail expansion when the power is greater than 10, growing larger than will fit in an equation space. "unfactor quick" only expands products of sums, and not exponentiated sums.
The opposite of unfactoring is factoring. Careful and neat factoring is always done by the simplify command.
"unfactor fraction" by itself expands algebraic fractions by also expanding division of sums, multiplying out each fraction with a sum in the numerator into the sum of smaller fractions with the same denominator for each term in numerator. See the example under the fraction command.
"unfactor power" does only power expansion; that is, (a*b)^(n+m) is transformed to (a^n)*(a^m)*(b^n)*(b^m).
Syntax: variables ["c" or "java" or "integer"] [equation-number-range]
Show all variable names used within the specified expressions, from most frequent to least frequently occurring. The programming language options output the variable definitions required to make code from the specified equations. This does not initialize any variables, it only defines them as needed for a C or Java compiler. This command is not necessary for generating Python code.
Syntax: version
Shows the version number, compilation options used, maximum possible memory usage, and license summary, for the currently running version of Mathomatic. The maximum memory usage displayed is the amount of RAM used when all equation spaces have been filled. It does not include stack size (which varies) or executable (code) size.
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