<html> <head> </head> <body> <tt> <p> <a href="Eigenmath.pdf"><img src="man.jpg"></a> <p> <hr> <p> <table> <tr> <td valign="top"> <tt> <ul> <li><a href="#abs">abs</a> <li><a href="#adj">adj</a> <li><a href="#and">and</a> <li><a href="#arccos">arccos</a> <li><a href="#arccosh">arccosh</a> <li><a href="#arcsin">arcsin</a> <li><a href="#arcsinh">arcsinh</a> <li><a href="#arctan">arctan</a> <li><a href="#arctanh">arctanh</a> <li><a href="#arg">arg</a> <li><a href="#ceiling">ceiling</a> <li><a href="#check">check</a> <li><a href="#choose">choose</a> <li><a href="#circexp">circexp</a> <li><a href="#coeff">coeff</a> <li><a href="#cofactor">cofactor</a> <li><a href="#conj">conj</a> <li><a href="#contract">contract</a> <li><a href="#cos">cos</a> <li><a href="#cosh">cosh</a> <li><a href="#cross">cross</a> <li><a href="#curl">curl</a> </ul> </td> <td valign="top"> <tt> <ul> <li><a href="#d">d</a> <li><a href="#defint">defint</a> <li><a href="#deg">deg</a> <li><a href="#denominator">denominator</a> <li><a href="#det">det</a> <li><a href="#dim">dim</a> <li><a href="#do">do</a> <li><a href="#dot">dot</a> <li><a href="#draw">draw</a> <li><a href="#erf">erf</a> <li><a href="#erfc">erfc</a> <li><a href="#eval">eval</a> <li><a href="#exp">exp</a> <li><a href="#expcos">expcos</a> <li><a href="#expsin">expsin</a> <li><a href="#factor">factor</a> <li><a href="#factorial">factorial</a> <li><a href="#filter">filter</a> <li><a href="#float">float</a> <li><a href="#floor">floor</a> <li><a href="#for">for</a> <li><a href="#gcd">gcd</a> </ul> </td> <td valign="top"> <tt> <ul> <li><a href="#hermite">hermite</a> <li><a href="#hilbert">hilbert</a> <li><a href="#imag">imag</a> <li><a href="#inner">inner</a> <li><a href="#integral">integral</a> <li><a href="#inv">inv</a> <li><a href="#isprime">isprime</a> <li><a href="#laguerre">laguerre</a> <li><a href="#lcm">lcm</a> <li><a href="#legendre">legendre</a> <li><a href="#log">log</a> <li><a href="#mag">mag</a> <li><a href="#mod">mod</a> <li><a href="#not">not</a> <li><a href="#nroots">nroots</a> <li><a href="#numerator">numerator</a> <li><a href="#or">or</a> <li><a href="#outer">outer</a> <li><a href="#polar">polar</a> <li><a href="#prime">prime</a> <li><a href="#print">print</a> <li><a href="#product">product</a> </ul> </td> <td valign="top"> <tt> <ul> <li><a href="#quote">quote</a> <li><a href="#quotient">quotient</a> <li><a href="#rank">rank</a> <li><a href="#rationalize">rationalize</a> <li><a href="#real">real</a> <li><a href="#rect">rect</a> <li><a href="#roots">roots</a> <li><a href="#simplify">simplify</a> <li><a href="#sin">sin</a> <li><a href="#sinh">sinh</a> <li><a href="#sqrt">sqrt</a> <li><a href="#stop">stop</a> <li><a href="#subst">subst</a> <li><a href="#sum">sum</a> <li><a href="#tan">tan</a> <li><a href="#tanh">tanh</a> <li><a href="#taylor">taylor</a> <li><a href="#test">test</a> <li><a href="#transpose">transpose</a> <li><a href="#unit">unit</a> <li><a href="#zero">zero</a> </ul> </td> </tr> </table> <p> <h1><tt><a name="abs">abs(<i>x</i>)</a></tt></h1> Returns the absolute value or vector length of x. <p> <h1><tt><a name="adj">adj(<i>m</i>)</a></tt></h1> Returns the adjunct of matrix m. The inverse of m is equal to adj(m) divided by det(m). <p><h1><tt><a name="and">and(<i>a,b,...</i>)</a></tt></h1> Logical-and of predicate expressions. <p> <h1><tt><a name="arccos">arccos(<i>x</i>)</a></tt></h1> Returns the inverse cosine of x. <p> <h1><tt><a name="arccosh">arccosh(<i>x</i>)</a></tt></h1> Returns the inverse hyperbolic cosine of x. <p> <h1><tt><a name="arcsin">arcsin(<i>x</i>)</a></tt></h1> Returns the inverse sine of x. <p> <h1><tt><a name="arcsinh">arcsinh(<i>x</i>)</a></tt></h1> Returns the inverse hyperbolic sine of x. <p> <h1><tt><a name="arctan">arctan(<i>x</i>)</a></tt></h1> Returns the inverse tangent of x. <p> <h1><tt><a name="arctanh">arctanh(<i>x</i>)</a></tt></h1> Returns the inverse hyperbolic tangent of x. <p> <h1><tt><a name="arg">arg(<i>z</i>)</a></tt></h1> Returns the angle of complex z. <p> <h1><tt><a name="ceiling">ceiling(<i>x</i>)</a></tt></h1> Returns the smallest integer not less than x. <p> <h1><tt><a name="check">check(<i>x</i>)</a></tt></h1> If x is true then continue, else stop. <p> <h1><tt><a name="choose">choose(<i>n,k</i>)</a></tt></h1> Returns the number of combinations of n items taken k at a time. <p> <h1><tt><a name="circexp">circexp(<i>x</i>)</a></tt></h1> Returns expression x with circular and hyperbolic functions converted to exponential forms. Sometimes this will simplify an expression. <p> <h1><tt><a name="coeff">coeff(<i>p,x,n</i>)</a></tt></h1> Returns the coefficient of x to the n in polynomial p. The x argument can be omitted for polynomials in x. <p> <h1><tt><a name="cofactor">cofactor(<i>m,i,j</i>)</a></tt></h1> Returns the cofactor of m for row i and column j. <p> <h1><tt><a name="conj">conj(<i>z</i>)</a></tt></h1> Returns the complex conjugate of z. <p> <h1><tt><a name="contract">contract(<i>a,i,j</i>)</a></tt></h1> Returns "a" summed over indices i and j. If i and j are omitted then 1 and 2 are used. contract(m) is equivalent to the trace of matrix m. <p> <h1><tt><a name="cos">cos(<i>x</i>)</a></tt></h1> Returns the cosine of x. <p> <h1><tt><a name="cosh">cosh(<i>x</i>)</a></tt></h1> Returns the hyperbolic cosine of x. <p> <h1><tt><a name="cross">cross(<i>u,v</i>)</a></tt></h1> Returns the cross product of vectors u and v. <p> <h1><tt><a name="curl">curl(<i>u</i>)</a></tt></h1> Returns the curl of vector u. <p> <h1><a name="d">d(<i>f,x</i>)</a></h1> Returns the partial derivative of f with respect to x. <p> <h1><a name="defint">defint(<i>f,x,a,b</i>)</h1> Returns the definite integral of f with respect to x evaluated from "a" to b. The argument list can be extended for multiple integrals. For example, defint(f,x,a,b,y,c,d). <p> <h1><tt><a name="deg">deg(<i>p,x</i>)</a></tt></h1> Returns the degree of polynomial p(x). <p> <h1><tt><a name="denominator">denominator(<i>x</i>)</a></tt></h1> Returns the denominator of expression x. <p> <h1><tt><a name="det">det(<i>m</i>)</a></tt></h1> Returns the determinant of matrix m. <p> <h1><tt><a name="dim">dim(<i>a,n</i>)</a></tt></h1> Returns the cardinality of the nth index of tensor "a". <p> <h1><tt><a name="do">do(<i>a,b,...</i>)</a></tt></h1> Evaluates each argument from left to right. Returns the result of the last argument. <p> <h1><tt><a name="dot">dot(<i>a,b,...</i>)</a></tt></h1> Returns the dot or inner product of tensors. <p> <h1><tt><a name="draw">draw(<i>f,x</i>)</a></tt></h1> Draws a graph of f(x). Drawing ranges can be set with xrange and yrange. <!-- <p> <h1><tt><a name="eigen">eigen(<i>m</i>)</a></tt></h1> <h1><tt>eigenval(<i>m</i>)</tt></h1> <h1><tt>eigenvec(<i>m</i>)</tt></h1> These functions compute eigenvalues and eigenvectors numerically. Matrix m must be both numerical and symmetric. The eigenval function returns a matrix with the eigenvalues along the diagonal. The eigenvec function returns a matrix with the eigenvectors arranged as row vectors. The eigen function does not return anything but stores the eigenvalue matrix in D and the eigenvector matrix in Q. <p> Example 1. Check the relation AX = lambda X where lambda is an eigenvalue and X is the associated eigenvector. <pre> <i>Enter</i> A = hilbert(3) eigen(A) lambda = D[1,1] X = Q[1] dot(A,X) - lambda X <i>Result</i> -1.16435e-14 -6.46705e-15 -4.55191e-15 </pre> <p> Example 2: Check the relation A = Q<sup>T</sup>DQ. <pre> <i>Enter</i> A - dot(transpose(Q),D,Q) <i>Result</i> 6.27365e-12 -1.58236e-11 1.81902e-11 -1.58236e-11 -1.95365e-11 2.56514e-12 1.81902e-11 2.56514e-12 1.32627e-11 </pre> --> <p> <h1><tt><a name="erf">erf(<i>x</i>)</a></tt></h1> Error function of x. <p> <h1><tt><a name="erfc">erfc(<i>x</i>)</a></tt></h1> Complementary error function of x. <p> <h1><tt><a name="eval">eval(<i>f,x,a</i>)</a></tt></h1> Returns f evaluated at x=a. <p> <h1><tt><a name="exp">exp(<i>x</i>)</a></tt></h1> Returns the exponential of x. <p> <h1><tt><a name="expcos">expcos(<i>x</i>)</a></tt></h1> Returns the exponential cosine of x. <p> <h1><tt><a name="expsin">expsin(<i>x</i>)</a></tt></h1> Returns the exponential sine of x. <p> <h1><tt><a name="factor">factor(<i>n</i>)</a></tt></h1> Factors integer n. <p> <h1><tt>factor(<i>p,x</i>)</tt></h1> Factors polynomial p(x). The x can be omitted for polynomials in x. The polynomial should be factorable over integers. The argument list can be extended. For example, factor(p,x,y) factors p over x and then over y. <p> <h1><tt><a name="factorial">factorial(<i>x</i>)</a></tt></h1> Can be entered as x! <p> <h1><tt><a name="filter">filter(<i>f,a,b,...</i>)</a></tt></h1> Returns f excluding any terms containing a, b, etc. <p> <h1><tt><a name="float">float(<i>x</i>)</a></tt></h1> Converts rational numbers and integers to floating point values. The symbol pi is also converted. <p> <h1><tt><a name="floor">floor(<i>x</i>)</a></tt></h1> Returns the largest integer not greater than x. <p> <h1><tt><a name="for">for(<i>i,j,k,a,b,...</i>)</a></tt></h1> For i equals j through k evaluate a, b, etc. <p> <h1><tt><a name="gcd">gcd(<i>a,b,...</i>)</a></tt></h1> Returns the greatest common divisor. <p> <h1><tt><a name="hermite">hermite(<i>x,n</i>)</a></tt></h1> Returns the nth Hermite polynomial in x. <p> <h1><tt><a name="hilbert">hilbert(<i>n</i>)</a></tt></h1> Returns an n by n Hilbert matrix. <p> <h1><tt><a name="imag">imag(<i>z</i>)</a></tt></h1> Returns the imaginary part of complex z. <p> <h1><tt><a name="inner">inner(<i>a,b,...</i>)</a></tt></h1> Returns the inner product of tensors. Same as the dot product. <p> <h1><tt><a name="integral">integral(<i>f,x</i>)</a></tt></h1> Returns the integral of f with respect to x. <p> <h1><tt><a name="inv">inv(<i>m</i>)</a></tt></h1> Returns the inverse of matrix m. <p> <h1><tt><a name="isprime">isprime(<i>n</i>)</a></tt></h1> Returns 1 if n is a prime number, returns zero otherwise. <p> <h1><tt><a name="laguerre">laguerre(<i>x,n,a</i>)</a></tt></h1> Returns the nth Laguerre polynomial in x. If "a" is omitted then a=0 is used. <p> <h1><tt><a name="lcm">lcm(<i>a,b,...</i>)</a></tt></h1> Returns the least common multiple. <p> <h1><tt><a name="legendre">legendre(<i>x,n,m</i>)</a></tt></h1> Returns the nth Legendre polynomial in x. If m is omitted then m=0 is used. <p> <h1><tt><a name="log">log(<i>x</i>)</a></tt></h1> Returns the natural logarithm of x. <p> <h1><tt><a name="mag">mag(<i>z</i>)</a></tt></h1> Returns the magnitude of complex z. <p> <h1><tt><a name="mod">mod(<i>a,b</i>)</a></tt></h1> Returns the remainder of the result of "a" divided by b. <p> <h1><tt><a name="not">not(<i>x</i>)</a></tt></h1> Returns the logical negation of x. <p> <h1><tt><a name="nroots">nroots(<i>p,x</i>)</a></tt></h1> Returns the numerical roots of polynomial p(x). The argument x can be omitted. If it is omitted then the computer will guess the free variable. <p> <h1><tt><a name="numerator">numerator(<i>x</i>)</a></tt></h1> Returns the numerator of expression x. <p> <h1><tt><a name="or">or(<i>a,b,...</i>)</a></tt></h1> Logical-or of predicate expressions. <p> <h1><tt><a name="outer">outer(<i>a,b,...</i>)</a></tt></h1> Returns the outer product of tensors. Also known as the tensor product. <p> <h1><tt><a name="polar">polar(<i>z</i>)</a></tt></h1> Returns complex z in polar form. <p> <h1><tt><a name="prime">prime(<i>n</i>)</a></tt></h1> Returns the nth prime number. The domain of n is 1 to 10000. <p> <h1><tt><a name="print">print(<i>a,b,...</i>)</a></tt></h1> Evaluate expressions and print the results. Useful for printing from inside a "for" loop. <p> <h1><tt><a name="product">product(<i>i,j,k,f</i>)</a></tt></h1> For i equals j through k evaluate f. Returns the product of all f. <p> <h1><tt><a name="quote">quote(<i>x</i>)</a></tt></h1> Returns expression x without evaluating it first. <p><h1><tt><a name="quotient">quotient(<i>p,q,x</i>)</a></tt></h1> Returns the quotient of polynomial p(x) over q(x). The last argument can be omitted for polynomials in x. The remainder can be calculated by p-q*quotient(p,q). <p> <h1><tt><a name="rank">rank(<i>a</i>)</a></tt></h1> Returns the number of indices that tensor "a" has. <p> <h1><tt><a name="rationalize">rationalize(<i>x</i>)</a></tt></h1> Returns x with everything over a common denominator. <p> <h1><tt><a name="real">real(<i>z</i>)</a></tt></h1> Returns the real part of complex z. <p> <h1><tt><a name="rect">rect(<i>z</i>)</a></tt></h1> Returns complex z in rectangular form. <p> <h1><tt><a name="roots">roots(<i>p,x</i>)</a></tt></h1> Returns the values of x such that p(x)=0. The polynomial p should be factorable over integers. Returns a vector for multiple roots. <p> <h1><tt><a name="simplify">simplify(<i>x</i>)</a></tt></h1> Returns x in a simpler form. <p> <h1><tt><a name="sin">sin(<i>x</i>)</a></tt></h1> Returns the sine of x. <p> <h1><tt><a name="sinh">sinh(<i>x</i>)</a></tt></h1> Returns the hyperbolic sine of x. <p> <h1><tt><a name="sqrt">sqrt(<i>x</i>)</a></tt></h1> Returns the square root of x. <p> <h1><tt><a name="stop">stop()</a></tt></h1> In a script, it does what it says. <p> <h1><tt><a name="subst">subst(<i>a,b,c</i>)</a></tt></h1> Substitutes "a" for b in c and returns the result. <p> <h1><tt><a name="sum">sum(<i>i,j,k,f</i>)</a></tt></h1> For i equals j through k evaluate f. Returns the sum of all f. <p> <h1><tt><a name="tan">tan(<i>x</i>)</a></tt></h1> Returns the tangent of <i>x</i>. <p> <h1><tt><a name="tanh">tanh(<i>x</i>)</a></tt></h1> Returns the hyperbolic tangent of <i>x</i>. <p> <h1><tt><a name="taylor">taylor(<i>f,x,n,a</i>)</a></tt></h1> Returns the Taylor expansion of f(x) around x=a. If "a" is omitted then a=0 is used. The argument n is the degree of the expansion. <p><h1><tt><a name="test">test(<i>a,b,c,d,...</i>)</a></tt></h1> If "a" is true then b is returned else if c is true then d is returned, etc. If the number of arguments is odd then the last argument is returned when all else fails. <p> <h1><tt><a name="transpose">transpose(<i>a,i,j</i>)</a></tt></h1> Returns the transpose of "a" with respect to indices i and j. If i and j are omitted then 1 and 2 are used. Hence a matrix can be transposed with a single argument. <p> <h1><tt><a name="unit">unit(<i>n</i>)</a></tt></h1> Returns an n by n identity matrix. <p> <h1><tt><a name="zero">zero(<i>i,j,...</i>)</a></tt></h1> Returns a null tensor with dimensions i, j, etc. Useful for creating a tensor and then setting the component values. <p> <a href="http://sourceforge.net"><img src="http://sflogo.sourceforge.net/sflogo.php?group_id=103462&type=2" width="125" height="37" border="0" alt="SourceForge.net Logo" /></a> </body> </html>