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