1746 lines
35 KiB
HTML
1746 lines
35 KiB
HTML
<tt>
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<table cellpadding="10" cellspacing="0" border="1">
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<tr>
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<td valign="top">
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<pre>
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<a href="#abs">abs</a> absolute value
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<a href="#adj">adj</a> adjunct of matrix
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<a href="#and">and</a> logical connective
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<a href="#arccos">arccos</a>
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<a href="#arccosh">arccosh</a>
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<a href="#arcsin">arcsin</a>
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<a href="#arcsinh">arcsinh</a>
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<a href="#arctan">arctan</a>
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<a href="#arctanh">arctanh</a>
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<a href="#arg">arg</a> argument of complex
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<a href="#binomial">binomial</a>
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<a href="#break">break</a> break out of a loop
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<a href="#ceiling">ceiling</a> ceiling function
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<a href="#check">check</a> stop if not zero
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<a href="#condense">condense</a> find subexpressions
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<a href="#conj">conj</a> complex conjugate
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<a href="#contract">contract</a> contract across tensor indices
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<a href="#cos">cos</a>
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<a href="#cosh">cosh</a>
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<a href="#d">d</a> derivative and gradient
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<a href="#denominator">denominator</a> denominator of expression
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<a href="#det">det</a> determinant of a matrix
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<a href="#dim">dim</a> dimension of tensor index
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<a href="#display">display</a> display an expression
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<a href="#do">do</a> evaluate multiple expressions
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<a href="#dot">dot</a> inner product of tensors
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<a href="#draw">draw</a> draw a graph
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<a href="#eigen">eigen</a>
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<a href="#erf">erf</a> error function
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<a href="#erfc">erfc</a> complementary error function
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<a href="#eval">eval</a> normalize an expression
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<a href="#exp">exp</a> exponential function
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<a href="#expcos">expcos</a> exponential cosine
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<a href="#expsin">expsin</a> exponential sine
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<a href="#factor">factor</a> factor a number or polynomial
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<a href="#factorial">factorial</a> factorial function
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<a href="#filter">filter</a> remove subexpressions
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<a href="#float">float</a> convert to floating point
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<a href="#floor">floor</a> floor function
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<a href="#for">for</a> for loop
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<a href="#gcd">gcd</a> greatest common denominator
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<a href="#hermite">hermite</a> hermite polynomial
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<a href="#hilbert">hilbert</a> hilbert matrix
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<a href="#imag">imag</a> imaginary part of complex
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<a href="#inner">inner</a> inner product of tensors
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<a href="#integral">integral</a>
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<a href="#inv">inv</a> invert a matrix
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<a href="#isprime">isprime</a>
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<a href="#laguerre">laguerre</a> laguerre polynomial
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<a href="#lcm">lcm</a> least common multiple
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<a href="#legendre">legendre</a> legendre polynomial
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<a href="#log">log</a> natural logarithm
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<a href="#mag">mag</a> magnitude of complex
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<a href="#mod">mod</a> modulo function
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<a href="#numerator">numerator</a> numerator of expression
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<a href="#outer">outer</a> outer product of tensors
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<a href="#prime">prime</a>
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<a href="#print">print</a>
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<a href="#product">product</a> multiply over an index
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<a href="#prog">prog</a> evaluate with scoped variables
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<a href="#quote">quote</a> do not evaluate
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<a href="#rank">rank</a> rank of tensor
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<a href="#rationalize">rationalize</a> combine fractions
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<a href="#real">real</a> real part of complex
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<a href="#rect">rect</a> complex to rectangular
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<a href="#return">return</a> early return from a function
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<a href="#roots">roots</a> find roots of a polynomial
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<a href="#simfac">simfac</a> simplify factorials
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<a href="#simplify">simplify</a> simplify an expression
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<a href="#sin">sin</a>
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<a href="#sinh">sinh</a>
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<a href="#sqrt">sqrt</a> square root
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<a href="#stop">stop</a> stop running a script
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<a href="#subst">subst</a> substitute expressions
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<a href="#sum">sum</a> sum over an index
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<a href="#tan">tan</a>
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<a href="#tanh">tanh</a>
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<a href="#taylor">taylor</a> taylor series
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<a href="#test">test</a> conditional evaluation
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<a href="#trace">trace</a> trace of matrix
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<a href="#transpose">transpose</a> transpose tensor indices
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<a href="#unit">unit</a> unit matrix
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<a href="#wedge">wedge</a> wedge product of tensors
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<a href="#zero">zero</a> new zero tensor
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</pre>
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</td>
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<td valign="top">
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<tt>
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Getting-started Examples
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<ul>
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<li>
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Enter x^2 then click the Draw button.
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<li>
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Enter x^2 then click the Integral button, then click the Draw button.
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<li>
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Enter 900 then click the Factor button.
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<a href="pics/factor-number.html">pic</a>
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<li>
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Enter 125 x^3 - 1 then click the Factor button, then click the Simplify button.
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<a href="pics/factor-poly.html">pic</a>
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</ul>
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<p>
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Scripts
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<ul>
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<li>
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Click the Edit Script button, enter a script, then click the Run button.
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Each line of the script is evaluated like an interactive command.
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<a href="pics/script-howto.html">pic</a>
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<li>
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Click <a href="examples/quantum-harmonic-oscillator.txt">here</a>
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for a sample script that can be pasted into the Edit Script window.
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</ul>
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<p>
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Trivia
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<ul>
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<li>
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Press the up and down arrow keys to choose a previously entered command.
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<li>
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To use the symbol e for the natural number, first define it as e=exp(1).
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<li>
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Most functions can be used without arguments.
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The previous result is used as a default argument.
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<li>
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The simplify function converts tensor zero to scalar zero.
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<li>
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A=B evaluates B and binds the result to the symbol A.
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On the other hand, the function definition A(X)=B does not evaluate B until A
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itself is evaluated.
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</ul>
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</td>
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</tr>
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</table>
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<tt>
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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 (vector length, magnitude) of <i>x</i>.
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<pre>
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<i>Enter</i>
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abs(a - b) + abs(b - a)
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<i>Result</i>
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2 abs(a - b)
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<i>Enter</i>
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abs(3 + 4 i)
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<i>Result</i>
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5
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<i>Enter</i>
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abs((a,b,c))
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<i>Result</i>
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1/2
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2 2 2
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(a + b + c )
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<i>Enter</i>
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A = (1,i,-i)
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sqrt(dot(A,conj(A))) - abs(A)
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<i>Result</i>
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0
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<i>Enter</i>
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A = ((1,2),(3,4))
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abs(A)
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<i>Result</i>
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stop: abs(tensor) with tensor rank > 1
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</pre>
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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 <i>m</i>.
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<pre>
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<i>Enter</i>
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A = ((a,b),(c,d))
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adj(A)
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<i>Result</i>
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d -b
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-c a
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</pre>
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The inverse of a matrix is equal to the adjunct divided by the determinant.
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<pre>
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<i>Enter</i>
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inv(A) - adj(A) / det(A)
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<i>Result</i>
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0
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</pre>
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<p>
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<h1><tt><a name="and">and(<i>a,b,...</i>)</a></tt></h1>
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Logical-and function.
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<pre>
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<i>Enter</i>
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and(1<2,2<3)
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<i>Result</i>
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true
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</pre>
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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 <i>x</i>.
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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 <i>x</i>.
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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 <i>x</i>.
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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 <i>x</i>.
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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 <i>x</i>.
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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 <i>x</i>.
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<p>
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<h1><tt><a name="arg">arg(z)</a></tt></h1>
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Returns the argument (angle) of complex z.
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Symbols in expression z are presumed to be positive and real.
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<pre>
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<i>Enter</i>
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arg(1 + exp(i pi/3))
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<i>Result</i>
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1/6 pi
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</pre>
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<p>
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<h1><tt><a name="binomial">binomial(<i>n,k</i>)</a></tt></h1>
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Returns the binomial coefficient.
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<pre>
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<i>Enter</i>
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binomial(10,5)
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<i>Result</i>
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252
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</pre>
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<p>
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<h1><tt><a name="break">break(<i>x</i>)</a></tt></h1>
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Causes an immediate return from a <tt>for</tt> function.
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Expression <i>x</i> is evaluated and returned as the <tt>for</tt> function
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value.
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A <tt>break</tt> with no argument returns the symbol <tt>nil</tt>.
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A <tt>break</tt> can be evaluated at any function level.
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For example, <tt>for()</tt> can evaluate <tt>f()</tt> which evaluates
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<tt>g()</tt> which evaluates <tt>break()</tt>.
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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 <i>x</i>.
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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 zero then continue, else stop.
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The check function also accepts a predicate expression such as A<B,
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A=B, etc.
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<p>
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<h1><tt><a name="condense">condense(<i>x</i>)</a></tt></h1>
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Attempts to simplify x by factoring common terms.
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<pre>
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<i>Enter</i>
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2 a (x + 1)
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<i>Result</i>
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2 a + 2 a x
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<i>Enter</i>
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condense(last)
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<i>Result</i>
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2 a (x + 1)
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</pre>
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<p>
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<h1><tt><a name="conj">conj(<i>x</i>)</a></tt></h1>
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Returns the complex conjugate of <i>x</i>.
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<pre>
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<i>Enter</i>
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conj(3+4i)
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<i>Result</i>
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3 - 4 i
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</pre>
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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 the contraction of tensor <i>a</i> across indices <i>i</i>
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and <i>j</i>.
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If <i>i</i> and <i>j</i> are omitted then indices 1 and 2 are used.
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The following example shows how contract adds diagonal elements.
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<pre>
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<i>Enter</i>
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A = ((a,b),(c,d))
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contract(A,1,2)
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<i>Result</i>
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a + d
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</pre>
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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 <i>x</i>.
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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 <i>x</i>.
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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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The second argument can be omitted in which case the computer will guess which
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symbol to use.
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Returns the gradient of f when x is a vector.
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Note that gradient raises the rank of f by 1.
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<pre>
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<i>Enter</i>
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d(x^2,x)
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<i>Result</i>
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2 x
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</pre>
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For tensor f the derivative of each element is computed.
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<pre>
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<i>Enter</i>
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d((x,x^2),x)
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<i>Result</i>
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1
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2 x
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</pre>
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Functions with no arguments are treated as dependent on any variable.
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<pre>
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<i>Enter</i>
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d(f(),(x,y))
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<i>Result</i>
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d(f(),x)
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d(f(),y)
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</pre>
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Since partial derivatives commute, multi-derivatives are ordered to produce a
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canonical form.
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<pre>
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<i>Enter</i>
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d(d(f(),y),x)
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<i>Result</i>
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d(d(f(),x),y)
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</pre>
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The following table shows the various forms that can be used to compute
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multiderivatives.
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<pre>
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Form Equivalent form
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d(f,2) d(d(f,x),x)
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d(f,x,2) d(d(f,x),x)
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d(f,x,x) d(d(f,x),x)
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d(f,x,2,y,2) d(d(d(d(f,x),x),y),y)
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d(f,x,x,y,y) d(d(d(d(f,x),x),y),y)
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</pre>
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<p>
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<h1><tt><a name="denominator">denominator(x)</a></tt></h1>
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Returns the denominator of expression x.
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<pre>
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<i>Enter</i>
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denominator(a/b)
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<i>Result</i>
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b
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</pre>
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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 <i>m</i>.
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<pre>
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<i>Enter</i>
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A = ((a,b),(c,d))
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det(A)
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<i>Result</i>
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a d - b c
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</pre>
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<p>
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<h1><tt><a name="dim">dim(<i>a,i</i>)</a></tt></h1>
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Returns the dimension of the <i>i</i>th index of tensor <i>a</i>.
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If <i>i</i> is omitted then the dimension of the first index is returned.
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<pre>
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<i>Enter</i>
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A = (1,2,3,4)
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dim(A)
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<i>Result</i>
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4
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<i>Enter</i>
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A = ((1,2,3),(4,5,6))
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dim(A,1)
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<i>Result</i>
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2
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<i>Enter</i>
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dim(A,2)
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<i>Result</i>
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3
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</pre>
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<p>
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<h1><tt><a name="display">display(<i>x</i>)</a></tt></h1>
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Evaluates expression <i>x</i> and displays the result using Times and Symbol
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fonts.
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User symbols are scanned for the keywords shown below.
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Each keyword is replaced with its associated Greek letter glyph.
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Multiglyph symbols are displayed using subscripts.
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This function can be selected as the default display mode by setting tty = 0.
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<p>
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<table>
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<tr>
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<td>Gamma</td><td>Γ</td>
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<td width="10"></td>
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<td>alpha</td><td>α</td>
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<td width="10"></td>
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<td>mu</td><td>μ</td>
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</tr>
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<tr>
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<td>Delta</td><td>Δ</td>
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<td></td>
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<td>beta</td><td>β</td>
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<td></td>
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<td>nu</td><td>ν</td>
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</tr>
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<tr>
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<td>Theta</td><td>Θ</td>
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<td></td>
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<td>gamma</td><td>γ</td>
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<td></td>
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<td>xi</td><td>ξ</td>
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</tr>
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<tr>
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<td>Lambda</td><td>Λ</td>
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<td></td>
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<td>delta</td><td>δ</td>
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<td></td>
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<td>pi</td><td>π</td>
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</tr>
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<tr>
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<td>Xi</td><td>Ξ</td>
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<td></td>
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<td>epsilon</td><td>ε</td>
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<td></td>
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<td>rho</td><td>ρ</td>
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</tr>
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<tr>
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<td>Pi</td><td>Π</td>
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<td></td>
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<td>zeta</td><td>ζ</td>
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<td></td>
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<td>sigma</td><td>σ</td>
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</tr>
|
|
|
|
<tr>
|
|
<td>Sigma</td><td>Σ</td>
|
|
<td></td>
|
|
<td>eta</td><td>η</td>
|
|
<td></td>
|
|
<td>tau</td><td>τ</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td>Upsilon</td><td>Υ</td>
|
|
<td></td>
|
|
<td>theta</td><td>θ</td>
|
|
<td></td>
|
|
<td>upsilon</td><td>υ</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td>Phi</td><td>Φ</td>
|
|
<td></td>
|
|
<td>iota</td><td>ι</td>
|
|
<td></td>
|
|
<td>phi</td><td>φ</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td>Psi</td><td>Ψ</td>
|
|
<td></td>
|
|
<td>kappa</td><td>κ</td>
|
|
<td></td>
|
|
<td>chi</td><td>χ</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td>Omega</td><td>Ω</td>
|
|
<td></td>
|
|
<td>lambda</td><td>λ</td>
|
|
<td></td>
|
|
<td>psi</td><td>ψ</td>
|
|
</tr>
|
|
|
|
<tr>
|
|
<td></td><td></td>
|
|
<td></td>
|
|
<td></td><td></td>
|
|
<td></td>
|
|
<td>omega</td><td>ω</td>
|
|
</tr>
|
|
|
|
</table>
|
|
<p>
|
|
<h1><tt><a name="do">do(<i>a,b,...</i>)</a></tt></h1>
|
|
Evaluates each statement in sequence.
|
|
Returns the value of the last statement.
|
|
<p>
|
|
<h1><tt><a name="dot">dot(<i>a,b,...</i>)</a></tt></h1>
|
|
Returns the dot product of tensors (aka inner product).
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
A = (A1,A2,A3)
|
|
|
|
B = (B1,B2,B3)
|
|
|
|
dot(A,B)
|
|
|
|
<i>Result</i>
|
|
|
|
A1 B1 + A2 B2 + A3 B3
|
|
</pre>
|
|
The dot product is equivalent to an outer product followed by a contraction
|
|
across the inner indices.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
A = hilbert(10)
|
|
|
|
dot(A,A) - contract(outer(A,A),2,3)
|
|
|
|
<i>Result</i>
|
|
|
|
0
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="draw">draw(<i>f,x</i>)</a></tt></h1>
|
|
Draws a graph of <i>f</i>.
|
|
The second argument can be omitted in which case the computer will
|
|
guess what variable to use.
|
|
Parametric drawing occurs when <i>f</i> returns a vector.
|
|
Ranges are set with <tt>xrange</tt> and <tt>yrange</tt>.
|
|
The defaults are <tt>xrange = (-10,10)</tt> and
|
|
<tt>yrange = (-10,10)</tt>.
|
|
The parametric variable range is set with <tt>trange</tt>.
|
|
The default is <tt>trange = (-pi,pi)</tt>.
|
|
<p>
|
|
Example 1.
|
|
<pre>
|
|
<i>Enter</i>
|
|
draw(5(cos(t),sin(t)))
|
|
|
|
<i>Result</i>
|
|
<a href="pics/circle.html">pic</a>
|
|
</pre>
|
|
<p>
|
|
Example 2.
|
|
<pre>
|
|
<i>Enter</i>
|
|
draw(5(cos(3t),sin(5t)))
|
|
|
|
<i>Result</i>
|
|
<a href="pics/lissajous.html">pic</a>
|
|
</pre>
|
|
<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.
|
|
<pre>
|
|
<i>Enter</i>
|
|
draw(erf(x))
|
|
|
|
<i>Result</i>
|
|
<a href="pics/erf.html">pic</a>
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="erfc">erfc(<i>x</i>)</a></tt></h1>
|
|
Complementary error function of x.
|
|
<pre>
|
|
<i>Enter</i>
|
|
draw(erfc(x))
|
|
|
|
<i>Result</i>
|
|
<a href="pics/erfc.html">pic</a>
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="eval">eval(<i>x</i>)</a></tt></h1>
|
|
Returns the evaluation of expression <i>x</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
A = quote(sin(pi/6))
|
|
|
|
A
|
|
|
|
<i>Result</i>
|
|
|
|
1
|
|
sin(--- pi)
|
|
6
|
|
|
|
<i>Enter</i>
|
|
|
|
eval(A)
|
|
|
|
<i>Result</i>
|
|
|
|
1
|
|
---
|
|
2
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="exp">exp(<i>x</i>)</a></tt></h1>
|
|
Returns the exponential of <i>x</i>.
|
|
The expression exp(1) should be used to represent the natural number <i>e</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
exp(1.0)
|
|
|
|
<i>Result</i>
|
|
|
|
2.71828
|
|
|
|
<i>Enter</i>
|
|
|
|
exp(a) exp(b)
|
|
|
|
<i>Result</i>
|
|
|
|
exp(a + b)
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="expcos">expcos(<i>x</i>)</a></tt></h1>
|
|
Returns the exponential cosine of <i>x</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
expcos(x)
|
|
|
|
<i>Result</i>
|
|
|
|
1 1
|
|
--- exp(-i x) + --- exp(i x)
|
|
2 2
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="expsin">expsin(<i>x</i>)</a></tt></h1>
|
|
Returns the exponential sine of <i>x</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
expsin(x)
|
|
|
|
<i>Result</i>
|
|
|
|
1 1
|
|
--- i exp(-i x) - --- i exp(i x)
|
|
2 2
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="factor">factor(<i>n</i>)</a></tt></h1>
|
|
<h1><tt>factor(<i>p,x</i>)</tt></h1>
|
|
The first form returns the prime factors for integer <i>n</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
factor(12345)
|
|
|
|
<i>Result</i>
|
|
|
|
3 5 823
|
|
</pre>
|
|
The second form factors polynomial <i>p</i> in <i>x</i>.
|
|
The argument <i>x</i> can be omitted in which case the computer will
|
|
guess which symbol to use.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
factor(x^3 + x^2 + x + 1)
|
|
|
|
<i>Result</i>
|
|
|
|
2
|
|
(1 + x) (1 + x )
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="factorial">factorial(<i>x</i>)</a></tt></h1>
|
|
Returns the factorial of <i>x</i>.
|
|
The syntax <i>x</i>! can also be used.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
factorial(100)
|
|
|
|
<i>Result</i>
|
|
|
|
93326215443944152681699238856266700490715968264381621468592963895217599993229915
|
|
608941463976156518286253697920827223758251185210916864000000000000000000000000
|
|
|
|
<i>Enter</i>
|
|
|
|
factorial(100) - 100!
|
|
|
|
<i>Result</i>
|
|
|
|
0
|
|
</pre>
|
|
<h1><tt><a name="filter">filter(<i>f,a,b,...</i>)</a></tt></h1>
|
|
Returns <i>f</i> with terms containing <i>a</i> (or <i>b</i> or...) removed.
|
|
Useful for implementing a "poor man's" Fourier transform.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
Y = A exp(-i k x) + B exp(i k x)
|
|
|
|
filter(Y exp(-i k x), x)
|
|
|
|
<i>Result</i>
|
|
|
|
B
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="float">float(<i>x</i>)</a></tt></h1>
|
|
Converts rational numbers and integers in <i>x</i> to floating
|
|
point values. The symbol <i>pi</i> is also converted.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
float(100!)
|
|
|
|
<i>Result</i>
|
|
|
|
9.33262e+157
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="floor">floor(<i>x</i>)</a></tt></h1>
|
|
Returns the largest integer not greater than <i>x</i>.
|
|
<p>
|
|
<h1><tt><a name="for">for(<i>a,i,j,b</i>)</a></tt></h1>
|
|
For <i>a</i> equals <i>i</i> through <i>j</i> evaluate <i>b</i>.
|
|
Normally <tt>for()</tt> returns the symbol <tt>nil</tt>.
|
|
A <tt>break()</tt> function can be used to return a different value.
|
|
The variable <i>a</i> has local scope within the <tt>for</tt> function.
|
|
The variable <i>a</i> remains unmodified after <tt>for</tt> returns.
|
|
The expressions <i>i</i> and <i>j</i> must evaluate to integers.
|
|
Usually <i>b</i> is a <tt>do()</tt> function.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
for(k,1,4,print(1/k,tab(10),1/k^2))
|
|
|
|
<i>Result</i>
|
|
|
|
1 1
|
|
|
|
1 1
|
|
--- ---
|
|
2 4
|
|
|
|
1 1
|
|
--- ---
|
|
3 9
|
|
|
|
1 1
|
|
--- ----
|
|
4 16
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="gcd">gcd(<i>a,b</i>)</a></tt></h1>
|
|
Returns the greatest common divisor of <i>a</i> and <i>b</i>.
|
|
<p>
|
|
<h1><tt><a name="hermite">hermite(<i>x,n</i>)</a></tt></h1>
|
|
Returns the <i>n</i>th Hermite polynomial in <i>x</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
hermite(x,3)
|
|
|
|
<i>Result</i>
|
|
|
|
3
|
|
-12 x + 8 x
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="hilbert">hilbert(<i>n</i>)</a></tt></h1>
|
|
Returns a Hilbert matrix of order <i>n</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
hilbert(3)
|
|
|
|
<i>Result</i>
|
|
|
|
1 1
|
|
1 --- ---
|
|
2 3
|
|
|
|
1 1 1
|
|
--- --- ---
|
|
2 3 4
|
|
|
|
1 1 1
|
|
--- --- ---
|
|
3 4 5
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="imag">imag(z)</a></tt></h1>
|
|
Returns the imaginary part of complex z.
|
|
Symbols in expression z are presumed to be positive and real.
|
|
<pre>
|
|
<i>Enter</i>
|
|
imag(1 + exp(i pi/3))
|
|
|
|
<i>Result</i>
|
|
1/2 3^(1/2)
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="inner">inner(<i>a,b,...</i>)</a></tt></h1>
|
|
Returns the inner product of tensors.
|
|
This is the same function as the dot product.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
A = (A1,A2,A3)
|
|
|
|
B = (B1,B2,B3)
|
|
|
|
inner(A,B)
|
|
|
|
<i>Result</i>
|
|
|
|
A1 B1 + A2 B2 + A3 B3
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="integral">integral(<i>f,x</i>)</a></tt></h1>
|
|
Returns the integral of <i>f</i> with respect to <i>x</i>.
|
|
The second argument can be omitted in which case the computer will guess which
|
|
symbol to use.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
integral(log(x),x)
|
|
|
|
<i>Result</i>
|
|
|
|
-x + x log(x)
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="inv">inv(<i>m</i>)</a></tt></h1>
|
|
Returns the inverse of matrix <i>m</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
A = ((a,b),(c,d))
|
|
|
|
inv(A)
|
|
|
|
<i>Result</i>
|
|
|
|
d b
|
|
----------- - -----------
|
|
a d - b c a d - b c
|
|
|
|
c a
|
|
- ----------- -----------
|
|
a d - b c a d - b c
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="isprime">isprime(<i>n</i>)</a></tt></h1>
|
|
Returns 1 if integer <i>n</i> is a prime number. Returns 0 if <i>n</i> is not
|
|
a prime number.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
isprime(9007199254740991)
|
|
|
|
<i>Result</i>
|
|
|
|
0
|
|
|
|
<i>Enter</i>
|
|
|
|
isprime(2^53 - 111)
|
|
|
|
<i>Result</i>
|
|
|
|
1
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="laguerre">laguerre(<i>x,n,a</i>)</a></tt></h1>
|
|
Returns the <i>n</i>th Laguerre polynomial in <i>x</i>.
|
|
If the argument <i>a</i> is omitted or <i>a</i> equals zero
|
|
then the function returns the non-associated Laguerre polynomial.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
laguerre(x,2)
|
|
|
|
<i>Result</i>
|
|
|
|
1 2
|
|
1 - 2 x + --- x
|
|
2
|
|
|
|
<i>Enter</i>
|
|
|
|
laguerre(x,2,a)
|
|
|
|
<i>Result</i>
|
|
|
|
3 1 2 1 2
|
|
1 + --- a - 2 x - a x + --- a + --- x
|
|
2 2 2
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="lcm">lcm(<i>a,b</i>)</a></tt></h1>
|
|
Returns the least common multiple of <i>a</i> and <i>b</i>.
|
|
The least common multiple is the smallest value or expression divisible
|
|
by both <i>a</i> and <i>b</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
lcm(4,6)
|
|
|
|
<i>Result</i>
|
|
|
|
12
|
|
|
|
<i>Enter</i>
|
|
|
|
lcm(4 x, 6 x y)
|
|
|
|
<i>Result</i>
|
|
|
|
12 x y
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="legendre">legendre(<i>x,n,m</i>)</a></tt></h1>
|
|
Returns the <i>n</i>th Legendre polynomial in <i>x</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
legendre(x,2)
|
|
|
|
<i>Result</i>
|
|
|
|
1 3 2
|
|
- --- + --- x
|
|
2 2
|
|
|
|
<i>Enter</i>
|
|
|
|
legendre(x,2,0)
|
|
|
|
<i>Result</i>
|
|
|
|
1 3 2
|
|
- --- + --- x
|
|
2 2
|
|
|
|
<i>Enter</i>
|
|
|
|
legendre(x,2,1)
|
|
|
|
<i>Result</i>
|
|
|
|
1/2
|
|
2
|
|
-3 x (1 - x )
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="log">log(<i>x</i>)</a></tt></h1>
|
|
Returns the natural logarithm of <i>x</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
log(10.0)
|
|
|
|
<i>Result</i>
|
|
|
|
2.30259
|
|
|
|
<i>Enter</i>
|
|
|
|
log(-10.0)
|
|
|
|
<i>Result</i>
|
|
|
|
2.30259 + i π
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="mag">mag(z)</a></tt></h1>
|
|
Returns the magnitude of complex z.
|
|
Symbols in expression z are presumed to be positive and real.
|
|
<pre>
|
|
<i>Enter</i>
|
|
mag(1 + exp(i pi/3))
|
|
|
|
<i>Result</i>
|
|
3^(1/2)
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="mod">mod(<i>a,b</i>)</a></tt></h1>
|
|
Returns the remainder of <i>a</i> divided by <i>b</i>.
|
|
<p>
|
|
<h1><tt><a name="numerator">numerator(<i>x</i>)</a></tt></h1>
|
|
Returns the numerator of expression x.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
numerator(a/b)
|
|
|
|
<i>Result</i>
|
|
|
|
a
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="outer">outer(<i>a,b,...</i>)</a></tt></h1>
|
|
Returns the outer product of tensors (aka tensor product).
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
A = (A1,A2,A3)
|
|
|
|
B = (B1,B2,B3)
|
|
|
|
outer(A,B)
|
|
|
|
<i>Result</i>
|
|
|
|
A1 B1 A1 B2 A1 B3
|
|
|
|
A2 B1 A2 B2 A2 B3
|
|
|
|
A3 B1 A3 B2 A3 B3
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="prime">prime(<i>n</i>)</a></tt></h1>
|
|
Returns the <i>n</i>th prime number.
|
|
The value of <i>n</i> must be greater than zero and less than 10,001.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
prime(1)
|
|
|
|
<i>Result</i>
|
|
|
|
2
|
|
|
|
<i>Enter</i>
|
|
|
|
prime(10000)
|
|
|
|
<i>Result</i>
|
|
|
|
104729
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="print">print(<i>a,b,...</i>)</a></tt></h1>
|
|
The <tt>print</tt> function evaluates and prints each expression.
|
|
The printing is done in tty mode.
|
|
The symbol <tt>nil</tt> is returned as the function value.
|
|
Spaces and other text can be printed by using quoted strings for print
|
|
arguments.
|
|
This function can be selected as the default printing mode by setting the
|
|
symbol tty = 1.
|
|
<p>
|
|
<h1><tt><a name="product">product(<i>a,i,j,b</i>)</a></tt></h1>
|
|
For <i>a</i> equals <i>i</i> through <i>j</i> evaluate <i>b</i>.
|
|
Returns the product of all <i>b</i>.
|
|
The variable <i>a</i> has local scope within the product function,
|
|
<i>a</i> remains unchanged after the product function returns.
|
|
The expressions <i>i</i> and <i>j</i> should evaluate to integers.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
product(k,1,3,1/(1-(1/prime(k)^s)))
|
|
|
|
<i>Result</i>
|
|
|
|
1
|
|
----------------------------------
|
|
1 1 1
|
|
(1 - ----) (1 - ----) (1 - ----)
|
|
s s s
|
|
2 3 5
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="prog">prog(<i>a,b,...,f</i>)</a></tt></h1>
|
|
The <tt>prog</tt> function evaluates <i>f</i>.
|
|
The result of the evaluation of <i>f</i> is returned as the <tt>prog</tt>
|
|
value.
|
|
The variables <i>a,b,...</i> have local scope within <tt>prog</tt>.
|
|
The variables <i>a,b,...</i> remain unmodified after <tt>prog</tt> returns.
|
|
Usually <i>f</i> is a <tt>do</tt> function.
|
|
<p>
|
|
<h1><tt><a name="quote">quote(<i>x</i>)</a></tt></h1>
|
|
Returns expression <i>x</i> without evaluating symbols or functions.
|
|
Can be used to clear symbolic values.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
n = 3
|
|
|
|
n
|
|
|
|
<i>Result</i>
|
|
|
|
3
|
|
|
|
<i>Enter</i>
|
|
|
|
n = quote(n)
|
|
|
|
n
|
|
|
|
<i>Result</i>
|
|
|
|
n
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="rank">rank(<i>a</i>)</a></tt></h1>
|
|
Returns the rank (number of indices) of tensor <i>a</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
U = (u1,u2,u3,u4)
|
|
|
|
rank(U)
|
|
|
|
<i>Result</i>
|
|
|
|
1
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="rationalize">rationalize(<i>x</i>)</a></tt></h1>
|
|
Puts terms in <i>x</i> over a common denominator.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
rationalize(1/x + 1/y)
|
|
|
|
<i>Result</i>
|
|
|
|
x + y
|
|
-------
|
|
x y
|
|
</pre>
|
|
Rationalize can often simplify expressions.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
A = ((a,b),(c,d))
|
|
|
|
B = inv(A)
|
|
|
|
dot(A,B)
|
|
|
|
<i>Result</i>
|
|
|
|
a d b c
|
|
----------- - ----------- 0
|
|
a d - b c a d - b c
|
|
|
|
a d b c
|
|
0 ----------- - -----------
|
|
a d - b c a d - b c
|
|
|
|
<i>Enter</i>
|
|
|
|
rationalize(last)
|
|
|
|
<i>Result</i>
|
|
|
|
1 0
|
|
|
|
0 1
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="real">real(z)</a></tt></h1>
|
|
Returns the real part of complex z.
|
|
Symbols in expression z are presumed to be positive and real.
|
|
<pre>
|
|
<i>Enter</i>
|
|
real(1 + exp(i pi/3))
|
|
|
|
<i>Result</i>
|
|
3/2
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="rect">rect(z)</a></tt></h1>
|
|
Returns complex z in rectangular form.
|
|
Symbols in expression z are presumed to be positive and real.
|
|
<pre>
|
|
<i>Enter</i>
|
|
rect(1 + exp(i pi/3))
|
|
|
|
<i>Result</i>
|
|
3/2 + 1/2 i 3^(1/2)
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="return">return(<i>x</i>)</a></tt></h1>
|
|
Evaluation of <tt>return</tt> causes an immediate exit from <tt>prog</tt>.
|
|
Expression <i>x</i> is evaluated and returned as the <tt>prog</tt> value.
|
|
A <tt>return</tt> with no argument returns the symbol <tt>nil</tt>.
|
|
A <tt>return</tt> can be evaluated at any function level.
|
|
For example, <tt>prog()</tt> can evaluate <tt>f()</tt> which evaluates
|
|
<tt>g()</tt> which evaluates <tt>return()</tt>.
|
|
<p>
|
|
<h1><tt><a name="roots">roots(<i>p,x</i>)</a></tt></h1>
|
|
Finds the values of <i>x</i> for which the polynomial <i>p(x)</i> equals zero.
|
|
Graphically, the roots of <i>p</i> are the <i>x</i> coordinates where the curve
|
|
<i>p(x)</i> crosses the horizontal line.
|
|
The argument <i>p</i> may be expressed as an equality.
|
|
The argument <i>x</i> can be omitted if it is literally
|
|
<i>x</i>, <i>y</i>, <i>z</i>, <i>t</i> or <i>r</i>.
|
|
If there is more than one root then a vector containing the roots is returned.
|
|
If <i>p</i> cannot be factored then <tt>roots</tt> returns the symbol
|
|
<tt>nil</tt>.
|
|
Note: The symbol <tt>nil</tt> is not normally printed so no result is visible
|
|
when <i>p</i> cannot be factored.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
(x - 1/2) (x - 1/3) (x + 1/4) / x^3
|
|
|
|
<i>Result</i>
|
|
|
|
1 1 7
|
|
1 + ------- - ------- - ------
|
|
3 2 12 x
|
|
24 x 24 x
|
|
|
|
<i>Enter</i>
|
|
|
|
roots(last,x)
|
|
|
|
<i>Result</i>
|
|
|
|
1
|
|
- ---
|
|
4
|
|
|
|
1
|
|
---
|
|
3
|
|
|
|
1
|
|
---
|
|
2
|
|
|
|
<i>Enter</i>
|
|
|
|
roots(a x = b)
|
|
|
|
<i>Result</i>
|
|
|
|
b
|
|
---
|
|
a
|
|
|
|
<i>Enter</i>
|
|
|
|
roots(a x^2 + b x + c)
|
|
|
|
<i>Result</i>
|
|
|
|
1/2
|
|
2
|
|
b (-4 a c + b )
|
|
- ----- - ------------------
|
|
2 a 2 a
|
|
|
|
1/2
|
|
2
|
|
b (-4 a c + b )
|
|
- ----- + ------------------
|
|
2 a 2 a
|
|
</pre>
|
|
<h1><tt><a name="simfac">simfac(<i>x</i>)</a></tt></h1>
|
|
Simplifies factorials in expression x according to
|
|
the following rules.
|
|
<pre>
|
|
This is converted to this
|
|
|
|
n! / n (n - 1)!
|
|
|
|
n (n - 1)! n!
|
|
|
|
(n + 1)! / n! n + 1
|
|
|
|
(n + 2)! / n! (n + 1) (n + 2)
|
|
</pre>
|
|
Example 1.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
F(n,k) = k binomial(n,k)
|
|
|
|
(F(n,k) + F(n,k-1)) / F(n+1,k)
|
|
|
|
<i>Result</i>
|
|
|
|
k! n! n! (1 - k + n)! k! n!
|
|
-------------------- + -------------------- - ----------------------
|
|
(-1 + k)! (1 + n)! (1 + n)! (-k + n)! k (-1 + k)! (1 + n)!
|
|
|
|
<i>Enter</i>
|
|
|
|
simfac
|
|
|
|
<i>Result</i>
|
|
|
|
1 1 - k + n 1
|
|
------- + ----------- - -------
|
|
1 + n 1 + n 1 + n
|
|
|
|
<i>Enter</i>
|
|
|
|
simplify
|
|
|
|
<i>Result</i>
|
|
|
|
n
|
|
-------
|
|
1 + n
|
|
</pre>
|
|
Example 2. It may be necessary to rationalize or condense an expression first.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
(n + 1) / (n + 1)!
|
|
|
|
<i>Result</i>
|
|
|
|
n 1
|
|
---------- + ----------
|
|
(1 + n)! (1 + n)!
|
|
|
|
<i>Enter</i>
|
|
|
|
simfac
|
|
|
|
<i>Result</i>
|
|
|
|
n 1
|
|
---------- + ----------
|
|
(1 + n)! (1 + n)!
|
|
|
|
<i>Enter</i>
|
|
|
|
rationalize
|
|
|
|
<i>Result</i>
|
|
|
|
1 + n
|
|
----------
|
|
(1 + n)!
|
|
|
|
<i>Enter</i>
|
|
|
|
simfac
|
|
|
|
<i>Result</i>
|
|
|
|
1
|
|
----
|
|
n!
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="simplify">simplify(<i>x</i>)</a></tt></h1>
|
|
Evaluates expression x and then attempts to simplify the result.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
(A-B)/(B-A)
|
|
|
|
<i>Result</i>
|
|
|
|
A B
|
|
-------- - --------
|
|
-A + B -A + B
|
|
|
|
<i>Enter</i>
|
|
|
|
simplify(last)
|
|
|
|
<i>Result</i>
|
|
|
|
-1
|
|
|
|
<i>Enter</i>
|
|
|
|
A = ((A11,A12),(A21,A22))
|
|
|
|
det(A) inv(A) - adj(A)
|
|
|
|
<i>Result</i>
|
|
|
|
((-A22 + A11 A22^2 / (A11 A22 - A12 A21) - A12 A21 A22 / (A11 A22 - A12 A21),
|
|
A12 - A11 A12 A22 / (A11 A22 - A12 A21) + A12^2 A21 / (A11 A22 - A12 A21)),
|
|
(A21 - A11 A21 A22 / (A11 A22 - A12 A21) + A12 A21^2 / (A11 A22 - A12 A21),
|
|
-A11 - A11 A12 A21 / (A11 A22 - A12 A21) + A11^2 A22 / (A11 A22 - A12 A21)))
|
|
|
|
<i>Enter</i>
|
|
|
|
simplify(last)
|
|
|
|
<i>Result</i>
|
|
|
|
0
|
|
</pre>
|
|
The <tt>simplify</tt> function can return factored
|
|
(unexpanded)
|
|
expressions.
|
|
Factored expressions can fail in tests for equality.
|
|
The <tt>eval</tt> function can be used to expand factored expressions.
|
|
<p>
|
|
<h1><tt><a name="sin">sin(<i>x</i>)</a></tt></h1>
|
|
Returns the sine of <i>x</i>.
|
|
<p>
|
|
<h1><tt><a name="sinh">sinh(<i>x</i>)</a></tt></h1>
|
|
Returns the hyperbolic sine of <i>x</i>.
|
|
<p>
|
|
<h1><tt><a name="sqrt">sqrt(<i>x</i>)</a></tt></h1>
|
|
Returns the square root of <i>x</i>.
|
|
<p>
|
|
<h1><tt><a name="stop">stop()</a></tt></h1>
|
|
When evaluated in a script, the stop function exits run mode and
|
|
returns to interactive mode.
|
|
<p>
|
|
<h1><tt><a name="subst">subst(<i>a,b,c</i>)</a></tt></h1>
|
|
Substitutes <i>a</i> for <i>b</i> in <i>c</i> and returns the result.
|
|
Note that this operation can return a denormalized expression.
|
|
Use <tt>eval</tt> to normalize the result of <tt>subst</tt>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
f = x^2
|
|
|
|
subst(sqrt(x),x,f)
|
|
|
|
<i>Result</i>
|
|
|
|
2
|
|
1/2
|
|
(x )
|
|
|
|
<i>Enter</i>
|
|
|
|
eval(last)
|
|
|
|
<i>Result</i>
|
|
|
|
x
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="sum">sum(<i>a,i,j,b</i>)</a></tt></h1>
|
|
For <i>a</i> equals <i>i</i> through <i>j</i> evaluate <i>b</i>.
|
|
Returns the sum of all <i>b</i>.
|
|
The variable <i>a</i> has local scope within the sum function,
|
|
<i>a</i> remains unchanged after the sum function returns.
|
|
The expressions <i>i</i> and <i>j</i> should evaluate to integers.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
sum(k,1,3,1/k^s)
|
|
|
|
<i>Result</i>
|
|
|
|
1 1
|
|
1 + ---- + ----
|
|
s s
|
|
2 3
|
|
</pre>
|
|
<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 <i>f</i> at <i>a</i>.
|
|
If <i>a</i> is omitted then zero is used.
|
|
The argument <i>x</i> is the free variable in <i>f</i>
|
|
and <i>n</i> is the power of the expansion.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
taylor(1/cos(x),x,6)
|
|
|
|
<i>Result</i>
|
|
|
|
1 2 5 4 61 6
|
|
1 + --- x + ---- x + ----- x
|
|
2 24 720
|
|
</pre>
|
|
<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 no
|
|
conditions are true.
|
|
If the number of arguments is even then the symbol void is returned when no
|
|
conditions are true.
|
|
The usual relational operators can be used.
|
|
<pre>
|
|
<i>Enter</i>
|
|
f(a,b) = test(a<b, -1, a=b, 0, a>b, 1)
|
|
f(1,2)
|
|
|
|
<i>Result</i>
|
|
-1
|
|
|
|
<i>Enter</i>
|
|
f(a,b)
|
|
|
|
<i>Result</i>
|
|
void
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="trace">trace(<i>m</i>)</a></tt></h1>
|
|
Returns the trace of matrix <i>m</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
A = ((a,b),(c,d))
|
|
|
|
trace(A)
|
|
|
|
<i>Result</i>
|
|
|
|
a + d
|
|
</pre>
|
|
Note that trace is equivalent to contract.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
trace(A) - contract(A,1,2)
|
|
|
|
<i>Result</i>
|
|
|
|
0
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="transpose">transpose(<i>a,i,j</i>)</a></tt></h1>
|
|
Returns the transpose of tensor <i>a</i> across indices <i>i</i> and
|
|
<i>j</i>.
|
|
If <i>i</i> and <i>j</i> are omitted then indices 1 and 2 are used.
|
|
<p>
|
|
<h1><tt><a name="unit">unit(<i>n</i>)</a></tt></h1>
|
|
Returns a unit matrix with dimension <i>n</i>.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
unit(4)
|
|
|
|
<i>Result</i>
|
|
|
|
1 0 0 0
|
|
|
|
0 1 0 0
|
|
|
|
0 0 1 0
|
|
|
|
0 0 0 1
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="wedge">wedge(<i>u,v</i>)</a></tt></h1>
|
|
<h1><tt>wedge(<i>u,v,w</i>)</tt></h1>
|
|
Returns the wedge product of tensors.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
u = (u1,u2,u3,u4)
|
|
|
|
v = (v1,v2,v3,v4)
|
|
|
|
wedge(u,v)
|
|
|
|
<i>Result</i>
|
|
|
|
0 u1 v2 - u2 v1 u1 v3 - u3 v1 u1 v4 - u4 v1
|
|
|
|
-u1 v2 + u2 v1 0 u2 v3 - u3 v2 u2 v4 - u4 v2
|
|
|
|
-u1 v3 + u3 v1 -u2 v3 + u3 v2 0 u3 v4 - u4 v3
|
|
|
|
-u1 v4 + u4 v1 -u2 v4 + u4 v2 -u3 v4 + u4 v3 0
|
|
|
|
<i>Enter</i>
|
|
|
|
wedge(u,v) + wedge(v,u)
|
|
|
|
<i>Result</i>
|
|
|
|
0
|
|
</pre>
|
|
<p>
|
|
<h1><tt><a name="zero">zero(<i>i,j,...</i>)</a></tt></h1>
|
|
Returns a zero tensor with dimensions <i>i,j,...</i>
|
|
The zero function is useful for creating a tensor and then setting the component values.
|
|
<pre>
|
|
<i>Enter</i>
|
|
|
|
A = zero(2,2)
|
|
A
|
|
|
|
<i>Result</i>
|
|
|
|
0 0
|
|
|
|
0 0
|
|
|
|
<i>Enter</i>
|
|
|
|
A[1,2] = a
|
|
A
|
|
|
|
<i>Result</i>
|
|
|
|
0 a
|
|
|
|
0 0
|
|
</pre>
|
|
|
|
<p>
|
|
<hr>
|