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Quelle  chap4_mj.html   Sprache: HTML

 
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<div class="ChapSects"><a href="chap4_mj.html#X7D12102B824C97B4">4 <span class="Heading">Vectors</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X788B78DE82579F3D">4.1 <span class="Heading">Creation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X83BF7BBC84478E77">4.1-1 CVec</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X83E8D01B82498337">4.1-2 CVec</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X826347BB7B81A06D">4.1-3 CVec</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X83E046D17AB12FF0">4.1-4 CVec</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X86642DDE797387E7">4.1-5 CVec</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7CC92B9E8286B025">4.1-6 CVec</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B9566C1867C5EA6">4.1-7 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X78E5FA4881DE6D1A">4.1-8 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X871E42B97E3664EB">4.1-9 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81C506327D635947">4.1-10 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79F5D11E830B9768">4.1-11 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X87D5ABC1835C4A31">4.1-12 CVecClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X876613FC7A9A276C">4.1-13 ZeroSameMutability</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7BA2BD517C058720">4.1-14 ShallowCopy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X84EBC4427913251F">4.1-15 Randomize</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X79862FF37B75BF48">4.2 <span class="Heading">Arithmetic</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X788A3D6482290F7D">4.2-1 <span class="Heading">Handling of scalars in arithmetic operations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7C2F565480B1A4A1"><code>4.2-2 \+</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7CFEB1507E279EF2"><code>4.2-3 \-</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7DD3036C847E12DC">4.2-4 AdditiveInverseSameMutability</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7A34F9AA8330EBB9">4.2-5 AdditiveInverseMutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8191F74E797B4A3D"><code>4.2-6 \*</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X820D829E87A82A80">4.2-7 AddRowVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X846D1EDA7B5B1B39">4.2-8 MultVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F4B8B747E88C59B">4.2-9 ScalarProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X86115BE97842EC3B">4.2-10 ZeroMutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7FB1A37D7F8603B9">4.2-11 ZeroVector</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7C68FFEC84A86786">4.3 <span class="Heading">Element access and slicing</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7A6A222E87452C27">4.3-1 ELM_LIST</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X78F1C2AE7A8DD868">4.3-2 ASS_LIST</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7D12C17C873DE3B4">4.3-3 ELMS_LIST</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X78878E5E839265A6">4.3-4 CVEC_Slice</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X80EC354D78D7B5A6">4.3-5 CopySubVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X84347D8B7DFA790C">4.3-6 CVEC_Concatenation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X81F3C9697E5636B6">4.4 <span class="Heading">Comparison of Vectors and other information</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8155313286F88093">4.4-1 =</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X790D69A88340E510">4.4-2 LT</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X78952A3C788B7DB3">4.4-3 IsZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F42E5AD87EC9D5A">4.4-4 PositionNonZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8370979D7F451279">4.4-5 PositionLastNonZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X780769238600AFD1">4.4-6 Length</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X80A1A9C386FD0E1B">4.5 <span class="Heading">Changing representation, Unpacking</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7BD632E57E1EB300">4.5-1 Unpack</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X850FB74483F87453">4.5-2 IntegerRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7FDFEF7A7E850151">4.5-3 NumberFFVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7E055C467B1B54D8">4.5-4 CVecNumber</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X853A85AE878B0CBF">4.6 <span class="Heading">Access to the base field</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X854192428704857E">4.6-1 BaseDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X842335E47DC49702">4.6-2 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81B99EEC7C134213">4.6-3 Characteristic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81E3BBBB80096DC2">4.6-4 DegreeFFE</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X823EFBDB7ED49554">4.7 <span class="Heading">Hashing techniques for <code class="code">cvec</code>s</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F0228A18327F87D">4.7-1 CVEC_HashFunctionForCVecs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82C8DE4C78B80B1E">4.7-2 ChooseHashFunction</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">Vectors</span></h3>

<p>See Section <a href="chap3_mj.html#X7ED5B368830755AF"><span class="RefLink">3.2</span></a> for a documentation of the data format of <code class="code">cvec</code>s and its restrictions.</p>

<p><a id="X788B78DE82579F3D" name="X788B78DE82579F3D"></a></p>

<h4>4.1 <span class="Heading">Creation</span></h4>

<p>Note that many functions described later in this chapter implicitly create new <code class="code">cvec</code>s, such that it is usually only in the beginning of a calculation necessary to create <code class="code">cvec</code>s explicitly.</p>

<p><a id="X83BF7BBC84478E77" name="X83BF7BBC84478E77"></a></p>

<h5>4.1-1 CVec</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">arg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a new mutable <code class="code">cvec</code></p>

<p>Creates a new <code class="code">cvec</code>. See the method descriptions for details.</p>

<p><a id="X83E8D01B82498337" name="X83E8D01B82498337"></a></p>

<h5>4.1-2 CVec</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">cvecclass</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a new mutable <code class="code">cvec</code></p>

<p><var class="Arg">cvecclass</var> must be a <code class="code">cvecclass</code>. Creates a new <code class="code">cvec</code> containing only zeroes. For an explanation of the term <code class="code">cvecclass</code> see Section <a href="chap3_mj.html#X7ED5B368830755AF"><span class="RefLink">3.2</span></a> and <code class="func">CVecClass</code> (<a href="chap4_mj.html#X7B9566C1867C5EA6"><span class="RefLink">4.1-7</span></a>).</p>

<p><a id="X826347BB7B81A06D" name="X826347BB7B81A06D"></a></p>

<h5>4.1-3 CVec</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">coeffs</var>, <var class="Arg">cvecclass</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a new mutable <code class="code">cvec</code></p>

<p>This method takes a coefficient list and a <code class="code">cvecclass</code> as arguments. Assume the vector will be over <span class="SimpleMath">\(GF(p,d)\)</span> with <span class="SimpleMath">\(q=p^d\)</span>. For the coefficient list <var class="Arg">coeffs</var> there exist several possibilities, partially depending on the base field size. The easiest way is to use <strong class="pkg">GAP</strong> finite field elements, which will be put into the new vector in the same order. If <span class="SimpleMath">\(d=1\)</span>, one can always use <strong class="pkg">GAP</strong> immediate integers between <span class="SimpleMath">\(0\)</span> and <span class="SimpleMath">\(p-1\)</span>, the vector will then contain the corresponding cosets in <span class="SimpleMath">\(GF(p)=Z/pZ\)</span>. If <span class="SimpleMath">\(q\)</span> is small enough to be a <strong class="pkg">GAP</strong> immediate integer, then one can use <strong class="pkg">GAP</strong> immediate integers that are equal to the <span class="SimpleMath">\(p\)</span>-adic expansion using the coefficients of the representing polynomial as coefficients. That is, if an element in <span class="SimpleMath">\(GF(p,d)\)</span> is represented by the polynomial <span class="SimpleMath">\(\sum_{{i=0}}^{{d-1}} a_i x^i\)</span> with <span class="SimpleMath">\(a_i \in \{0,\ldots,p-1\}\)</span>, then one has to put <span class="SimpleMath">\(\sum_{{i=0}}^{{d-1}} a_i p^i\)</span> as integer into the coefficient list <var class="Arg">coeffs</var>. If <span class="SimpleMath">\(q\)</span> is larger, then <var class="Arg">coeffs</var> must be a list of lists of length <span class="SimpleMath">\(d\)</span> and contains for each field element of <span class="SimpleMath">\(GF(p,d)\)</span> in the vector a list of its <span class="SimpleMath">\(d\)</span> coefficients of the representing polynomial. For an explanation of the term <code class="code">cvecclass</code> see Section <a href="chap3_mj.html#X7ED5B368830755AF"><span class="RefLink">3.2</span></a> and <code class="func">CVecClass</code> (<a href="chap4_mj.html#X7B9566C1867C5EA6"><span class="RefLink">4.1-7</span></a>). Of course, the length of the list <var class="Arg">coeffs</var> must match the length of the vector given in the <code class="code">cvecclass</code>.</p>

<p><a id="X83E046D17AB12FF0" name="X83E046D17AB12FF0"></a></p>

<h5>4.1-4 CVec</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">coeffs</var>, <var class="Arg">p</var>, <var class="Arg">d</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a new mutable <code class="code">cvec</code></p>

<p>This method takes a coefficient list and two positive integers <var class="Arg">p</var> and <var class="Arg">d</var> as arguments. A new <code class="code">cvec</code> over <span class="SimpleMath">\(GF(p,d)\)</span> will be created. Let <span class="SimpleMath">\(q := p^d\)</span>.</p>

<p>For a description of the possible values of the coefficient list <var class="Arg">coeffs</var> see the description of the method <code class="func">CVec</code> (<a href="chap4_mj.html#X826347BB7B81A06D"><span class="RefLink">4.1-3</span></a>).</p>

<p><a id="X86642DDE797387E7" name="X86642DDE797387E7"></a></p>

<h5>4.1-5 CVec</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">v</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a new <code class="code">cvec</code></p>

<p><var class="Arg">v</var> must be a <strong class="pkg">GAP</strong> compressed vector either over <span class="SimpleMath">\(GF(2)\)</span> or <span class="SimpleMath">\(GF(p,d)\)</span> with <span class="SimpleMath">\(p^d \leq 256\)</span>. Creates a new <code class="code">cvec</code> containing the same numbers as <var class="Arg">v</var> over the field which the <code class="code">Field</code> operation returns for <var class="Arg">v</var>.</p>

<p><a id="X7CC92B9E8286B025" name="X7CC92B9E8286B025"></a></p>

<h5>4.1-6 CVec</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVec</code>( <var class="Arg">coeffs</var>, <var class="Arg">f</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: a new mutable <code class="code">cvec</code></p>

<p>This method takes a coefficient list and a finite field <var class="Arg">f</var> as arguments. A new <code class="code">cvec</code> over <var class="Arg">f</var> will be created. Let <span class="SimpleMath">\(q := \)</span><code class="code">Size(</code><var class="Arg">f</var><code class="code">)</code>.</p>

<p>For a description of the possible values of the coefficient list <var class="Arg">coeffs</var> see the description of the method <code class="func">CVec</code> (<a href="chap4_mj.html#X826347BB7B81A06D"><span class="RefLink">4.1-3</span></a>).</p>

<p>After having encountered the concept of a <code class="code">cvecclass</code> already a few times it is time to learn how to create one. The following operation is used first to create new <code class="code">cvecclass</code>es and second to ask a <code class="code">cvec</code> for its class. In addition, it is used for <code class="code">csca</code>s.</p>

<p><a id="X7B9566C1867C5EA6" name="X7B9566C1867C5EA6"></a></p>

<h5>4.1-7 CVecClass</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVecClass</code>( <var class="Arg">arg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <code class="code">cvecclass</code></p>

<p>Creates new <code class="code">cvecclass</code>es and asks <code class="code">cvec</code>s for their class. See the following method descriptions for details. For an explanation of the term <code class="code">cvecclass</code> see Section <a href="chap3_mj.html#X7ED5B368830755AF"><span class="RefLink">3.2</span></a>.</p>

<p>>

h51java.lang.StringIndexOutOfBoundsException: Range [19, 11) out of bounds for length 24

<div ="">br>span=""&bspa =chap4_mj#">41- />
<>: a code=""></code<pjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53

<p>All threeargumentsmust class>< /< class"">&;;><=.#X7CC92B9E8286B025416</><>

<> an  t< classcode" its">&;&bsp<>a java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 134

>==X871E42B97E3664EBp

h5h5 =<span class="ContSS<""&;;<panhrefhtml>.- CVec>sjava.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 129

<div class="func"><spanclass="ContSS"<br/< class=""> </><a ="chap4_mj.#X7BA2BD517C058720">4.-4ShallowCopya</spanjava.lang.StringIndexOutOfBoundsException: Index 137 out of bounds for length 137
<<p ="ContSS>br />span "nocss&&;<<a".#X84EBC4427913251F>.-5 spanjava.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135

<p>The argument ="ContSSBlock>

s/>

<h510 <hjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

<div class="func"><table class="func< ="ContSS < class ="><< class=">;&;</span=".html#X7A34F9AA8330EBB9>.- AdditiveInverseMutable/>/span>
<p>Returns a codeclasscode>cvecclass/code>/p>

<p>The first class=""><br><span="nocss" pan=java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 137

<p><a id="
  
.html">4.-


<h5>4span="ContSS#X876613FC7A9A276C"41-3ZeroSameMutability>span

< classfunctableclass width0>trjava.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 136
<eturns code ="></code></>

<p>The argument <var span=""> /< class"&; />>431ELM_LISTa>/>

<p><a ="X87D5ABC1835C4A31 name="X87D5ABC1835C4A31<a>pjava.lang.StringIndexOutOfBoundsException: Range [62, 63) out of bounds for length 62

<h5>4.1-12 CVecClass class"nbsp />< href"chap4_mj#X78878E5E839265A643-4CVEC_Slice/>

<div class=func><table classfunc" width="100%"
func827+< vclass">v, cvec<>


java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0

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<p><a id="X7CFEB1507E279EF2" name =ContSSbr /

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<Returns

<p>Forp><a idp>This method a coefficientlist and < class"code">cvecclass/code  .  the

<p>a id=

<h5424 AdditiveInverseSameMutability/h5

<div="< =func" width
< class"func">tableclass"func "10"tp>: anew code><code<pjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
<pinverse class""/varas new class"ode">/></>

<p>For>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<class">table class=func"width"10%>
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<>25AdditiveInverseMutable>

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<p>Returns nothing<p

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<<idjava.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0

<h5>4.2-9 ScalarProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ScalarProducth541-12CVecClass<h5
<p>Returns: the scalar product

<p>Bothp>a id="X876613FC7A9A276C" name"X876613FC7A9A276C"></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p> "><

<h5>4.2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

div""table"
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<>4.-11 ZeroVector</h5

<div class""><table="func"width10%">tr>td class"tdleft class=func>&8227;ZeroVector/code( <varclass"rg">
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<p><var

<p><a id="
<tableclass"unc width"10%>tr class="tdleft"><odeclass"func>&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h4p><iv class""width="00">r<code"func">#27 Randomizecode(<classArg><var<var classArg<var</>td="tdright">nbsp )/d<tr></div

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<p><a id="X7A6A222E87452C27" name="X7A6A222E87452C27"></a></p>

<h5>4.3-1 <p><a id="X79862FF37B75" name="X79862FF37B75BF48></a><p>

<div class="func"><table">Arithmetic
<p>Returns: a finite element

<) element accessof. <var="Arg>v mustbeaArgcvecvar varArg<var bepositive  thano<class">v.Thefinite field element att position  name="X78F1C2AE7A8DD868"></a></p>

<h5>4.3-2 ASS_LIST</h5>

<div class="func"><table class==p>Integersbetween <span="SimpleMath">\(0\java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 0
<p><p>Returns

<p>This is <p>Returns:  zerocode="code"cvec>  thesame <code ="code>cvecclass<table"func width=10%

td class"">>&82; Randomize

<p>Note that the list slice access syntax <q><var class="Arg">v</var>{<var class="Arg">l<pReturns the < class"rg>

<p>Note that there<divclass=func"width10"
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<p><>aidversevarArg"vvar as anew code class=code>cvec/ode><>

<h5<Ofcourse standard infixspan=SimpleMath(\<span<panSimpleMath>\<span and class"\*\/>( and expected by usingthe methodsbelow start thissection a on usageof operations involving vectors.



<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CVEC_Slice</codejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returnsnothing>

<p>varclass="Arg">rc and class">dst/var> must be ">cvec/>s the same field field elementsfrom positions th 0

<>4.3-CopySubVector/>

iv"func>func =0%<>td">&82 CopySubVector> var="Arg>rc =Argdrcpossdstposs/> )tdright&;ethod<td<tablejava.lang.StringIndexOutOfBoundsException: Range [312, 313) out of bounds for length 312
<p>Returns: nothing</p>

<p>Implements the operation <code class="func">CopySubVector</code> (<a href="../../../doc/ref/chap26_mj.html

Returns the scalar multiple s</ar><spanclass>:the <span="SimpleMath>(textit{v+{\})/> asa ode class""cvec/code>/p>

<p><a id="X84347D8B7DFA790C" name="X84347D8B7DFA790C"></a></p>

<

<div class="func"
<p>: a  <codeclass=codecvec>by concatenating arguments/pjava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83

<p>This

<p><a <p><a id

<h4>4.4h524AdditiveInverseSameMutability<h5

<p< ="X8155313286F88093 =X8155313286F88093>/



<h5>4.4-1 <p>var classp>: the inverse <arclass"v/varas new "code><c>p>

<div=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns: <code class="code">true</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p> <code="code>ifthe< class=code">cvec>s var="Arg>and< ="/  .Thebetheand  ./>

<p><a id="X790D69A88340E510" name="X790D69A88340E510"></a></p>

<h5>4.4-2 LT</h5

<div
eclass"true/ode codeclass""false/>/>

<p>turnsodeclasscode>truecode if<codecodecvec/ <arlass"vis < class="Argw/>.  mustbe the fieldand have length order is efficient is<mnot> with ordering lists  fieldelements!This shouldonlyb used binarypurposes that operationcode="code>LT<

<p> id="" nameX78952A3C788B7DB3a/p

>p>Returns: the multiple <var class="rg>s class="SimpleMath">\(cdot)ar class="Argp

<div>="java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<: code=java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0

<Returnscode="true
if the s to lists respectjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 220
<:positive

<p>Returns the>eturns </p>

<p><a id="X8370979D7F451279" name="X8370979D7F451279"></a></p>

<h5>4.4-5 PositionLastNonZero</h5>

<p>f <var=Arg<orArgto<"impleMath">()/pan it defaults < class\\/pan code class=code(v/ respectively>
<Returns integer

<  index lastin < classcvec/>< class/ar is equalto. Ifvector equal zero then classSimpleMath0</pan ./>

<p><a id="X780769238600AFD1" name="X780769238600AFD1"></a></p>

<h5>4.4-6 Length</h5>

< ="func">table class
<p>Returns: a non-negative integer</p>

Returns  code="< Arg>

pa=X80A1A9C386FD0E1B=

<h4p> that the list syntaxq< class"[class"">] triggersacall the < class=func>ELM_LIST

p>id"name"7pReturns: acopy thezero class=code<code in same class">
><var>

<>4.5-Unpack>

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<

<p<>The <var="peturns a mutable the zero < =""cvec/ode>overthe same field thanbut with length><arp>

<p><>Note thatthe  

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<p><a id="X7FDFEF7A7E850151" name="X7FDFEF7A7E850151"></a></p>

<h5>4.5-3 NumberFFVector</h5

div=func
<:  integer

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<>4

<div
<div java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
<:<=">/de>/>

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<h5>4.7-2 ChooseHashFunction</h5

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