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85 lines
6.3 KiB
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<!-- Common Lisp HyperSpec (TM), version 7.0 generated by Kent M. Pitman on Mon, 11-Apr-2005 2:31am EDT -->
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<HEAD>
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<TITLE>CLHS: Function SQRT, ISQRT</TITLE>
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<H1><A REV=MADE HREF="http://www.lispworks.com/"><IMG WIDTH=80 HEIGHT=65 ALT="[LISPWORKS]" SRC="../Graphics/LWSmall.gif" ALIGN=Bottom></A><A REL=TOP HREF="../Front/index.htm"><IMG WIDTH=237 HEIGHT=65 ALT="[Common Lisp HyperSpec (TM)]" SRC="../Graphics/CLHS_Sm.gif" ALIGN=Bottom></A> <A REL=PREV HREF="f_signum.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Previous]" SRC="../Graphics/Prev.gif" ALIGN=Bottom></A><A REL=UP HREF="c_number.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Up]" SRC="../Graphics/Up.gif" ALIGN=Bottom></A><A REL=NEXT HREF="t_rnd_st.htm"><IMG WIDTH=40 HEIGHT=40 ALT="[Next]" SRC="../Graphics/Next.gif" ALIGN=Bottom></A></H1>
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<HR>
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<A NAME="sqrt"><A NAME="isqrt"><I>Function</I> <B>SQRT, ISQRT</B></A></A> <P>
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<P><B>Syntax:</B><P>
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<P>
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<B>sqrt</B> <I>number</I> => <I>root</I><P>
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<B>isqrt</B> <I>natural</I> => <I>natural-root</I><P>
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<P>
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<P><B>Arguments and Values:</B><P>
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<P>
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<I>number</I>, <I>root</I>---a <A REL=DEFINITION HREF="26_glo_n.htm#number"><I>number</I></A>. <P>
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<I>natural</I>, <I>natural-root</I>---a non-negative <A REL=DEFINITION HREF="26_glo_i.htm#integer"><I>integer</I></A>. <P>
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<P><B>Description:</B><P>
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<P>
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<A REL=DEFINITION HREF="#sqrt"><B>sqrt</B></A> and <A REL=DEFINITION HREF="#isqrt"><B>isqrt</B></A> compute square roots. <P>
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<A REL=DEFINITION HREF="#sqrt"><B>sqrt</B></A> returns the <A REL=DEFINITION HREF="26_glo_p.htm#principal"><I>principal</I></A> square root of <I>number</I>. If the <I>number</I> is not a <A REL=DEFINITION HREF="26_glo_c.htm#complex"><I>complex</I></A> but is negative, then the result is a <A REL=DEFINITION HREF="26_glo_c.htm#complex"><I>complex</I></A>. <P>
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<A REL=DEFINITION HREF="#isqrt"><B>isqrt</B></A> returns the greatest <A REL=DEFINITION HREF="26_glo_i.htm#integer"><I>integer</I></A> less than or equal to the exact positive square root of <I>natural</I>. <P>
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If <I>number</I> is a positive <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A>, it is <A REL=DEFINITION HREF="26_glo_i.htm#implementation-dependent"><I>implementation-dependent</I></A> whether <I>root</I> is a <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A> or a <A REL=DEFINITION HREF="26_glo_f.htm#float"><I>float</I></A>. If <I>number</I> is a negative <A REL=DEFINITION HREF="26_glo_r.htm#rational"><I>rational</I></A>, it is <A REL=DEFINITION HREF="26_glo_i.htm#implementation-dependent"><I>implementation-dependent</I></A> whether <I>root</I> is a <A REL=DEFINITION HREF="26_glo_c.htm#complex_rational"><I>complex rational</I></A> or a <A REL=DEFINITION HREF="26_glo_c.htm#complex_float"><I>complex float</I></A>. <P>
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The mathematical definition of complex square root (whether or not minus zero is supported) follows: <P>
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<TT>(sqrt </TT><I>x</I><TT>) = (exp (/ (log </TT><I>x</I><TT>) 2))</TT> <P>
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<P>
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The branch cut for square root lies along the negative real axis, continuous with quadrant II. The range consists of the right half-plane, including the non-negative imaginary axis and excluding the negative imaginary axis. <P>
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<P><B>Examples:</B><P>
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<P>
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<PRE>
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(sqrt 9.0) => 3.0
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(sqrt -9.0) => #C(0.0 3.0)
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(isqrt 9) => 3
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(sqrt 12) => 3.4641016
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(isqrt 12) => 3
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(isqrt 300) => 17
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(isqrt 325) => 18
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(sqrt 25)
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=> 5
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OR=> 5.0
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(isqrt 25) => 5
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(sqrt -1) => #C(0.0 1.0)
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(sqrt #c(0 2)) => #C(1.0 1.0)
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</PRE>
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</TT> <P>
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<P><B>Side Effects:</B> None.
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<P>
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<P><B>Affected By:</B> None.
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<P>
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<P><B>Exceptional Situations:</B><P>
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<P>
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The <A REL=DEFINITION HREF="26_glo_f.htm#function"><I>function</I></A> <A REL=DEFINITION HREF="#sqrt"><B>sqrt</B></A> should signal <A REL=DEFINITION HREF="e_tp_err.htm#type-error"><B>type-error</B></A> if its argument is not a <A REL=DEFINITION HREF="26_glo_n.htm#number"><I>number</I></A>. <P>
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The <A REL=DEFINITION HREF="26_glo_f.htm#function"><I>function</I></A> <A REL=DEFINITION HREF="#isqrt"><B>isqrt</B></A> should signal <A REL=DEFINITION HREF="e_tp_err.htm#type-error"><B>type-error</B></A> if its argument is not a non-negative <A REL=DEFINITION HREF="26_glo_i.htm#integer"><I>integer</I></A>. <P>
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The functions <A REL=DEFINITION HREF="#sqrt"><B>sqrt</B></A> and <A REL=DEFINITION HREF="#isqrt"><B>isqrt</B></A> might signal <A REL=DEFINITION HREF="e_arithm.htm#arithmetic-error"><B>arithmetic-error</B></A>. <P>
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<P><B>See Also:</B><P>
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<P>
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<A REL=DEFINITION HREF="f_exp_e.htm#exp"><B>exp</B></A>, <A REL=DEFINITION HREF="f_log.htm#log"><B>log</B></A>, <A REL=CHILD HREF="12_acc.htm">Section 12.1.3.3 (Rule of Float Substitutability)</A> <P>
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<P><B>Notes:</B><P>
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<P>
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<PRE>
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(isqrt x) == (values (floor (sqrt x)))
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</PRE>
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</TT> but it is potentially more efficient. <P>
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<P><HR>The following <A REL=META HREF="../Front/X3J13Iss.htm">X3J13 cleanup issue</A>, <I>not part of the specification</I>, applies to this section:<P><UL><LI> <A REL=CHILD HREF="../Issues/iss192.htm">IEEE-ATAN-BRANCH-CUT:SPLIT</A><P></UL><HR>
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<A REL=COPYRIGHT HREF="../Front/Help.htm#Legal"><I>Copyright 1996-2005, LispWorks Ltd. All rights reserved.</I></A><P>
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