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<H1><A NAME="SECTION03910000000000000000">
Notes</A>
</H1>

<P>
<DL COMPACT>
<DT>1.
<DD>This index<A NAME="21811"></A>  
lists related pairs of real and complex routines together,
in the same style as in Appendix A.
<P>
<DT>2.
<DD>Routines are listed in alphanumeric order
of the real (single precision) routine name (which always begins with S-).
(See subsection&nbsp;<A HREF="node24.html#subsecnaming">2.2.3</A> for details of the LAPACK naming scheme.)

<P>
<DT>3.
<DD>A few complex routines have no real equivalents, and they are listed
first; routines listed in italics (for example, <I>CROT</I>), have real
equivalents in the Level 1 or Level 2 BLAS.

<P>
<DT>4.
<DD>Double precision routines are not listed here;
they have names beginning with D- instead of
S-, or Z- instead of C-. 
The only exceptions to this simple rule are that 
the double precision versions of ICMAX1, SCSUM1 and CSRSCL
are named IZMAX1, DZSUM1 and ZDRSCL.

<P>
<DT>5.
<DD>A few routines in the list have names that are independent of data type: 
ILAENV, LSAME, LSAMEN and XERBLA.

<P>
<DT>6.
<DD>This index gives only a brief description of the purpose of each
routine. For a precise description consult the leading comments in the code,
which have been written in the same style as for the driver and
computational routines.

<P>
</DL>

<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER" COLSPAN=2>Routine</TD>
<TD ALIGN="CENTER" COLSPAN=1>Description</TD>
</TR>
<TR><TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">CLACGV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Conjugates a complex vector.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">CLACRM</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Performs a matrix multiplication <IMG
 WIDTH="97" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img953.png"
 ALT="$C~=~A \ast B$">,
where A is complex, B is
real, and C is complex.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">CLACRT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Performs the transformation 
<!-- MATH
 $\left( \begin{array}{cc} c & s \\-s & c \end{array} \right) \; \left( \begin{array}{c} x \\y \end{array} \right)$
 -->
<IMG
 WIDTH="150" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img954.png"
 ALT="$\left( \begin{array}{cc} c &amp; s \\ -s &amp; c \end{array} \right) \; \left( \begin{array}{c} x \\ y \end{array} \right) $">,
where <B><I>c</I></B>, <B><I>s</I></B>, <B><I>x</I></B>, and <B><I>y</I></B> are complex.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">CLAESY</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix,
and checks that the norm of the matrix of eigenvectors is larger than
a threshold value.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT"><I> CROT</I></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies a plane rotation with real cosine and complex sine
to a pair of complex vectors.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT"><I> CSPMV </I></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the matrix-vector product 
<!-- MATH
 $y = \alpha Ax + \beta y$
 -->
<IMG
 WIDTH="112" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img955.png"
 ALT="$y = \alpha Ax + \beta y$">,
where <IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img49.png"
 ALT="$\alpha$">
and <IMG
 WIDTH="15" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img50.png"
 ALT="$\beta$">
are complex scalars, 
<B><I>x</I></B> and <B><I>y</I></B> are complex vectors and
<B><I>A</I></B> is a complex symmetric matrix in packed storage.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT"><I> CSPR  </I></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Performs the symmetric rank-1 update 
<!-- MATH
 $A = \alpha x x^T  + A$
 -->
<IMG
 WIDTH="117" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img956.png"
 ALT="$A = \alpha x x^T + A$">,
where <IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img49.png"
 ALT="$\alpha$">
is a complex scalar, 
<B><I>x</I></B> is a complex vector and 
<B><I>A</I></B> is a complex symmetric matrix in packed storage.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT"><I> CSROT </I></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies a plane rotation with real cosine and sine
to a pair of complex vectors.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT"><I> CSYMV </I></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the matrix-vector product 
<!-- MATH
 $y = \alpha Ax + \beta y$
 -->
<IMG
 WIDTH="112" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img955.png"
 ALT="$y = \alpha Ax + \beta y$">,
where <IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img49.png"
 ALT="$\alpha$">
and <IMG
 WIDTH="15" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img50.png"
 ALT="$\beta$">
are complex scalars, 
<B><I>x</I></B> and <B><I>y</I></B> are complex vectors and
<B><I>A</I></B> is a complex symmetric matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT"><I> CSYR </I></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Performs the symmetric rank-1 update 
<!-- MATH
 $A = \alpha x x^T  + A$
 -->
<IMG
 WIDTH="117" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img956.png"
 ALT="$A = \alpha x x^T + A$">,
where <IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img49.png"
 ALT="$\alpha$">
is a complex scalar, 
<B><I>x</I></B> is a complex vector and 
<B><I>A</I></B> is a complex symmetric matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT"><I> ICMAX1</I></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Finds the index of the element whose real part has maximum absolute value 
(similar to the Level 1 BLAS ICAMAX,
but using the absolute value of the real part).</TD>
</TR>
<TR><TD ALIGN="LEFT">ILAENV<A NAME="21834"></A></TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Environmental enquiry function which returns values for tuning
algorithmic performance.</TD>
</TR>
<TR><TD ALIGN="LEFT">LSAME<A NAME="21835"></A></TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Tests two characters for equality regardless of case.</TD>
</TR>
<TR><TD ALIGN="LEFT">LSAMEN<A NAME="21836"></A></TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Tests two character strings for equality regardless of case.</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT"><I> SCSUM1</I></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Forms the 1-norm of a complex vector 
(similar to the Level 1 BLAS SCASUM,
but using the true absolute value).</TD>
</TR>
<TR><TD ALIGN="LEFT">SGBTF2</TD>
<TD ALIGN="LEFT">CGBTF2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes an <B><I>LU</I></B> factorization of a general band matrix,
using partial pivoting with row interchanges 
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SGEBD2</TD>
<TD ALIGN="LEFT">CGEBD2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Reduces a general rectangular matrix to  real bidiagonal form 
by an orthogonal/unitary transformation
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SGEHD2</TD>
<TD ALIGN="LEFT">CGEHD2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Reduces a general matrix to upper Hessenberg form 
by an orthogonal/unitary similarity transformation
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SGELQ2</TD>
<TD ALIGN="LEFT">CGELQ2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes an <B><I>LQ</I></B> factorization of a general rectangular matrix
(unblocked algorithm).</TD>
</TR>
</TABLE>
</DIV>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER" COLSPAN=2>Routine</TD>
<TD ALIGN="CENTER" COLSPAN=1>Description</TD>
</TR>
<TR><TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">SGEQL2</TD>
<TD ALIGN="LEFT">CGEQL2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes a <B><I>QL</I></B> factorization of a general rectangular matrix
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SGEQR2</TD>
<TD ALIGN="LEFT">CGEQR2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes a <B><I>QR</I></B> factorization of a general rectangular matrix
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SGERQ2</TD>
<TD ALIGN="LEFT">CGERQ2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes an <B><I>RQ</I></B> factorization of a general rectangular matrix
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SGESC2</TD>
<TD ALIGN="LEFT">CGESC2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves a system of linear equations 
<!-- MATH
 $A \ast X = scale \ast RHS$
 -->
<IMG
 WIDTH="169" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img957.png"
 ALT="$A \ast X = scale \ast RHS$">
using
the <B><I>LU</I></B> factorization with complete pivoting computed by xGETC2.</TD>
</TR>
<TR><TD ALIGN="LEFT">SGETC2</TD>
<TD ALIGN="LEFT">CGETC2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes an <B><I>LU</I></B> factorization with complete pivoting of the
general <B><I>n</I></B>-by-<B><I>n</I></B> matrix A</TD>
</TR>
<TR><TD ALIGN="LEFT">SGETF2</TD>
<TD ALIGN="LEFT">CGETF2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes an <B><I>LU</I></B> factorization of a general matrix,
using partial pivoting with row interchanges
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SGTTS2</TD>
<TD ALIGN="LEFT">CGTTS2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves one of the systems of equations <IMG
 WIDTH="87" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img958.png"
 ALT="$A \ast X = B$">
or  
<!-- MATH
 $A^H \ast X = B$
 -->
<IMG
 WIDTH="100" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img959.png"
 ALT="$A^H \ast X = B$">,
with a tridiagonal matrix A using the <B><I>LU</I></B> factorization computed
by SGTTRF/CGTTRF.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLABAD</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the square root of the underflow and overflow thresholds
if the exponent-range is very large.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLABRD</TD>
<TD ALIGN="LEFT">CLABRD</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Reduces the first <B><I>nb</I></B> rows and columns of a general rectangular matrix <B><I>A</I></B>
to real bidiagonal form by an orthogonal/unitary transformation,
and returns auxiliary matrices 
which are needed to apply the transformation to the unreduced part of <B><I>A</I></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLACON</TD>
<TD ALIGN="LEFT">CLACON</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Estimates the 1-norm of a square matrix,
using reverse communication for evaluating matrix-vector products.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLACPY</TD>
<TD ALIGN="LEFT">CLACPY</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Copies all or part of one two-dimensional array to another.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLADIV</TD>
<TD ALIGN="LEFT">CLADIV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Performs complex division in real arithmetic, 
avoiding unnecessary overflow.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAE2</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the eigenvalues of a 2-by-2 symmetric matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAEBZ</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the number of eigenvalues of a real symmetric tridiagonal matrix 
which are less than or equal to a given value,
and performs other tasks required by the routine SSTEBZ<A NAME="21850"></A>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAED0</TD>
<TD ALIGN="LEFT">CLAED0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by xSTEDC.
Computes all eigenvalues and corresponding eigenvectors of an
unreduced symmetric tridiagonal matrix using the divide and conquer
method.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAED1</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SSTEDC.
Computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix.
Used when the original matrix is tridiagonal.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAED2</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SSTEDC.
Merges eigenvalues and deflates secular equation.
Used when the original matrix is tridiagonal.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAED3</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SSTEDC.
Finds the roots of the secular equation and updates the eigenvectors.
Used when the original matrix is tridiagonal.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAED4</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SSTEDC.
Finds a single root of the secular equation.</TD>
</TR>
</TABLE>
</DIV>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER" COLSPAN=2>Routine</TD>
<TD ALIGN="CENTER" COLSPAN=1>Description</TD>
</TR>
<TR><TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>complex</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAED5</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SSTEDC.
Solves the 2-by-2 secular equation.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAED6</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SSTEDC.
Computes one Newton step in solution of secular equation.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAED7</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAED7</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SSTEDC.
Computes the updated eigensystem of a diagonal
matrix after modification by a rank-one symmetric matrix.
Used when the original matrix is dense.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAED8</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAED8</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by xSTEDC.
Merges eigenvalues and deflates secular equation.
Used when the original matrix is dense.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAED9</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SSTEDC.
Finds the roots of the secular equation and updates the eigenvectors.
Used when the original matrix is dense.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAEDA</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SSTEDC.
Computes the Z vector determining the rank-one modification of the
diagonal matrix.
Used when the original matrix is dense.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAEIN</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAEIN</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes a specified right or left eigenvector of an upper Hessenberg matrix
by inverse iteration.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAEV2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAEV2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian 
matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAEXC</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Swaps adjacent diagonal blocks of a real upper quasi-triangular matrix 
in Schur canonical form,
by an orthogonal similarity transformation.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAG2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the eigenvalues of a 2-by-2 generalized eigenvalue problem
<IMG
 WIDTH="94" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img960.png"
 ALT="$A~-~w\ast B$">,
with scaling as necessary to avoid over-/underflow.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAGS2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes 2-by-2 orthogonal matrices <B><I>U</I></B>, <B><I>V</I></B>, and <B><I>Q</I></B>, and applies
them to matrices <B><I>A</I></B> and <B><I>B</I></B> such that the rows of the transformed
<B><I>A</I></B> and <B><I>B</I></B> are parallel.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAGTF</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes an <B><I>LU</I></B> factorization of a matrix 
<!-- MATH
 $(T - \lambda I)$
 -->
<IMG
 WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img961.png"
 ALT="$(T - \lambda I)$">,
where <B><I>T</I></B> is a general tridiagonal matrix, and <IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img23.png"
 ALT="$\lambda$">
a scalar,
using partial pivoting with row interchanges.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAGTM</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAGTM</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Performs a matrix-matrix product of the form 
<!-- MATH
 $C = \alpha A B + \beta C$
 -->
<IMG
 WIDTH="126" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img962.png"
 ALT="$C = \alpha A B + \beta C$">,
where <B><I>A</I></B> is a tridiagonal matrix, 
<B><I>B</I></B> and <B><I>C</I></B> are rectangular matrices, 
and <IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img49.png"
 ALT="$\alpha$">
and <IMG
 WIDTH="15" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img50.png"
 ALT="$\beta$">
are scalars, which may be 0, 1, or <B>-1</B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAGTS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves the system of equations 

<!-- MATH
 $(T - \lambda I) x = y$
 -->
<IMG
 WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img963.png"
 ALT="$(T - \lambda I) x = y$">
or 
<!-- MATH
 $(T - \lambda I)^T x = y$
 -->
<IMG
 WIDTH="124" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img964.png"
 ALT="$(T - \lambda I)^T x = y$">,
where <B><I>T</I></B> is a general tridiagonal matrix and <IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img23.png"
 ALT="$\lambda$">
a scalar,
using the <B><I>LU</I></B> factorization computed by SLAGTF.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAGV2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the Generalized Schur factorization of a real 2-by-2
matrix pencil <B>(<I>A</I>,<I>B</I>)</B> where <B><I>B</I></B> is upper triangular.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAHQR</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAHQR</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, 
using the double-shift/single-shift <B><I>QR</I></B> algorithm.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAHRD</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAHRD</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Reduces the first <B><I>nb</I></B> columns of a general rectangular matrix <B><I>A</I></B>
so that elements below the <B><I>k</I><SUP><I>th</I></SUP></B> subdiagonal are zero,
by an orthogonal/unitary transformation,
and returns auxiliary matrices 
which are needed to apply the transformation to the unreduced part of <B><I>A</I></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAIC1</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAIC1</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies one step of incremental condition estimation.</TD>
</TR>
</TABLE>
</DIV>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER" COLSPAN=2>Routine</TD>
<TD ALIGN="CENTER" COLSPAN=1>Description</TD>
</TR>
<TR><TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>complex</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">SLALN2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves a 1-by-1 or 2-by-2 system of equations of the form  

<!-- MATH
 $(\gamma A - \lambda D ) x = \sigma b$
 -->
<IMG
 WIDTH="139" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img965.png"
 ALT="$(\gamma A - \lambda D ) x = \sigma b$">
or 
<!-- MATH
 $(\gamma A^T - \lambda D) x = \sigma b$
 -->
<IMG
 WIDTH="149" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img966.png"
 ALT="$(\gamma A^T - \lambda D) x = \sigma b$">,
where <B><I>D</I></B> is a diagonal matrix,
<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img23.png"
 ALT="$\lambda$">,
<B><I>b</I></B> and <B><I>x</I></B> may be complex, 
and <IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img967.png"
 ALT="$\sigma$">
is a scale factor set to avoid overflow.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLALS0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLALS0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by xGELSD.
Applies back the multiplying factors of either the left or the right
singular vector matrix of a diagonal matrix appended by a row to
the right hand side matrix <B><I>B</I></B> in solving the least squares problem
using the divide and conquer SVD approach.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLALSA</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLALSA</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by xGELSD.
An intermediate step in solving the least squares problem
by computing the SVD of the coefficient matrix in compact form.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLALSD</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLALSD</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by xGELSD.
Uses the singular value decomposition of <B><I>A</I></B> to solve the least
squares problem of finding <B><I>X</I></B> to minimize the Euclidean norm of each
column of <IMG
 WIDTH="85" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img968.png"
 ALT="$A \ast X-B$">.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAMCH<A NAME="21878"></A><A NAME="21879"></A></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Determines machine parameters for floating-point arithmetic.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAMRG</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Creates a permutation list which will merge the entries of
two independently sorted sets into a single set which is sorted
in ascending order.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANGB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANGB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a general band matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANGE</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANGE</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a general rectangular matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANGT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANGT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element, 
of a general tridiagonal matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANHS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANHS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of an upper Hessenberg matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANSB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANSB CLANHB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a real symmetric/complex symmetric/complex Hermitian band matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANSP</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANSP CLANHP</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a real symmetric/complex symmetric/complex Hermitian matrix
in packed storage.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANST</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANHT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a symmetric/Hermitian tridiagonal matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANSY</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANSY CLANHE</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a real symmetric/complex symmetric/complex Hermitian matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANTB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANTB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a triangular band matrix.</TD>
</TR>
</TABLE>
</DIV>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER" COLSPAN=2>Routine</TD>
<TD ALIGN="CENTER" COLSPAN=1>Description</TD>
</TR>
<TR><TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>complex</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANTP</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANTP</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a triangular matrix in packed storage.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANTR</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLANTR</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns the value of the 1-norm, Frobenius norm, infinity-norm,  
or the largest absolute value of any element,
of a triangular matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLANV2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the Schur factorization of a real 2-by-2 nonsymmetric matrix 
in Schur canonical form.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAPLL</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAPLL</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Measures the linear dependence of two vectors <B><I>X</I></B> and <B><I>Y</I></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAPMT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAPMT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Performs a forward or backward permutation of the columns of a matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAPY2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns 
<!-- MATH
 $\sqrt{x^2 + y^2}$
 -->
<IMG
 WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img969.png"
 ALT="$ \sqrt{x^2 + y^2}$">,
avoiding unnecessary overflow or harmful underflow.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAPY3</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns 
<!-- MATH
 $\sqrt{x^2 + y^2 + z^2}$
 -->
<IMG
 WIDTH="114" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img970.png"
 ALT="$ \sqrt{x^2 + y^2 + z^2}$">,
avoiding unnecessary overflow or harmful underflow.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAQGB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAQGB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Scales a general band matrix,
using row and column scaling factors computed by SGBEQU<A NAME="21895"></A>/CGBEQU<A NAME="21896"></A>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAQGE</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAQGE</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Scales a general rectangular matrix, 
using row and column scaling factors computed by SGEEQU<A NAME="21897"></A>/CGEEQU<A NAME="21898"></A>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAQP2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAQP2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes a <B><I>QR</I></B> factorization with column pivoting of the block

<!-- MATH
 $A(OFFSET+1:M,1:N)$
 -->
<B><I>A</I>(<I>OFFSET</I>+1:<I>M</I>,1:<I>N</I>)</B>.  The block 
<!-- MATH
 $A(1:OFFSET,1:N)$
 -->
<B><I>A</I>(1:<I>OFFSET</I>,1:<I>N</I>)</B> is accordingly pivoted,
but not factorized.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAQPS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAQPS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes a step of <B><I>QR</I></B> factorization with column pivoting
of a real <B><I>M</I></B>-by-<B><I>N</I></B> matrix <B><I>A</I></B> by using Level 3 Blas.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAQSB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAQSB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Scales a symmetric/Hermitian band matrix,
using scaling factors computed by SPBEQU<A NAME="21899"></A>/CPBEQU<A NAME="21900"></A>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAQSP</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAQSP</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Scales a symmetric/Hermitian matrix in packed storage,
using scaling factors computed by SPPEQU<A NAME="21901"></A>/CPPEQU<A NAME="21902"></A>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAQSY</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAQSY</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Scales a symmetric/Hermitian matrix,
using scaling factors computed by SPOEQU<A NAME="21903"></A>/CPOEQU<A NAME="21904"></A>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAQTR</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves a real quasi-triangular system of equations,
or a complex quasi-triangular system of special form,
in real arithmetic.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAR1V</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAR1V</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the (scaled) r<B><SUP><I>th</I></SUP></B> column of the inverse of
the sumbmatrix in rows B1 through BN of the tridiagonal matrix

<!-- MATH
 $L D L^T - \sigma I$
 -->
<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img971.png"
 ALT="$L D L^T - \sigma I$">.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAR2V</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAR2V</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies a vector of plane rotations with real cosines and real/complex sines
from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARF</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARF</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies an elementary reflector to a general rectangular matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARFB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARFB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies a block reflector or its transpose/conjugate-transpose 
to a general rectangular matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARFG</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARFG</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Generates an elementary reflector (Householder matrix).</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARFT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARFT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Forms the triangular factor <B><I>T</I></B> of a block reflector 
<!-- MATH
 $H = I - V T V^H$
 -->
<B><I>H</I> = <I>I</I> - <I>V T V</I><SUP><I>H</I></SUP></B>.</TD>
</TR>
</TABLE>
</DIV>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER" COLSPAN=2>Routine</TD>
<TD ALIGN="CENTER" COLSPAN=1>Description</TD>
</TR>
<TR><TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>complex</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARFX</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARFX</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies an elementary reflector to a general rectangular matrix,
with loop unrolling when the reflector has order <IMG
 WIDTH="40" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img972.png"
 ALT="$\leq 10$">.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARGV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARGV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Generates a vector of plane rotations with real cosines and real/complex 
sines.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARNV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARNV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns a vector of random numbers from a uniform or normal distribution.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARRB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Given the relatively robust representation(RRR) <B><I>L D L</I><SUP><I>T</I></SUP></B>, SLARRB
does ``limited'' bisection to locate the eigenvalues of <B><I>L D L</I><SUP><I>T</I></SUP></B>,
W(IFIRST) through W(ILAST), to more accuracy.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARRE</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Given the tridiagonal matrix <B><I>T</I></B>, SLARRE sets ``small'' off-diagonal
elements to zero, and for each unreduced block <B><I>T</I><SUB><I>i</I></SUB></B>, it finds
the numbers <IMG
 WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img36.png"
 ALT="$\sigma _ i $">,
the base 
<!-- MATH
 $T_i - \sigma_i I~=~L_i D_i L_i^T$
 -->
<IMG
 WIDTH="160" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img973.png"
 ALT="$T_i - \sigma_i I~=~L_i D_i L_i^T$">
representations and the eigenvalues of each <B><I>L</I><SUB><I>i</I></SUB> <I>D</I><SUB><I>i</I></SUB> <I>L</I><SUB><I>i</I></SUB><SUP><I>T</I></SUP></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARRF</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Finds a new relatively robust representation

<!-- MATH
 $L D L^T - \Sigma I~=~L(+) D(+) L(+)^T$
 -->
<IMG
 WIDTH="262" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img974.png"
 ALT="$L D L^T - \Sigma I~=~L(+) D(+) L(+)^T$">
such that at least one of the
eigenvalues of 
<!-- MATH
 $L(+) D(+) L(+)^T$
 -->
<B><I>L</I>(+) <I>D</I>(+) <I>L</I>(+)<SUP><I>T</I></SUP></B> is relatively isolated.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARRV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARRV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the eigenvectors of the tridiagonal matrix
<B><I>T</I>&nbsp;=&nbsp;L <I>D L</I><SUP><I>T</I></SUP></B> given <B><I>L</I></B>, <B><I>D</I></B> and the eigenvalues of <B><I>L D L</I><SUP><I>T</I></SUP></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARTG</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARTG</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Generates a plane rotation with real cosine and real/complex sine.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARTV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARTV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies a vector of plane rotations with real cosines and real/complex sines
to the elements of a pair of vectors.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARUV</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Returns a vector of <B><I>n</I></B> random real numbers from a uniform (0,1) distribution 
(<IMG
 WIDTH="64" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img975.png"
 ALT="$n \leq 128$">).</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARZ</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARZ</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies an elementary reflector (as returned by xTZRZF) to a general
matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARZB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARZB</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies a block reflector or its transpose/conjugate-transpose to a 
general matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLARZT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLARZT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Forms the triangular factor <B><I>T</I></B> of a block reflector 
<!-- MATH
 $H = I - V T V^H$
 -->
<B><I>H</I> = <I>I</I> - <I>V T V</I><SUP><I>H</I></SUP></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAS2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the singular values of a 2-by-2 triangular matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASCL</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLASCL</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Multiplies a general rectangular matrix by a real scalar defined as

<!-- MATH
 $c_{to}/c_{from}$
 -->
<B><I>c</I><SUB><I>to</I></SUB>/<I>c</I><SUB><I>from</I></SUB></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASD0</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Computes via a divide and conquer method the singular values
of a real upper bidiagonal <B><I>n</I></B>-by-<B><I>m</I></B> matrix with diagonal <B><I>D</I></B> and offdiagonal
<B><I>E</I></B>, where <B><I>M</I> = <I>N</I> + <I>SQRE</I></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASD1</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Computes the SVD of an upper bidiagonal <B><I>N</I></B>-by-<B><I>M</I></B> matrix,
where 
<!-- MATH
 $N = NL + NR + 1$
 -->
<B><I>N</I> = <I>NL</I> + <I>NR</I> + 1</B> and <B><I>M</I> = <I>N</I> + <I>SQRE</I></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASD2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Merges the two sets of singular values together into a
single sorted set, and then it tries to deflate the size of the problem.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASD3</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Finds all the square roots of the roots of the secular
equation, as defined by the values in <B><I>D</I></B> and <B><I>Z</I></B>, and then
updates the singular vectors by matrix multiplication.</TD>
</TR>
</TABLE>
</DIV>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER" COLSPAN=2>Routine</TD>
<TD ALIGN="CENTER" COLSPAN=1>Description</TD>
</TR>
<TR><TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>complex</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASD4</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Computes the square root of the I-th updated
eigenvalue of a positive symmetric rank-one modification to
a positive diagonal matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASD5</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Computes the square root of the I-th eigenvalue
of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASD6</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Computes the SVD of an updated upper bidiagonal matrix
obtained by merging two smaller ones by appending a row.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASD7</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Merges the two sets of singular values together into a
single sorted set, and then it tries to deflate the size of the problem.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASD8</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Finds the square roots of the roots of the secular
equation, and stores, for each element in <B><I>D</I></B>, the distance to its two
nearest poles (elements in <B><I>DSIGMA</I></B>).</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASD9</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Finds the square roots of the roots of the secular
equation, and stores, for each element in <B><I>D</I></B>, the distance to its two
nearest poles (elements in <B><I>DSIGMA</I></B>).</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASDA</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Computes the singular value decomposition (SVD) of a
real upper bidiagonal <B><I>N</I></B>-by-<B><I>M</I></B> matrix with diagonal <B><I>D</I></B> and offdiagonal <B><I>E</I></B>,
where <B><I>M</I> = <I>N</I> + <I>SQRE</I></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASDQ</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Computes the singular value decomposition (SVD) of a real
(upper or lower) bidiagonal matrix with diagonal <B><I>D</I></B> and
offdiagonal <B><I>E</I></B>, accumulating the transformations if desired.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASDT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSDC.  Creates a tree of subproblems for bidiagonal divide and
conquer.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASET</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLASET</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Initializes the off-diagonal elements of a matrix to <IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img49.png"
 ALT="$\alpha$">
and the diagonal elements to <IMG
 WIDTH="15" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img50.png"
 ALT="$\beta$">.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASQ1</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSQR.
Computes the singular values of a real <B><I>n</I></B>-by-<B><I>n</I></B> bidiagonal
matrix with diagonal D and offdiagonal E.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASQ2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSQR and SSTEGR.
Computes all the eigenvalues of the symmetric positive
definite tridiagonal matrix associated with the qd array Z to high
relative accuracy.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASQ3</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSQR.
Checks for deflation, computes a shift (TAU) and calls dqds.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASQ4</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSQR.
Computes an approximation TAU to the smallest eigenvalue
using values of d from the previous transform.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASQ5</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSQR and SSTEGR.
Computes one dqds transform in ping-pong form.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASQ6</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Used by SBDSQR and SSTEGR.
computes one dqds transform in ping-pong form.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASR</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLASR</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Applies a sequence of plane rotations to a general rectangular
matrix.</TD>
</TR>
</TABLE>
</DIV>

<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER" COLSPAN=2>Routine</TD>
<TD ALIGN="CENTER" COLSPAN=1>Description</TD>
</TR>
<TR><TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>complex</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASRT</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Sorts numbers in increasing or decreasing order using Quick Sort,
reverting to Insertion sort on arrays of size <IMG
 WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img976.png"
 ALT="$\leq$">
20.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASSQ</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLASSQ</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Updates a sum of squares represented in scaled form.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASV2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the singular value decomposition of a 2-by-2 triangular matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASWP</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLASWP</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Performs a sequence of row interchanges on a general rectangular matrix.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASY2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves the Sylvester matrix equation 
<!-- MATH
 $A X \pm X B = \sigma C$
 -->
<IMG
 WIDTH="132" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img977.png"
 ALT="$A X \pm X B = \sigma C$">
where <B><I>A</I></B> and <B><I>B</I></B> are of order 1 or 2, 
and may be transposed,
and <IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img967.png"
 ALT="$\sigma$">
is a scale factor.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLASYF</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLASYF CLAHEF</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes a partial factorization of 
a real symmetric/complex symmetric/complex Hermitian indefinite matrix,
using the diagonal pivoting<A NAME="21947"></A> method.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLATBS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLATBS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves a triangular banded system of equations

<!-- MATH
 $A  x = \sigma b$
 -->
<IMG
 WIDTH="69" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img978.png"
 ALT="$A x = \sigma b$">,

<!-- MATH
 $A^T  x = \sigma b$
 -->
<IMG
 WIDTH="79" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img979.png"
 ALT="$A^T x = \sigma b$">,
or  
<!-- MATH
 $A^H  x = \sigma b$
 -->
<IMG
 WIDTH="82" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img980.png"
 ALT="$A^H x = \sigma b$">,
where <IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img967.png"
 ALT="$\sigma$">
is a scale factor set to prevent overflow.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLATDF</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLATDF</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Uses the <B><I>LU</I></B> factorization of the <B><I>n</I></B>-by-<B><I>n</I></B> matrix computed by
SGETC2 and computes a contribution to the reciprocal Dif-estimate.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLATPS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLATPS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves a triangular system of equations

<!-- MATH
 $A  x = \sigma b$
 -->
<IMG
 WIDTH="69" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img978.png"
 ALT="$A x = \sigma b$">,

<!-- MATH
 $A^T  x = \sigma b$
 -->
<IMG
 WIDTH="79" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img979.png"
 ALT="$A^T x = \sigma b$">,
or  
<!-- MATH
 $A^H  x = \sigma b$
 -->
<IMG
 WIDTH="82" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img980.png"
 ALT="$A^H x = \sigma b$">,
where <B><I>A</I></B> is held in packed storage,
and <IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img967.png"
 ALT="$\sigma$">
is a scale factor set to prevent overflow.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLATRD</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLATRD</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Reduces the first <B><I>nb</I></B> rows and columns of a symmetric/Hermitian matrix <B><I>A</I></B>
to real tridiagonal form by an orthogonal/unitary similarity transformation,
and returns auxiliary matrices
which are needed to apply the transformation to the unreduced part of <B><I>A</I></B>.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLATRS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLATRS</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves a triangular system of equations

<!-- MATH
 $A  x = \sigma b$
 -->
<IMG
 WIDTH="69" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img978.png"
 ALT="$A x = \sigma b$">,

<!-- MATH
 $A^T  x = \sigma b$
 -->
<IMG
 WIDTH="79" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img979.png"
 ALT="$A^T x = \sigma b$">,
or  
<!-- MATH
 $A^H  x = \sigma b$
 -->
<IMG
 WIDTH="82" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img980.png"
 ALT="$A^H x = \sigma b$">,
where <IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img967.png"
 ALT="$\sigma$">
is a scale factor set to prevent overflow.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLATRZ</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLATRZ</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Factors an upper trapezoidal matrix by means of orthogonal/unitary
transformations.</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAUU2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAUU2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the product <B><I>U  U</I><SUP><I>H</I></SUP></B> or <B><I>L</I><SUP><I>H</I></SUP> <I>L</I></B>, 
where <B><I>U</I></B> and <B><I>L</I></B> are upper or lower triangular matrices
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SLAUUM</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CLAUUM</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the product <B><I>U  U</I><SUP><I>H</I></SUP></B> or <B><I>L</I><SUP><I>H</I></SUP> <I>L</I></B>, 
where <B><I>U</I></B> and <B><I>L</I></B> are upper or lower triangular matrices.</TD>
</TR>
<TR><TD ALIGN="LEFT">SORG2L</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CUNG2L</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Generates all or part of the orthogonal/unitary matrix <B><I>Q</I></B> 
from a <B><I>QL</I></B> factorization determined by SGEQLF<A NAME="21948"></A>/CGEQLF
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SORG2R</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CUNG2R</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Generates all or part of the orthogonal/unitary matrix <B><I>Q</I></B> 
from a <B><I>QR</I></B> factorization determined by SGEQRF<A NAME="21949"></A>/CGEQRF
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SORGL2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CUNGL2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Generates all or part of the orthogonal/unitary matrix <B><I>Q</I></B> 
from an <B><I>LQ</I></B> factorization determined by SGELQF<A NAME="21950"></A>/CGELQF
(unblocked algorithm).</TD>
</TR>
</TABLE>
</DIV>

<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER" COLSPAN=2>Routine</TD>
<TD ALIGN="CENTER" COLSPAN=1>Description</TD>
</TR>
<TR><TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>complex</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">SORGR2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CUNGR2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Generates all or part of the orthogonal/unitary matrix <B><I>Q</I></B> 
from an <B><I>RQ</I></B> factorization determined by SGERQF<A NAME="21964"></A>/CGERQF
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SORM2L</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CUNM2L</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Multiplies a general matrix by the orthogonal/unitary matrix 
from a <B><I>QL</I></B> factorization determined by SGEQLF<A NAME="21965"></A>/CGEQLF
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SORM2R</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CUNM2R</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Multiplies a general matrix by the orthogonal/unitary matrix 
from a <B><I>QR</I></B> factorization determined by SGEQRF<A NAME="21966"></A>/CGEQRF
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SORML2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CUNML2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Multiplies a general matrix by the orthogonal/unitary matrix 
from an <B><I>LQ</I></B> factorization determined by SGELQF<A NAME="21967"></A>/CGELQF
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SORMR2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CUNMR2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Multiplies a general matrix by the orthogonal/unitary matrix 
from an <B><I>RQ</I></B> factorization determined by SGERQF<A NAME="21968"></A>/CGERQF
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SORMR3</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CUNMR3</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Multiplies a general matrix by the orthogonal/unitary matrix 
from an <B><I>RZ</I></B> factorization determined by STZRZF<A NAME="21969"></A>/CTZRZF
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SPBTF2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CPBTF2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the Cholesky factorization of 
a symmetric/Hermitian positive definite band matrix
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SPOTF2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CPOTF2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the Cholesky factorization of 
a symmetric/Hermitian positive definite matrix
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SPTTS2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CPTTS2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves a tridiagonal system of the form
<IMG
 WIDTH="87" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img958.png"
 ALT="$A \ast X = B$">
using the 
<!-- MATH
 $L \ast D \ast L^H$
 -->
<IMG
 WIDTH="89" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img981.png"
 ALT="$L \ast D \ast L^H$">
factorization of A computed by
SPTTRF/CPTTRF.</TD>
</TR>
<TR><TD ALIGN="LEFT">SRSCL</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CSRSCL</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Multiplies a vector by the reciprocal of a real scalar.</TD>
</TR>
<TR><TD ALIGN="LEFT">SSYGS2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CHEGS2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Reduces a symmetric/Hermitian definite generalized eigenproblem 
<IMG
 WIDTH="85" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img176.png"
 ALT="$Ax = \lambda Bx$">,
<IMG
 WIDTH="85" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img951.png"
 ALT="$ABx=\lambda x$">,
or <IMG
 WIDTH="85" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img952.png"
 ALT="$BAx=\lambda x$">,
to standard form, where <B><I>B</I></B> has been factorized by SPOTRF<A NAME="21970"></A>/CPOTRF
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SSYTD2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CHETD2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Reduces a symmetric/Hermitian matrix to 
real symmetric tridiagonal form 
by an orthogonal/unitary similarity transformation
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">SSYTF2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CSYTF2 CHETF2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the factorization of 
a real symmetric/complex symmetric/complex Hermitian indefinite matrix,
using the diagonal pivoting<A NAME="21971"></A> method
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">STGEX2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CTGEX2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Swaps adjacent diagonal blocks <B>(<I>A</I>11, <I>B</I>11)</B> and <B>(<I>A</I>22, <I>B</I>22)</B>
of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
<B>(<I>A</I>, <I>B</I>)</B> by an orthogonal/unitary equivalence transformation.</TD>
</TR>
<TR><TD ALIGN="LEFT">STGSY2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CTGSY2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Solves the generalized Sylvester equation (unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">STRTI2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>CTRTI2</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Computes the inverse of a triangular matrix
(unblocked algorithm).</TD>
</TR>
<TR><TD ALIGN="LEFT">XERBLA<A NAME="21972"></A></TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=54>&nbsp;</TD>
<TD ALIGN="LEFT" VALIGN="TOP" WIDTH=324>Error handling routine called by LAPACK routines 
if an input parameter has an invalid value.</TD>
</TR>
</TABLE>
</DIV>

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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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