1SLABRD(1)           LAPACK auxiliary routine (version 3.1)           SLABRD(1)
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NAME

6       SLABRD  - the first NB rows and columns of a real general m by n matrix
7       A to upper or lower bidiagonal form by an orthogonal transformation  Q'
8       * A * P, and returns the matrices X and Y which are needed to apply the
9       transformation to the unreduced part of A
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SYNOPSIS

12       SUBROUTINE SLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )
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14           INTEGER        LDA, LDX, LDY, M, N, NB
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16           REAL           A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ(  *  ),
17                          X( LDX, * ), Y( LDY, * )
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PURPOSE

20       SLABRD  reduces  the first NB rows and columns of a real general m by n
21       matrix A to upper or lower bidiagonal form by an orthogonal transforma‐
22       tion  Q'  * A * P, and returns the matrices X and Y which are needed to
23       apply the transformation to the unreduced part of A.
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25       If m >= n, A is reduced to upper bidiagonal form; if m <  n,  to  lower
26       bidiagonal form.
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28       This is an auxiliary routine called by SGEBRD
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ARGUMENTS

32       M       (input) INTEGER
33               The number of rows in the matrix A.
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35       N       (input) INTEGER
36               The number of columns in the matrix A.
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38       NB      (input) INTEGER
39               The number of leading rows and columns of A to be reduced.
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41       A       (input/output) REAL array, dimension (LDA,N)
42               On  entry,  the  m by n general matrix to be reduced.  On exit,
43               the first NB rows and columns of the  matrix  are  overwritten;
44               the rest of the array is unchanged.  If m >= n, elements on and
45               below the diagonal in the first  NB  columns,  with  the  array
46               TAUQ, represent the orthogonal matrix Q as a product of elemen‐
47               tary reflectors; and elements above the diagonal in  the  first
48               NB rows, with the array TAUP, represent the orthogonal matrix P
49               as a product of elementary reflectors.   If  m  <  n,  elements
50               below  the  diagonal  in  the  first NB columns, with the array
51               TAUQ, represent the orthogonal matrix Q as a product of elemen‐
52               tary  reflectors, and elements on and above the diagonal in the
53               first NB rows, with the array TAUP,  represent  the  orthogonal
54               matrix  P  as  a product of elementary reflectors.  See Further
55               Details.  LDA     (input) INTEGER The leading dimension of  the
56               array A.  LDA >= max(1,M).
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58       D       (output) REAL array, dimension (NB)
59               The  diagonal  elements of the first NB rows and columns of the
60               reduced matrix.  D(i) = A(i,i).
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62       E       (output) REAL array, dimension (NB)
63               The off-diagonal elements of the first NB rows and  columns  of
64               the reduced matrix.
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66       TAUQ    (output) REAL array dimension (NB)
67               The scalar factors of the elementary reflectors which represent
68               the orthogonal matrix Q. See Further Details.  TAUP    (output)
69               REAL array, dimension (NB) The scalar factors of the elementary
70               reflectors which represent the orthogonal matrix P. See Further
71               Details.   X        (output) REAL array, dimension (LDX,NB) The
72               m-by-nb matrix X required to update the unreduced part of A.
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74       LDX     (input) INTEGER
75               The leading dimension of the array X. LDX >= M.
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77       Y       (output) REAL array, dimension (LDY,NB)
78               The n-by-nb matrix Y required to update the unreduced  part  of
79               A.
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81       LDY     (input) INTEGER
82               The leading dimension of the array Y. LDY >= N.
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FURTHER DETAILS

85       The  matrices Q and P are represented as products of elementary reflec‐
86       tors:
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88          Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
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90       Each H(i) and G(i) has the form:
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92          H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'
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94       where tauq and taup are real scalars, and v and u are real vectors.
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96       If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m)  is  stored  on  exit  in
97       A(i:m,i);  u(1:i)  =  0,  u(i+1) = 1, and u(i+1:n) is stored on exit in
98       A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
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100       If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is  stored  on  exit  in
101       A(i+2:m,i);  u(1:i-1)  =  0,  u(i) = 1, and u(i:n) is stored on exit in
102       A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
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104       The elements of the vectors v and u together form the m-by-nb matrix  V
105       and  the nb-by-n matrix U' which are needed, with X and Y, to apply the
106       transformation to the unreduced part  of  the  matrix,  using  a  block
107       update of the form:  A := A - V*Y' - X*U'.
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109       The  contents  of  A  on exit are illustrated by the following examples
110       with nb = 2:
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112       m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
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114         (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
115         (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
116         (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
117         (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
118         (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
119         (  v1  v2  a   a   a  )
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121       where a denotes an element of the original matrix which  is  unchanged,
122       vi denotes an element of the vector defining H(i), and ui an element of
123       the vector defining G(i).
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128 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       SLABRD(1)