~ubuntu-branches/ubuntu/raring/scilab/raring-proposed

produces an upper triangular matrix <literal>R</literal> of the same dimension as <literal>X</literal> and an orthogonal (unitary in the complex case) matrix <literal>Q</literal> so that <literal>X = Q*R</literal>. <literal>[Q,R] = qr(X,"e")</literal> produces an "economy size": If <literal>X</literal> is m-by-n with m > n, then only the first n columns of <literal>Q</literal> are computed as well as the first n rows of <literal>R</literal>.

</para>

<para>

From <literal>Q*R = X</literal> , it follows that

the kth column of the matrix <literal>X</literal>, is expressed as a linear combination

of the k first columns of <literal>Q</literal> (with coefficients <literal> R(1,k), ..., R(k,k) </literal>). The k first columns of <literal>Q</literal> make an orthogonal basis

of the subspace spanned by the k first comumns of <literal>X</literal>. If column <literal>k</literal>

of <literal>X</literal> (i.e. <literal>X(:,k)</literal> ) is a linear combination of the first

<literal>p</literal> columns of <literal>X</literal>, then the entries <literal>R(p+1,k), ..., R(k,k)</literal>

are zero. It this situation, <literal>R</literal> is upper trapezoidal. If <literal>X</literal> has

rank <literal>rk</literal>, rows <literal>R(rk+1,:), R(rk+2,:), ...</literal> are zeros.

</para>

</listitem>

</varlistentry>

<para>

produces a (column) permutation matrix <literal>E</literal>, an upper

triangular <literal>R</literal> with decreasing diagonal elements and an

orthogonal (or unitary) <literal>Q</literal> so that <literal>X*E = Q*R</literal>.

If <literal>rk</literal> is the rank of <literal>X</literal>, the

<literal>rk</literal> first entries along the main diagonal of

100

101

are all different from zero. <literal>[Q,R,E] = qr(X,"e")</literal>

102

produces an "economy size":

103

If <literal>X</literal> is m-by-n with m > n, then only the first n

104

columns of <literal>Q</literal> are computed as well as the first n

105

rows of <literal>R</literal>.

106

</para>

107

</listitem>

108

</varlistentry>

109

110

111

112

<para>

113

returns <literal>rk</literal> = rank estimate of <literal>X</literal> i.e. <literal>rk</literal> is the number of diagonal elements in <literal>R</literal> which are larger than a given threshold <literal>tol</literal>.

114

</para>

115

</listitem>

116

</varlistentry>

117

118

119

120

<para>

121

returns <literal>rk</literal> = rank estimate of <literal>X</literal>

122

i.e. <literal>rk</literal> is the number of diagonal elements in

123

<literal>R</literal> which are larger than

124

<literal>tol=R(1,1)*%eps*max(size(R))</literal>. See <literal>rankqr</literal>

125

for a rank revealing QR factorization, using the condition number

126

of <literal>R</literal>.

127

</para>

128

</listitem>

129

</varlistentry>

130

</variablelist>

131

</refsection>

132

133

<title>Examples</title>

134

<programlisting role="example"><![CDATA[

135

// QR factorization, generic case

136

// X is tall (full rank)

137

X=rand(5,2);[Q,R]=qr(X); [Q'*X R]

138

139

//X is fat (full rank)

140

X=rand(2,3);[Q,R]=qr(X); [Q'*X R]

141

142

//Column 4 of X is a linear combination of columns 1 and 2:

143

X=rand(8,5);X(:,4)=X(:,1)+X(:,2); [Q,R]=qr(X); R, R(:,4)

144

145

//X has rank 2, rows 3 to $ of R are zero:

146

X=rand(8,2)*rand(2,5);[Q,R]=qr(X); R

147

148

//Evaluating the rank rk: column pivoting ==> rk first

149

//diagonal entries of R are non zero :

150

A=rand(5,2)*rand(2,5);

151

[Q,R,rk,E] = qr(A,1.d-10);

152

norm(Q'*A-R)

153

svd([A,Q(:,1:rk)]) //span(A) =span(Q(:,1:rk))

154

]]></programlisting>

155

</refsection>

156

157

158

159

160

161

</member>

162

163

164

</member>

165

166

167

</member>

168

169

170

</member>

171

172

173

</member>

174

</simplelist>

175

</refsection>

176

177

<title>Used Functions</title>

178

<para>

179

qr decomposition is based the Lapack routines DGEQRF, DGEQPF,

180

DORGQR for the real matrices and ZGEQRF, ZGEQPF, ZORGQR for the

181

complex case.

182

</para>

183

</refsection>

184

</refentry>

Older »