QR Call

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QR Call

produces the QR decomposition of a matrix by Householder transformations

CALL QR( q, r, piv, lindep, a<, ord><, b>);

The QR subroutine returns the following values:

q: specifies an orthogonal matrix Q that is the product of the Householder transformations applied to the m ×n matrix A, if the b argument is not specified. In this case, the min(m,n) Householder transformations are applied, and q is an m ×m matrix. If the b argument is specified, q is the m ×p matrix Q' B that has the transposed Householder transformations Q' applied on the p columns of the argument matrix B.
r: specifies a min(m,n) ×n upper triangular matrix R that is the upper part of the m ×n upper triangular matrix $\widetilde{R}$ of the QR decomposition of the matrix A. The matrix $\widetilde{R}$ of the QR decomposition can be obtained by vertical concatenation (using the IML operator //) of the (m - min(m,n)) ×n zero matrix to the result matrix R.
piv: specifies an n ×1 vector of permutations of the columns of A; that is, on return, the QR decomposition is computed, not of A, but of the permuted matrix whose columns are [A_piv[1] ... A_piv[n]]. The vector piv corresponds to an n ×n permutation matrix ${{\Pi}}$ .
lindep: is the number of linearly dependent columns in matrix A detected by applying the min(m,n) Householder transformations in the order specified by the argument vector piv.

The inputs to the QR subroutine are as follows:

a

specifies an m ×n matrix A that is to be decomposed into the product of the orthogonal matrix Q and the upper triangular matrix $\widetilde{R}$ .

ord

specifies an optional n ×1 vector that specifies the order of Householder transformations applied to matrix A, as follows:

ord[j] > 0: Column j of A is an initial column, meaning it has to be processed at the start in increasing order of ord[j].
ord[j] = 0: Column j of A is allowed to be permuted in order of decreasing residual Euclidean norm (pivoting).
ord[j] < 0: Column j of A is a final column, meaning it has to be processed at the end in decreasing order of ord[j].

The default is ord[j] = j, in which case the Householder transformations are done in the same order that the columns are stored in matrix A (without pivoting).

b

specifies an optional m ×p matrix B that is to be multiplied by the transposed m ×m matrix Q'. If b is specified, the result q contains the m ×p matrix Q' B. If b is not specified, the result q contains the m ×m matrix Q.

The QR subroutine decomposes an m ×n matrix A into the product of an m ×m orthogonal matrix Q and an m ×n upper triangular matrix $\widetilde{R}$ , so that

$A{{\Pi}}= Q\widetilde{R}, \hspace*{.2in} Q^' Q= Q{Q}^' = I_m$

by means of min(m,n) Householder transformations.

The m ×m orthogonal matrix Q is computed only if the last argument b is not specified, as follows:

   call qr(q,r,piv,lindep,a,ord);

In many applications, the number of rows, m, is very large. In these cases, the explicit computation of the m ×m matrix Q can require too much memory or time. In the usual case where m > n,

$A= [ * & * & * \ {}* & * & * \ {}* & * & * \ {}* & * & * \ {}* & * & * ] & & ... ... & * \ 0 & 0 & * ] \ Q= [ Q_1 \vdots Q_2 ], & & \widetilde{R} = [ R\ 0 ]$

where R is the result returned by the QR subroutine. The n columns of matrix Q₁ provide an orthonormal basis for the n columns of A and are called the range space of A. Since the m-n columns of Q₂ are orthogonal to the n columns of A, Q₂' A = 0, they provide an orthonormal basis for the orthogonal complement of the columns of A and are called the null space of A.

In the case where m < n,

$A= [ * & * & * & * & * \ {}* & * & * & * & * \ {}* & * & * & * & * ] & & Q= [... ...detilde{R} = R= [ * & * & * & * & * \ 0 & * & * & * & * \ 0 & 0 & * & * & * ]$

Specifying the argument ord as an n vector lets you specify a special order of the columns in matrix A on which the Householder transformations are applied. When you specify the ord argument, the columns of A can be divided into the following groups:

ord[j]>0: Column j of A is an initial column, meaning it has to be processed at the start in increasing order of ord[j]. This specification defines the first nl columns of A that are to be processed.
ord[j]=0: Column j of A is a pivot column, meaning it is to be processed in order of decreasing residual Euclidean norms. The pivot columns of A are processed after the nl initial columns and before the nu final columns.
ord[j]<0: Column j of A is a final column, meaning it has to be processed at the end in decreasing order of ord[j]. This specification defines the last nu columns of A that are to be processed. If n>m, some of these columns will not be processed at all.

There are two special cases:

If you do not specify the ord argument, the default values ord[j] = j are used. In this case, Householder transformations are done in the same order in which the columns are stored in A (without pivoting).
If you set all components of ord to zero, the Householder transformations are done in order of decreasing Euclidean norms of the columns of A.

The resulting n ×1 vector piv specifies the permutation of the columns of A on which the Householder transformations are applied; that is, on return, the QR decomposition is computed, not of A, but of the matrix with columns that are permuted in the order A_piv[1], ... , A_piv[n]. To check the QR decomposition, use the following statements to compute the three residual sum of squares, represented by the variables SS0, SS1, and SS2, which should be close to zero:

   m = nrow(a); n = ncol(a);
   call qr(q,r,piv,lindep,a,ord);
   ss0 = ssq(a[ ,piv] - q[,1:n] * r);
   ss1 = ssq(q * q` - i(m));
   ss2 = ssq(q` * q - i(m));

If the QR subroutine detects linearly dependent columns while processing matrix A, the column order given in the result vector piv can differ from an explicitly specified order in the argument vector ord. If a column of A is found to be linearly dependent on columns already processed, this column is swapped to the end of matrix A. The order of columns in the result matrix R corresponds to the order of columns processed in A. The swapping of a linearly dependent column of A to the end of the matrix corresponds to the swapping of the same column in R and leads to a zero row at the end of the upper triangular matrix R.

The scalar result lindep counts the number of linearly dependent columns that are detected in constructing the first min(m,n) Householder transformations in the order specified by the argument vector ord. The test of linear dependence depends on the size of the singularity criterion used; currently it is specified as 1E-8.

Solving the linear system Rx = Q' b with an upper triangular matrix R whose columns are permuted corresponding to the result vector piv leads to a solution x with permuted components. You can reorder the components of x by using the index vector piv at the left-hand side of an expression, as follows:

   call qr(qtb,r,piv,lindep,a,ord,b);
   x[piv] = inv(r) * qtb[1:n,1:p];

The following example solves the full rank linear least-squares problem. Specify the argument b as an m ×p matrix B, as follows:

   call qr(q,r,piv,lindep,a,ord,b);

When you specify the b argument, the QR call computes the matrix Q' B (instead of Q) as the result q. Now you can compute the p least-squares solutions x_k of an overdetermined linear system with an m ×n, m > n coefficient matrix A, rank(A) = n, and p right-hand sides b_k stored as the columns of the m ×p matrix B:

min_{x_k} | Ax_k - b_k |², k = 1, ... ,p

where | ·| is the Euclidean vector norm. This is accomplished by solving the p upper triangular systems with back-substitution:

$x_k = {{\Pi}}^' R^{-1}Q_1^' b_k, \hspace*{0.2in} k = 1, ... ,p$

For most applications, m, the number of rows of A, is much larger than n, the number of columns of A, or p, the number of right-hand sides. In these cases, you are advised not to compute the large m ×m matrix Q (which can consume too much memory and time) if you can solve your problem by computing only the smaller m ×p matrix Q' B implicitly. For an example, use the first five columns of the 6 ×6 Hilbert matrix A:

proc iml;
   a= {  36      -630      3360     -7560      7560     -2772,
       -630     14700    -88200    211680   -220500     83160,
       3360    -88200    564480  -1411200   1512000   -582120,
      -7560    211680  -1411200   3628800  -3969000   1552320,
       7560   -220500   1512000  -3969000   4410000  -1746360,
      -2772     83160   -582120   1552320  -1746360    698544 };
   b= { 463, -13860, 97020, -258720, 291060, -116424};
   n = 5; aa = a[,1:n];
   call qr(qtb,r,piv,lindep,aa,,b);
   if lindep=0 then x=inv(r)*qtb[1:n];
   print x;

Note that you are using only the first n rows, Q'₁B, of QTB. The IF-THEN statement of the preceding code may be replaced by the more efficient TRISOLV function, as follows:

   if lindep=0 then x=TRISOLV(1,r,qtb[1:n],piv);
   print x;

Both cases produce the following output:

For information on solving rank-deficient linear least-squares problems, see the RZLIND call.

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