F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08KCF (DGELSD)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08KCF (DGELSD) computes the minimum norm solution to a real linear least squares problem
 $minx b-Ax2 .$

## 2  Specification

 SUBROUTINE F08KCF ( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, IWORK, INFO)
 INTEGER M, N, NRHS, LDA, LDB, RANK, LWORK, IWORK(*), INFO REAL (KIND=nag_wp) A(LDA,*), B(LDB,*), S(*), RCOND, WORK(max(1,LWORK))
The routine may be called by its LAPACK name dgelsd.

## 3  Description

F08KCF (DGELSD) uses the singular value decomposition (SVD) of $A$, where $A$ is a real $m$ by $n$ matrix which may be rank-deficient.
Several right-hand side vectors $b$ and solution vectors $x$ can be handled in a single call; they are stored as the columns of the $m$ by $r$ right-hand side matrix $B$ and the $n$ by $r$ solution matrix $X$.
The problem is solved in three steps:
1. reduce the coefficient matrix $A$ to bidiagonal form with Householder transformations, reducing the original problem into a ‘bidiagonal least squares problem’ (BLS);
2. solve the BLS using a divide-and-conquer approach;
3. apply back all the Householder transformations to solve the original least squares problem.
The effective rank of $A$ is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     M – INTEGERInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{M}}\ge 0$.
2:     N – INTEGERInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrices $B$ and $X$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
4:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $m$ by $n$ coefficient matrix $A$.
On exit: the contents of A are destroyed.
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08KCF (DGELSD) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
6:     B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On entry: the $m$ by $r$ right-hand side matrix $B$.
On exit: B is overwritten by the $n$ by $r$ solution matrix $X$. If $m\ge n$ and ${\mathbf{RANK}}=n$, the residual sum of squares for the solution in the $i$th column is given by the sum of squares of elements $n+1,\dots ,m$ in that column.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08KCF (DGELSD) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}},{\mathbf{N}}\right)$.
8:     S($*$) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array S must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)\right)$.
On exit: the singular values of $A$ in decreasing order.
9:     RCOND – REAL (KIND=nag_wp)Input
On entry: used to determine the effective rank of $A$. Singular values ${\mathbf{S}}\left(i\right)\le {\mathbf{RCOND}}×{\mathbf{S}}\left(1\right)$ are treated as zero. If ${\mathbf{RCOND}}<0$, machine precision is used instead.
10:   RANK – INTEGEROutput
On exit: the effective rank of $A$, i.e., the number of singular values which are greater than ${\mathbf{RCOND}}×{\mathbf{S}}\left(1\right)$.
11:   WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.
12:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08KCF (DGELSD) is called.
The exact minimum amount of workspace needed depends on M, N and NRHS. As long as LWORK is at least
 $12⁢r+ 2⁢r× smlsiz+ 8⁢r× nlvl+ r× NRHS+ smlsiz+1 2 ,$
where $\mathit{smlsiz}$ is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about $25$), $\mathit{nlvl}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(0,\mathrm{int}\left({\mathrm{log}}_{2}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)/\left(\mathit{smlsiz}+1\right)\right)\right)+1\right)$ and $r=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)$, the code will execute correctly.
If ${\mathbf{LWORK}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the WORK array and the minimum size of the IWORK array, and returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK is issued.
Suggested value: for optimal performance, LWORK should generally be larger than the minimum required as set out above. Consider increasing LWORK by at least $\mathit{nb}×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{LWORK}}\ge 12r+2r×\mathit{smlsiz}+8r×\mathit{nlvl}+r×{\mathbf{NRHS}}+{\left(\mathit{smlsiz}+1\right)}^{2}\text{​ or ​}{\mathbf{LWORK}}=-1$.
13:   IWORK($*$) – INTEGER arrayWorkspace
Note: the dimension of the array IWORK must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{liwork}\right)$, where $\mathit{liwork}$ is at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,3×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)×\mathit{nlvl}+11×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)\right)$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{IWORK}}\left(1\right)$ returns the minimum $\mathit{liwork}$.
14:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
The algorithm for computing the SVD failed to converge; if ${\mathbf{INFO}}=i$, $i$ off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

## 7  Accuracy

See Section 4.5 of Anderson et al. (1999) for details.

## 8  Further Comments

The complex analogue of this routine is F08KQF (ZGELSD).

## 9  Example

This example solves the linear least squares problem
 $minx b-Ax2$
for the solution, $x$, of minimum norm, where
 $A = -0.09 -1.56 -1.48 -1.09 0.08 -1.59 0.14 0.20 -0.43 0.84 0.55 -0.72 -0.46 0.29 0.89 0.77 -1.13 1.06 0.68 1.09 -0.71 2.11 0.14 1.24 1.29 0.51 -0.96 -1.27 1.74 0.34 and b= 7.4 4.3 -8.1 1.8 8.7 .$
A tolerance of $0.01$ is used to determine the effective rank of $A$.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

### 9.1  Program Text

Program Text (f08kcfe.f90)

### 9.2  Program Data

Program Data (f08kcfe.d)

### 9.3  Program Results

Program Results (f08kcfe.r)