F08BPF (ZTPQRT) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08BPF (ZTPQRT)

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.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08BPF (ZTPQRT) computes the QR factorization of a complex m+n by n triangular-pentagonal matrix.

2  Specification

SUBROUTINE F08BPF ( M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK, INFO)
INTEGER  M, N, L, NB, LDA, LDB, LDT, INFO
COMPLEX (KIND=nag_wp)  A(LDA,*), B(LDB,*), T(LDT,*), WORK(*)
The routine may be called by its LAPACK name ztpqrt.

3  Description

F08BPF (ZTPQRT) forms the QR factorization of a complex m+n by n triangular-pentagonal matrix C,
C= A B  
where A is an upper triangular n by n matrix and B is an m by n pentagonal matrix consisting of an m-l by n rectangular matrix B1 on top of an l by n upper trapezoidal matrix B2:
B= B1 B2 .  
The upper trapezoidal matrix B2 consists of the first l rows of an n by n upper triangular matrix, where 0lminm,n. If l=0, B is m by n rectangular; if l=n and m=n, B is upper triangular.
A recursive, explicitly blocked, QR factorization (see F08APF (ZGEQRT)) is performed on the matrix C. The upper triangular matrix R, details of the unitary matrix Q, and further details (the block reflector factors) of Q are returned.
Typically the matrix A or B2 contains the matrix R from the QR factorization of a subproblem and F08BPF (ZTPQRT) performs the QR update operation from the inclusion of matrix B1.
For example, consider the QR factorization of an l by n matrix B^ with l<n: B^ = Q^R^ , R^ = R1^ R2^ , where R1^ is l by l upper triangular and R2^ is n-l by n rectangular (this can be performed by F08APF (ZGEQRT)). Given an initial least-squares problem B^ X^ = Y^  where X and Y are l by nrhs matrices, we have R^ X^ = Q^H Y^ .
Now, adding an additional m-l rows to the original system gives the augmented least squares problem
BX=Y  
where B is an m by n matrix formed by adding m-l rows on top of R^ and Y is an m by nrhs matrix formed by adding m-l rows on top of Q^HY^.
F08BPF (ZTPQRT) can then be used to perform the QR factorization of the pentagonal matrix B; the n by n matrix A will be zero on input and contain R on output.
In the case where B^ is r by n, rn, R^ is n by n upper triangular (forming A) on top of r-n rows of zeros (forming first r-n rows of B). Augmentation is then performed by adding rows to the bottom of B with l=0.

4  References

Elmroth E and Gustavson F (2000) Applying Recursion to Serial and Parallel QR Factorization Leads to Better Performance IBM Journal of Research and Development. (Volume 44) 4 605–624
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:     M – INTEGERInput
On entry: m, the number of rows of the matrix B.
Constraint: M0.
2:     N – INTEGERInput
On entry: n, the number of columns of the matrix B and the order of the upper triangular matrix A.
Constraint: N0.
3:     L – INTEGERInput
On entry: l, the number of rows of the trapezoidal part of B (i.e., B2).
Constraint: 0LminM,N.
4:     NB – INTEGERInput
On entry: the explicitly chosen block-size to be used in the algorithm for computing the QR factorization. See Section 9 for details.
Constraints:
  • NB1;
  • if N>0, NBN.
5:     ALDA* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: the n by n upper triangular matrix A.
On exit: the upper triangle is overwritten by the corresponding elements of the n by n upper triangular matrix R.
6:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08BPF (ZTPQRT) is called.
Constraint: LDAmax1,N.
7:     BLDB* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least max1,N.
On entry: the m by n pentagonal matrix B composed of an m-l by n rectangular matrix B1 above an l by n upper trapezoidal matrix B2.
On exit: details of the unitary matrix Q.
8:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08BPF (ZTPQRT) is called.
Constraint: LDBmax1,M.
9:     TLDT* – COMPLEX (KIND=nag_wp) arrayOutput
Note: the second dimension of the array T must be at least N.
On exit: further details of the unitary matrix Q. The number of blocks is b=kNB, where k=minm,n and each block is of order NB except for the last block, which is of order k-b-1×NB. For each of the blocks, an upper triangular block reflector factor is computed: T1,T2,,Tb. These are stored in the NB by n matrix T as T=T1|T2||Tb.
10:   LDT – INTEGERInput
On entry: the first dimension of the array T as declared in the (sub)program from which F08BPF (ZTPQRT) is called.
Constraint: LDTNB.
11:   WORK* – COMPLEX (KIND=nag_wp) arrayWorkspace
Note: the dimension of the array WORK must be at least NB×N.
12:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

7  Accuracy

The computed factorization is the exact factorization of a nearby matrix A+E, where
E2 = Oε A2 ,  
and ε is the machine precision.

8  Parallelism and Performance

F08BPF (ZTPQRT) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The total number of floating-point operations is approximately 23 n2 3m-n  if mn or 23 m2 3n-m  if m<n.
The block size, NB, used by F08BPF (ZTPQRT) is supplied explicitly through the interface. For moderate and large sizes of matrix, the block size can have a marked effect on the efficiency of the algorithm with the optimal value being dependent on problem size and platform. A value of NB=64minm,n is likely to achieve good efficiency and it is unlikely that an optimal value would exceed 340.
To apply Q to an arbitrary complex rectangular matrix C, F08BPF (ZTPQRT) may be followed by a call to F08BQF (ZTPMQRT). For example,
CALL ZTPMQRT('Left','Transpose',M,P,N,L,NB,B,LDB, &
              T,LDT,C,LDC,C(n+1,1),LDC,WORK,INFO)
forms C=QHC, where C is m+n by p.
To form the unitary matrix Q explicitly set p=m+n, initialize C to the identity matrix and make a call to F08BQF (ZTPMQRT) as above.

10  Example

This example finds the basic solutions for the linear least squares problems
minimize Axi - bi 2 ,   i=1,2  
where b1 and b2 are the columns of the matrix B,
A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i   and    
B= -2.09+1.93i 3.26-2.70i 3.34-3.53i -6.22+1.16i -4.94-2.04i 7.94-3.13i 0.17+4.23i 1.04-4.26i -5.19+3.63i -2.31-2.12i 0.98+2.53i -1.39-4.05i .  
A QR factorization is performed on the first 4 rows of A using F08APF (ZGEQRT) after which the first 4 rows of B are updated by applying QT using F08AQF (ZGEMQRT). The remaining row is added by performing a QR update using F08BPF (ZTPQRT); B is updated by applying the new QT using F08BQF (ZTPMQRT); the solution is finally obtained by triangular solve using R from the updated QR.

10.1  Program Text

Program Text (f08bpfe.f90)

10.2  Program Data

Program Data (f08bpfe.d)

10.3  Program Results

Program Results (f08bpfe.r)


F08BPF (ZTPQRT) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016