F01HBF (PDF version)
F01 Chapter Contents
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NAG Library Manual

NAG Library Routine Document

F01HBF

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

F01HBF computes the action of the matrix exponential etA, on the matrix B, where A is a complex n by n matrix, B is a complex n by m matrix and t is a complex scalar. It uses reverse communication for evaluating matrix products, so that the matrix A is not accessed explicitly.

2  Specification

SUBROUTINE F01HBF ( IREVCM, N, M, B, LDB, T, TR, B2, LDB2, X, LDX, Y, LDY, P, R, Z, CCOMM, COMM, ICOMM, IFAIL)
INTEGER  IREVCM, N, M, LDB, LDB2, LDX, LDY, ICOMM(2*N+40), IFAIL
REAL (KIND=nag_wp)  COMM(3*N+14)
COMPLEX (KIND=nag_wp)  B(LDB,*), T, TR, B2(LDB2,*), X(LDX,*), Y(LDY,*), P(N), R(N), Z(N), CCOMM(N*(M+2))

3  Description

etAB is computed using the algorithm described in Al–Mohy and Higham (2011) which uses a truncated Taylor series to compute the etAB without explicitly forming etA.
The algorithm does not explicity need to access the elements of A; it only requires the result of matrix multiplications of the form AX or AHY. A reverse communication interface is used, in which control is returned to the calling program whenever a matrix product is required.

4  References

Al–Mohy A H and Higham N J (2011) Computing the action of the matrix exponential, with an application to exponential integrators SIAM J. Sci. Statist. Comput. 33(2) 488-511
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

5  Parameters

Note:  this routine uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the parameter IREVCM. Between intermediate exits and re-entries, all parameters other than B2, X, Y, P and R must remain unchanged.
1:     IREVCM – INTEGERInput/Output
On initial entry: must be set to 0.
On intermediate exit: IREVCM=1, 2, 3, 4 or 5. The calling program must:
(a) if IREVCM=1: evaluate B2=AB, where B2 is an n by m matrix, and store the result in B2;
if IREVCM=2: evaluate Y=AX, where X and Y are n by 2 matrices, and store the result in Y;
if IREVCM=3: evaluate X=AHY and store the result in X;
if IREVCM=4: evaluate p=Az and store the result in P;
if IREVCM=5: evaluate r=AHz and store the result in R.
(b) call F01HBF again with all other parameters unchanged.
On final exit: IREVCM=0.
2:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
3:     M – INTEGERInput
On entry: the number of columns of the matrix B.
Constraint: M0.
4:     BLDB* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least M.
On initial entry: the n by m matrix B.
On intermediate exit: if IREVCM=1, contains the n by m matrix B.
On intermediate re-entry: must not be changed.
On final exit: the n by m matrix etAB.
5:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F01HBF is called.
Constraint: LDBN.
6:     T – COMPLEX (KIND=nag_wp)Input
On entry: the scalar t.
7:     TR – COMPLEX (KIND=nag_wp)Input
On entry: the trace of A. If this is not available then any number can be supplied (0 is a reasonable default); however, in the trivial case, n=1, the result eTRtB is immediately returned in the first row of B. See Section 9.
8:     B2LDB2* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B2 must be at least M.
On initial entry: need not be set.
On intermediate re-entry: if IREVCM=1, must contain AB.
On final exit: the array is undefined.
9:     LDB2 – INTEGERInput
On initial entry: the first dimension of the array B2 as declared in the (sub)program from which F01HBF is called.
Constraint: LDB2N.
10:   XLDX* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array X must be at least 2.
On initial entry: need not be set.
On intermediate exit: if IREVCM=2, contains the current n by 2 matrix X.
On intermediate re-entry: if IREVCM=3, must contain AHY.
On final exit: the array is undefined.
11:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F01HBF is called.
Constraint: LDXN.
12:   YLDY* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array Y must be at least 2.
On initial entry: need not be set.
On intermediate exit: if IREVCM=3, contains the current n by 2 matrix Y.
On intermediate re-entry: if IREVCM=2, must contain AX.
On final exit: the array is undefined.
13:   LDY – INTEGERInput
On entry: the first dimension of the array Y as declared in the (sub)program from which F01HBF is called.
Constraint: LDYN.
14:   PN – COMPLEX (KIND=nag_wp) arrayInput/Output
On initial entry: need not be set.
On intermediate re-entry: if IREVCM=4, must contain Az.
On final exit: the array is undefined.
15:   RN – COMPLEX (KIND=nag_wp) arrayInput/Output
On initial entry: need not be set.
On intermediate re-entry: if IREVCM=5, must contain AHz.
On final exit: the array is undefined.
16:   ZN – COMPLEX (KIND=nag_wp) arrayInput/Output
On initial entry: need not be set.
On intermediate exit: if IREVCM=4 or 5, contains the vector z.
On intermediate re-entry: must not be changed.
On final exit: the array is undefined.
17:   CCOMMN×M+2 – COMPLEX (KIND=nag_wp) arrayCommunication Array
18:   COMM3×N+14 – REAL (KIND=nag_wp) arrayCommunication Array
19:   ICOMM2×N+40 – INTEGER arrayCommunication Array
20:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
IFAIL=2
etAB has been computed using an IEEE double precision Taylor series, although the arithmetic precision is higher than IEEE double precision.
IFAIL=-1
On initial entry, IREVCM=value.
Constraint: IREVCM=0.
On intermediate re-entry, IREVCM=value.
Constraint: IREVCM=1, 2, 3, 4 or 5.
IFAIL=-2
On initial entry, N=value.
Constraint: N0.
IFAIL=-3
On initial entry, M=value.
Constraint: M0.
IFAIL=-5
On initial entry, LDB=value and N=value.
Constraint: LDBN.
IFAIL=-9
On initial entry, LDB2=value and N=value.
Constraint: LDB2N.
IFAIL=-11
On initial entry, LDX=value and N=value.
Constraint: LDXN.
IFAIL=-13
On initial entry, LDY=value and N=value.
Constraint: LDYN.
IFAIL=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.
IFAIL=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
IFAIL=-999
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7  Accuracy

For an Hermitian matrix A (for which AH=A) the computed matrix etAB is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for non-Hermitian matrices. See Section 4 of Al–Mohy and Higham (2011) for details and further discussion.

8  Parallelism and Performance

F01HBF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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

9.1  Use of TrA

The elements of A are not explicitly required by F01HBF. However, the trace of A is used in the preprocessing phase of the algorithm. If TrA is not available to the calling subroutine then any number can be supplied (0 is recommended). This will not affect the stability of the algorithm, but it may reduce its efficiency.

9.2  When to use F01HBF

F01HBF is designed to be used when A is large and sparse. Whenever a matrix multiplication is required, the routine will return control to the calling program so that the multiplication can be done in the most efficient way possible. Note that etAB will not, in general, be sparse even if A is sparse.
If A is small and dense then F01HAF can be used to compute etAB without the use of a reverse communication interface.
The real analog of F01HBF is F01GBF.

9.3  Use in Conjunction with NAG Library Routines

To compute etAB, the following skeleton code can normally be used:
revcm: Do 
  Call F01HBF(IREVCM,N,M,B,LDB,T,TR,B2,LDB2,X,LDX,Y,LDX,P,R,Z, &
             CCOMM,COMM,ICOMM,IFAIL)
  If (IREVCM == 0) Then 
      Exit revcm 
  Else If (IREVCM == 1) Then
      .. Code to compute B2=AB ..
  Else If (IREVCM == 2) Then
      .. Code to compute Y=AX ..
  Else If (IREVCM == 3) Then
      .. Code to compute X=A^H Y ..
  Else If (IREVCM == 4) Then
      .. Code to compute P=AZ ..
  Else If (IREVCM == 5) Then 
      .. Code to compute R=A^H Z ..
  End If
End Do revcm
The code used to compute the matrix products will vary depending on the way A is stored. If all the elements of A are stored explicitly, then F06ZAF (ZGEMM) can be used. If A is triangular then F06ZFF (ZTRMM) should be used. If A is Hermitian, then F06ZCF (ZHEMM) should be used. If A is symmetric, then F06ZTF (ZSYMM) should be used. For sparse A stored in coordinate storage format F11XNF and F11XSF can be used. For sparse A stored in compressed column storage format (CCS) the program text of Section 10 contains the routine matmul to perform matrix products.

10  Example

This example computes etAB where
A = 0.7+0.8i -0.2+0.0i 1.0+0.0i 0.6+0.5i 0.3+0.7i 0.7+0.0i 0.9+3.0i 1.0+0.8i 0.3+3.0i -0.7+0.0i 0.2+0.6i 0.7+0.5i 0.0+0.9i 4.0+0.0i 0.0+0.0i 0.2+0.0i ,  
B = 0.1+0.0i 1.2+0.1i 1.3+0.9i -0.2+2.0i 4.0+0.6i -1.0+0.8i 0.4+0.0i -0.9+0.0i  
and
t=1.1+0.0i .  
A is stored in compressed column storage format (CCS) and matrix multiplications are performed using the routine matmul.

10.1  Program Text

Program Text (f01hbfe.f90)

10.2  Program Data

Program Data (f01hbfe.d)

10.3  Program Results

Program Results (f01hbfe.r)


F01HBF (PDF version)
F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015