NAG Library Routine Document
F01HBF
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:
N≥0.
- 3: M – INTEGERInput
On entry: the number of columns of the matrix B.
Constraint:
M≥0.
- 4: B(LDB,*) – 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:
LDB≥max1,N.
- 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 8.
- 8: B2(LDB2,*) – 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:
LDB2≥max1,N.
- 10: X(LDX,*) – 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:
LDX≥max1,N.
- 12: Y(LDY,*) – 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:
LDY≥max1,N.
- 14: P(N) – 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: R(N) – 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: Z(N) – 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: CCOMM(N×M+2) – COMPLEX (KIND=nag_wp) arrayCommunication Array
- 18: COMM(3×N+14) – REAL (KIND=nag_wp) arrayCommunication Array
- 19: ICOMM(2×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=1
-
Note: this failure should not occur, and suggests that the routine has been called incorrectly. An unexpected internal error occurred when estimating a matrix norm.
- IFAIL=2
-
An unexpected internal error has occurred. Please contact
NAG.
- 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: N≥0.
- IFAIL=-3
-
On initial entry, M=value.
Constraint: M≥0.
- IFAIL=-5
-
On initial entry, LDB=value and N=value.
Constraint: LDB≥max1,N.
- IFAIL=-9
-
On initial entry, LDB2=value and N=value.
Constraint: LDB2≥max1,N.
- IFAIL=-11
-
On initial entry, LDX=value and N=value.
Constraint: LDX≥max1,N.
- IFAIL=-13
-
On initial entry, LDY=value and N=value.
Constraint: LDY≥max1,N.
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 Further Comments
8.1 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.
8.2 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.
8.3 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.
9 Example
This example computes
etAB where
and
9.1 Program Text
Program Text (f01hbfe.f90)
9.2 Program Data
Program Data (f01hbfe.d)
9.3 Program Results
Program Results (f01hbfe.r)