I have a small Hermitian matrix (6*6). I need to inverse this matrix. The program memory is bounded.
What method is optimal in this case?
Can you give any e-links?
You aren't clear on how memory bounded your program is,
so it's hard to say what algorithms are best.
That is a tiny matrix and, being 6x6 it is composed of
only 2x2 size 3x3 sub-blocks, or alternatively 3x3 size 2x2
sub-blocks. Thus you could use optimized 2x2 or 3x3 block
algorithms to help with your memory constraint.
Certainly such a target 6x6 complex
double precision COMPLEX*8 matrix takes only 576 bytes
of storage, though since it's Hermitian you can store its
lower or upper triangle in 288 bytes of memory.
Given that a few hundred bytes of code memory or data
memory is easy to expend whether you use a
code-intensive storage-reduced algorithm, or a
storage-intensive code-reduced algorithm, it's perhaps
application dependent as to what trade-off is best for you.
Furthermore, the optimum algorithm will depend on
whether you know of any additional structure of the
matrix besides it being Hermitian (e.g. sparseness,
is it positive definite, et. al.).
I'll provide some links and source codes for algorithms
to factor and invert either an indefinite or positive
definite Hermitian matrix using either packed or normal
storage and given either the upper or lower triangular
portion of the input Hermitian matrix as input to the
algorithm.
If you're going to port or rewrite the algorithms in some
memory conserving language like DSP code, C, assembler,
you'll probably find the LINPACK codes to be more explicit
since they make less use of subroutines, and the LAPACK
codes more algorithmically modular. Obviously one
would tend to just reimplement the code for one's
particular language and use case given your memory
constraints and for such a small and fixed problem size.
In general most of the work involves factoring the
input matrix, then simple algorithms can use the factored
matrix representations to calculate the inverse.
Generally the algorithms iterate over portions / blocks
of the input matrix, so, again, expanding the code for
the blocks / structure of your particular input will
yield some simplicity since it's a small and obviously
nicely blocked Hermitian matrix.
Of course if you're just trying to solve linear equations,
eigenproblems, get eigenvalues/eigenvectors, there
are other and usually better algorithms (e.g. in LAPACK/LINPACK) for such problem areas that don't
generally involve calculating the specific matrix inverse.
* ZHETRF computes the factorization of a complex Hermitian matrix A
* using the Bunch-Kaufman diagonal pivoting method. The form of the
* factorization is
*
* A = U*D*U**H or A = L*D*L**H
*
* where U (or L) is a product of permutation and unit upper (lower)
* triangular matrices, and D is Hermitian and block diagonal with
* 1-by-1 and 2-by-2 diagonal blocks.
*
* This is the blocked version of the algorithm, calling Level 3 BLAS.
* CHETRF, ZHETRF: Computes the factorization of a complex Hermitian
indefinite matrix, using the diagonal pivoting method.
* CHETRI, ZHETRI: Computes the inverse of a complex Hermitian
indefinite matrix, using the factorization computed by CHETRF.
hermitian indefinite packed storage
* CHPTRF, ZHPTRF: Computes the factorization of a complex Hermitian
indefinite matrix in packed storage, using the diagonal pivoting
method.
* CHPTRI, ZHPTRI: Computes the inverse of a complex Hermitian
indefinite matrix in packed storage, using the factorization computed
by CHPTRF.
* ZPOTRF computes the Cholesky factorization of a complex Hermitian
* positive definite matrix A.
*
* The factorization has the form
* A = U**H * U, if UPLO = 'U', or
* A = L * L**H, if UPLO = 'L',
* where U is an upper triangular matrix and L is lower triangular.
*
* This is the block version of the algorithm, calling Level 3 BLAS.
* SPOTRF, DPOTRF, CPOTRF, ZPOTRF: Computes the Cholesky factorization
of a symmetric or Hermitian positive definite matrix.
* SPOTRI, DPOTRI, CPOTRI, ZPOTRI: Computes the inverse of a symmetric
or Hermitian positive definite matrix, using the Cholesky
factorization computed by SPOTRF or CPOTRF.
hermitian positive definite packed storage
* SPPTRF, DPPTRF, CPPTRF, ZPPTRF: Computes the Cholesky factorization
of a symmetric or Hermitian positive definite matrix in packed
storage.
* SPPTRI, DPPTRI, CPPTRI, ZPPTRI: Computes the inverse of a symmetric
or Hermitian positive definite matrix in packed storage, using the
Cholesky factorization computed by SPPTRF or CPPTRF.
hermitian indefinite
single precision
file chifa.f chifa.f plus dependencies
gams D2d1a
for factors a complex Hermitian matrix
, by elimination with symmetric pivoting
prec complex
file chidi.f chidi.f plus dependencies
gams D2d1a, D3d1a
for computes the determinant, inertia and inverse of a complex
, Hermitian matrix using the factors from linpack/chifa
prec complex
hermitian indefinite
double precision
file zhifa.f zhifa.f plus dependencies
gams D2d1a
for factors a complex Hermitian matrix
, by elimination with symmetric pivoting
prec doublecomplex
file zhidi.f zhidi.f plus dependencies
gams D2d1a, D3d1a
for computes the determinant, inertia and inverse of a complex
, Hermitian matrix using the factors from linpack/zhifa)
prec doublecomplex
hermitian indefinite
single precision packed storage
file chpfa.f chpfa.f plus dependencies
gams D2d1a
for factors a complex Hermitian matrix
, stored in packed form by elimination with symmetric pivoting
prec complex
file chpdi.f chpdi.f plus dependencies
gams D2d1a, D3d1a
for computes the determinant, inertia and inverse of a
, complex Hermitian matrix using the factors from linpack/chpfa,
, where the matrix is stored in packed form
prec complex
hermitian indefinite
double precision packed storage
file zhpfa.f zhpfa.f plus dependencies
gams D2d1a
for factors a complex Hermitian matrix
, stored in packed form by elimination with symmetric pivoting
prec doublecomplex
file zhpdi.f zhpdi.f plus dependencies
gams D2d1a, D3d1a
for computes the determinant, inertia and inverse of a
, complex Hermitian matrix using the factors from linpack/zhpfa,
, where the matrix is stored in packed form
prec doublecomplex
I'm reviewing Meirovitch's "Methods of Analytical Dynamics," and I don't understand the commutation of the derivative from r to dr:
$$
\mathbf{F} \cdot d\mathbf{r} = m \ddot{\mathbf{r}} \cdot d\mathbf{r} = m\mathbf{\dot{r}} \cdot d\mathbf{\dot{r}}
$$
