如何确定 MATLAB 使用哪些例程来求解稀疏矩阵？
How can I determine which routines MATLAB uses to solve a sparse matrix?
我正在尝试求解形式为 A*x = b[= 的稀疏矩阵方程25=]，其中 A 是已知的稀疏方阵，b 是已知的列向量，x为待定列向量。解决这个问题的标准 MATLAB 语法是：
x = A\b;
在幕后，\ 运算符对于 "use whatever algorithm seems best to solve this equation." 是 shorthand MATLAB 相应地选择它认为是求解该方程的最佳算法并使用求解方程组该算法。
虽然这种一种符号适用于所有情况的方法在过去对我很有效，但我需要知道正在使用哪种算法来求解我的方程组。有谁知道我怎么能找到这个？也许有一种方法可以告诉 MATLAB 打印 any/all 被调用的函数，并为嵌套调用缩进？
我认为你应该使用来自 matlab 的精子 forum
help spparms
spparms - Set parameters for sparse matrix routines
This MATLAB function sets one or more of the tunable parameters used in the
sparse routines.
spparms('key',value)
spparms
values = spparms
[keys,values] = spparms
spparms(values)
value = spparms('key')
spparms('default')
spparms('tight')
Reference page for spparms
See also chol, colamd, lu, qr, symamd
像这样
>> A = sparse(rand(10).*round(rand(10)-0.2));
spparms('spumoni',2)
A\rand(10,1)
sp\: bandwidth = 9+1+7.
sp\: is A diagonal? no.
sp\: is band density (0.27) > bandden (0.50) to try banded solver? no.
sp\: is A triangular? no.
sp\: is A morally triangular? no.
sp\: is A a candidate for Cholesky (symmetric, real positive diagonal)? no.
sp\: use Unsymmetric MultiFrontal PACKage with automatic reordering.
UMFPACK V5.4.0 (May 20, 2009), Control:
Matrix entry defined as: double
Int (generic integer) defined as: UF_long
0: print level: 2
1: dense row parameter: 0.2
"dense" rows have > max (16, (0.2)*16*sqrt(n_col) entries)
2: dense column parameter: 0.2
"dense" columns have > max (16, (0.2)*16*sqrt(n_row) entries)
3: pivot tolerance: 0.1
4: block size for dense matrix kernels: 32
5: strategy: 0 (auto)
6: initial allocation ratio: 0.7
7: max iterative refinement steps: 2
12: 2-by-2 pivot tolerance: 0.01
13: Q fixed during numerical factorization: 0 (auto)
14: AMD dense row/col parameter: 10
"dense" rows/columns have > max (16, (10)*sqrt(n)) entries
Only used if the AMD ordering is used.
15: diagonal pivot tolerance: 0.001
Only used if diagonal pivoting is attempted.
16: scaling: 1 (divide each row by sum of abs. values in each row)
17: frontal matrix allocation ratio: 0.5
18: drop tolerance: 0
19: AMD and COLAMD aggressive absorption: 1 (yes)
The following options can only be changed at compile-time:
8: BLAS library used: Fortran BLAS. size of BLAS integer: 8
9: compiled for MATLAB
10: CPU timer is POSIX times ( ) routine.
11: compiled for normal operation (debugging disabled)
computer/operating system: Linux
size of int: 4 UF_long: 8 Int: 8 pointer: 8 double: 8 Entry: 8 (in bytes)
sp\: UMFPACK's factorization was successful.
sp\: UMFPACK's solve was successful.
UMFPACK V5.4.0 (May 20, 2009), Info:
matrix entry defined as: double
Int (generic integer) defined as: UF_long
BLAS library used: Fortran BLAS. size of BLAS integer: 8
MATLAB: yes.
CPU timer: POSIX times ( ) routine.
number of rows in matrix A: 10
number of columns in matrix A: 10
entries in matrix A: 26
memory usage reported in: 16-byte Units
size of int: 4 bytes
size of UF_long: 8 bytes
size of pointer: 8 bytes
size of numerical entry: 8 bytes
strategy used: unsymmetric
ordering used: colamd on A
modify Q during factorization: yes
prefer diagonal pivoting: no
pivots with zero Markowitz cost: 2
submatrix S after removing zero-cost pivots:
number of "dense" rows: 0
number of "dense" columns: 0
number of empty rows: 0
number of empty columns 0
submatrix S not square or diagonal not preserved
symbolic factorization defragmentations: 0
symbolic memory usage (Units): 238
symbolic memory usage (MBytes): 0.0
Symbolic size (Units): 57
Symbolic size (MBytes): 0
symbolic factorization CPU time (sec): 0.00
symbolic factorization wallclock time(sec): 0.00
matrix scaled: yes (divided each row by sum of abs values in each row)
minimum sum (abs (rows of A)): 1.21495e-01
maximum sum (abs (rows of A)): 2.36586e+00
symbolic/numeric factorization: upper bound actual %
variable-sized part of Numeric object:
initial size (Units) 171 161 94%
peak size (Units) 938 899 96%
final size (Units) 39 28 72%
Numeric final size (Units) 130 114 88%
Numeric final size (MBytes) 0.0 0.0 88%
peak memory usage (Units) 1189 1150 97%
peak memory usage (MBytes) 0.0 0.0 97%
numeric factorization flops 1.79000e+02 3.30000e+01 18%
nz in L (incl diagonal) 31 19 61%
nz in U (incl diagonal) 36 23 64%
nz in L+U (incl diagonal) 57 32 56%
largest front (# entries) 42 6 14%
largest # rows in front 7 3 43%
largest # columns in front 6 3 50%
initial allocation ratio used: 0.7
# of forced updates due to frontal growth: 0
nz in L (incl diagonal), if none dropped 19
nz in U (incl diagonal), if none dropped 23
number of small entries dropped 0
nonzeros on diagonal of U: 10
min abs. value on diagonal of U: 1.30e-01
max abs. value on diagonal of U: 9.70e-01
estimate of reciprocal of condition number: 1.35e-01
indices in compressed pattern: 12
numerical values stored in Numeric object: 29
numeric factorization defragmentations: 1
numeric factorization reallocations: 1
costly numeric factorization reallocations: 1
numeric factorization CPU time (sec): 0.16
numeric factorization wallclock time (sec): 0.17
numeric factorization mflops (CPU time): 0.00
numeric factorization mflops (wallclock): 0.00
solve flops: 2.58000e+02
iterative refinement steps taken: 0
iterative refinement steps attempted: 0
sparse backward error omega1: 2.11e-16
sparse backward error omega2: 0.00e+00
solve CPU time (sec): 0.00
solve wall clock time (sec): 0.00
total symbolic + numeric + solve flops: 2.91000e+02
ans =
-8.8364
29.2610
72.4619
51.8905
-42.4795
-46.4504
0.5000
5.6994
12.7503
45.2984
我正在尝试求解形式为 A*x = b[= 的稀疏矩阵方程25=]，其中 A 是已知的稀疏方阵，b 是已知的列向量，x为待定列向量。解决这个问题的标准 MATLAB 语法是：
x = A\b;
在幕后，\ 运算符对于 "use whatever algorithm seems best to solve this equation." 是 shorthand MATLAB 相应地选择它认为是求解该方程的最佳算法并使用求解方程组该算法。
虽然这种一种符号适用于所有情况的方法在过去对我很有效，但我需要知道正在使用哪种算法来求解我的方程组。有谁知道我怎么能找到这个？也许有一种方法可以告诉 MATLAB 打印 any/all 被调用的函数，并为嵌套调用缩进？
我认为你应该使用来自 matlab 的精子 forum
help spparms
spparms - Set parameters for sparse matrix routines
This MATLAB function sets one or more of the tunable parameters used in the
sparse routines.
spparms('key',value)
spparms
values = spparms
[keys,values] = spparms
spparms(values)
value = spparms('key')
spparms('default')
spparms('tight')
Reference page for spparms
See also chol, colamd, lu, qr, symamd
像这样
>> A = sparse(rand(10).*round(rand(10)-0.2));
spparms('spumoni',2)
A\rand(10,1)
sp\: bandwidth = 9+1+7.
sp\: is A diagonal? no.
sp\: is band density (0.27) > bandden (0.50) to try banded solver? no.
sp\: is A triangular? no.
sp\: is A morally triangular? no.
sp\: is A a candidate for Cholesky (symmetric, real positive diagonal)? no.
sp\: use Unsymmetric MultiFrontal PACKage with automatic reordering.
UMFPACK V5.4.0 (May 20, 2009), Control:
Matrix entry defined as: double
Int (generic integer) defined as: UF_long
0: print level: 2
1: dense row parameter: 0.2
"dense" rows have > max (16, (0.2)*16*sqrt(n_col) entries)
2: dense column parameter: 0.2
"dense" columns have > max (16, (0.2)*16*sqrt(n_row) entries)
3: pivot tolerance: 0.1
4: block size for dense matrix kernels: 32
5: strategy: 0 (auto)
6: initial allocation ratio: 0.7
7: max iterative refinement steps: 2
12: 2-by-2 pivot tolerance: 0.01
13: Q fixed during numerical factorization: 0 (auto)
14: AMD dense row/col parameter: 10
"dense" rows/columns have > max (16, (10)*sqrt(n)) entries
Only used if the AMD ordering is used.
15: diagonal pivot tolerance: 0.001
Only used if diagonal pivoting is attempted.
16: scaling: 1 (divide each row by sum of abs. values in each row)
17: frontal matrix allocation ratio: 0.5
18: drop tolerance: 0
19: AMD and COLAMD aggressive absorption: 1 (yes)
The following options can only be changed at compile-time:
8: BLAS library used: Fortran BLAS. size of BLAS integer: 8
9: compiled for MATLAB
10: CPU timer is POSIX times ( ) routine.
11: compiled for normal operation (debugging disabled)
computer/operating system: Linux
size of int: 4 UF_long: 8 Int: 8 pointer: 8 double: 8 Entry: 8 (in bytes)
sp\: UMFPACK's factorization was successful.
sp\: UMFPACK's solve was successful.
UMFPACK V5.4.0 (May 20, 2009), Info:
matrix entry defined as: double
Int (generic integer) defined as: UF_long
BLAS library used: Fortran BLAS. size of BLAS integer: 8
MATLAB: yes.
CPU timer: POSIX times ( ) routine.
number of rows in matrix A: 10
number of columns in matrix A: 10
entries in matrix A: 26
memory usage reported in: 16-byte Units
size of int: 4 bytes
size of UF_long: 8 bytes
size of pointer: 8 bytes
size of numerical entry: 8 bytes
strategy used: unsymmetric
ordering used: colamd on A
modify Q during factorization: yes
prefer diagonal pivoting: no
pivots with zero Markowitz cost: 2
submatrix S after removing zero-cost pivots:
number of "dense" rows: 0
number of "dense" columns: 0
number of empty rows: 0
number of empty columns 0
submatrix S not square or diagonal not preserved
symbolic factorization defragmentations: 0
symbolic memory usage (Units): 238
symbolic memory usage (MBytes): 0.0
Symbolic size (Units): 57
Symbolic size (MBytes): 0
symbolic factorization CPU time (sec): 0.00
symbolic factorization wallclock time(sec): 0.00
matrix scaled: yes (divided each row by sum of abs values in each row)
minimum sum (abs (rows of A)): 1.21495e-01
maximum sum (abs (rows of A)): 2.36586e+00
symbolic/numeric factorization: upper bound actual %
variable-sized part of Numeric object:
initial size (Units) 171 161 94%
peak size (Units) 938 899 96%
final size (Units) 39 28 72%
Numeric final size (Units) 130 114 88%
Numeric final size (MBytes) 0.0 0.0 88%
peak memory usage (Units) 1189 1150 97%
peak memory usage (MBytes) 0.0 0.0 97%
numeric factorization flops 1.79000e+02 3.30000e+01 18%
nz in L (incl diagonal) 31 19 61%
nz in U (incl diagonal) 36 23 64%
nz in L+U (incl diagonal) 57 32 56%
largest front (# entries) 42 6 14%
largest # rows in front 7 3 43%
largest # columns in front 6 3 50%
initial allocation ratio used: 0.7
# of forced updates due to frontal growth: 0
nz in L (incl diagonal), if none dropped 19
nz in U (incl diagonal), if none dropped 23
number of small entries dropped 0
nonzeros on diagonal of U: 10
min abs. value on diagonal of U: 1.30e-01
max abs. value on diagonal of U: 9.70e-01
estimate of reciprocal of condition number: 1.35e-01
indices in compressed pattern: 12
numerical values stored in Numeric object: 29
numeric factorization defragmentations: 1
numeric factorization reallocations: 1
costly numeric factorization reallocations: 1
numeric factorization CPU time (sec): 0.16
numeric factorization wallclock time (sec): 0.17
numeric factorization mflops (CPU time): 0.00
numeric factorization mflops (wallclock): 0.00
solve flops: 2.58000e+02
iterative refinement steps taken: 0
iterative refinement steps attempted: 0
sparse backward error omega1: 2.11e-16
sparse backward error omega2: 0.00e+00
solve CPU time (sec): 0.00
solve wall clock time (sec): 0.00
total symbolic + numeric + solve flops: 2.91000e+02
ans =
-8.8364
29.2610
72.4619
51.8905
-42.4795
-46.4504
0.5000
5.6994
12.7503
45.2984