?pbtrs
?pbtrs
Solves a system of linear equations with a Cholesky-factored symmetric (Hermitian) positive-definite band coefficient matrix.
Syntax
lapack_int
LAPACKE_spbtrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
float
*
ab
,
lapack_int
ldab
,
float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_dpbtrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
double
*
ab
,
lapack_int
ldab
,
double
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_cpbtrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
lapack_complex_float
*
ab
,
lapack_int
ldab
,
lapack_complex_float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_zpbtrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const
lapack_complex_double
*
ab
,
lapack_int
ldab
,
lapack_complex_double
*
b
,
lapack_int
ldb
);
Include Files
- mkl.h
Description
The routine solves for real data a system of linear equations with a symmetric positive-definite or, for complex data, Hermitian positive-definite band matrix
A*X
= B
A
, given the Cholesky factorization of A
:A = U T *U A = U H *U | if uplo ='U' |
A = L*L T A = L*L H | if uplo ='L' |
where
L
is a lower triangular matrix and U
is upper triangular. The system is solved with multiple right-hand sides stored in the columns of the matrix B
.Before calling this routine, you must call
?pbtrf
to compute the Cholesky factorization of A
in the band storage form.Input Parameters
- matrix_layout
- Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
- uplo
- Must be'U'or'L'.Indicates how the input matrixAhas been factored:If,uplo='U'Uis stored inab, whereA=UT*Ufor real matrices andA=UH*Ufor complex matrices.If,uplo='L'Lis stored inab, whereA=L*LTfor real matrices andA=L*LHfor complex matrices.
- n
- The order of matrixA;n≥0.
- kd
- The number of superdiagonals or subdiagonals in the matrixA;kd≥0.
- nrhs
- The number of right-hand sides;nrhs≥0.
- ab
- Arrayabis of size max (1,ldab*n).The arrayabcontains the Cholesky factor, as returned by the factorization routine, in band storage form.The arraybcontains the matrixBwhose columns are the right-hand sides for the systems of equations.
- b
- The arraybcontains the matrixBwhose columns are the right-hand sides for the systems of equations.The size ofbis at least max(1,ldb*nrhs) for column major layout and max(1,ldb*n) for row major layout.
- ldab
- The leading dimension of the arrayab;ldab≥kd+1.
- ldb
- The leading dimension ofb;.ldb≥max(1,n) for column major layout andldb≥nrhsfor row major layout
Output Parameters
- b
- Overwritten by the solution matrixX.
Return Values
This function returns a value
info
.If , the execution is successful.
info
=0If , parameter
info
= -i
i
had an illegal value.Application Notes
For each right-hand side
b
, the computed solution is the exact solution of a perturbed system of equations (
, where A
+ E
)x
= b
|E| ≤ c(kd + 1)ε P|UH||U| or |E| ≤ c(kd + 1)ε P|LH||L|
c
(k
)k
, and ε
is the machine precision.If is the true solution, the computed solution
x
0
x
satisfies this error bound:
where / || || = (
cond(
= || |A
,x
)A
-1
||A
| |x
| ||∞
x
||∞
≤
||A
-1
||∞
A
||∞
κ
∞
A
).Note that (.
cond(
can be much smaller than A
,x
)κ
∞
A
)The approximate number of floating-point operations for one right-hand side vector is 4
n
*kd
for real flavors and 16n
*kd
for complex flavors.To estimate the condition number (, call
κ
∞
A
)?pbcon
. To refine the solution and estimate the error, call
?pbrfs
.