SPBSTF(3F) SPBSTF(3F)
SPBSTF - compute a split Cholesky factorization of a real symmetric
positive definite band matrix A
SUBROUTINE SPBSTF( UPLO, N, KD, AB, LDAB, INFO )
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
REAL AB( LDAB, * )
SPBSTF computes a split Cholesky factorization of a real symmetric
positive definite band matrix A.
This routine is designed to be used in conjunction with SSBGST.
The factorization has the form A = S**T*S where S is a band matrix of
the same bandwidth as A and the following structure:
S = ( U )
( M L )
where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U', or
the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB (input/output) REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first kd+1 rows of the array. The j-th
column of A is stored in the j-th column of the array AB as
follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,jkd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
j<=i<=min(n,j+kd).
On exit, if INFO = 0, the factor S from the split Cholesky
factorization A = S**T*S. See Further Details. LDAB (input)
INTEGER The leading dimension of the array AB. LDAB >= KD+1.
Page 1
SPBSTF(3F) SPBSTF(3F)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the factorization could not be completed,
because the updated element a(i,i) was negative; the matrix A is
not positive definite.
FURTHER DETAILS
The band storage scheme is illustrated by the following example, when N =
7, KD = 2:
S = ( s11 s12 s13 )
( s22 s23 s24 )
( s33 s34 )
( s44 )
( s53 s54 s55 )
( s64 s65 s66 )
( s75 s76 s77 )
If UPLO = 'U', the array AB holds:
on entry: on exit:
* * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76 a11
a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
If UPLO = 'L', the array AB holds:
on entry: on exit:
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 a21
a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 * a31 a42
a53 a64 a64 * * s13 s24 s53 s64 s75 * *
Array elements marked * are not used by the routine.
SPBSTF(3F) SPBSTF(3F)
SPBSTF - compute a split Cholesky factorization of a real symmetric
positive definite band matrix A
SUBROUTINE SPBSTF( UPLO, N, KD, AB, LDAB, INFO )
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
REAL AB( LDAB, * )
SPBSTF computes a split Cholesky factorization of a real symmetric
positive definite band matrix A.
This routine is designed to be used in conjunction with SSBGST.
The factorization has the form A = S**T*S where S is a band matrix of
the same bandwidth as A and the following structure:
S = ( U )
( M L )
where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U', or
the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB (input/output) REAL array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first kd+1 rows of the array. The j-th
column of A is stored in the j-th column of the array AB as
follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,jkd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for
j<=i<=min(n,j+kd).
On exit, if INFO = 0, the factor S from the split Cholesky
factorization A = S**T*S. See Further Details. LDAB (input)
INTEGER The leading dimension of the array AB. LDAB >= KD+1.
Page 1
SPBSTF(3F) SPBSTF(3F)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the factorization could not be completed,
because the updated element a(i,i) was negative; the matrix A is
not positive definite.
FURTHER DETAILS
The band storage scheme is illustrated by the following example, when N =
7, KD = 2:
S = ( s11 s12 s13 )
( s22 s23 s24 )
( s33 s34 )
( s44 )
( s53 s54 s55 )
( s64 s65 s66 )
( s75 s76 s77 )
If UPLO = 'U', the array AB holds:
on entry: on exit:
* * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76 a11
a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
If UPLO = 'L', the array AB holds:
on entry: on exit:
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 a21
a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 * a31 a42
a53 a64 a64 * * s13 s24 s53 s64 s75 * *
Array elements marked * are not used by the routine.
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