Decomposition Operations<TReal, TComplex>.Band Cholesky Decompose Method
Definition
Assembly: Numerics.NET (in Numerics.NET.dll) Version: 9.0.3
Overload List
Band | Computes the Cholesky factorization of a real symmetric positive definite band matrix A. |
Band | Computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. |
Band | Computes the Cholesky factorization of a real symmetric positive definite band matrix A. |
Band | Computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. |
Band | Computes the Cholesky factorization of a real symmetric positive definite band matrix A. |
Band | Computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. |
BandCholeskyDecompose(MatrixTriangle, Int32, Int32, Array2D<TReal>, Int32)
Computes the Cholesky factorization of a real symmetric positive definite band matrix A.
public void BandCholeskyDecompose(
MatrixTriangle storedTriangle,
int n,
int kd,
Array2D<TReal> ab,
out int info
)
Parameters
- storedTriangle MatrixTriangle
= 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
- n Int32
The order of the matrix A. N >= 0.
- kd Int32
The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
- ab Array2D<TReal>
AB is TReal 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,j-kd)<=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 triangular factor U or L from the Cholesky factorization A = UT*U or A = L*LT of the band matrix A, in the same storage format as A.
The leading dimension of the array AB. LDAB >= KD+1.
- info Int32
= 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.
Remarks
The factorization has the form A = UT * U, if UPLO = 'U', or A = L * LT, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular.
Further Details:
The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U': On entry: On exit: * * a13 a24 a35 a46 * * u13 u24 u35 u46 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 Similarly, if UPLO = 'L' the format of A is as follows: On entry: On exit: a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * a31 a42 a53 a64 * * l31 l42 l53 l64 * * Array elements marked * are not used by the routine.
This method corresponds to the LAPACK routine ?PBTRF.
BandCholeskyDecompose(MatrixTriangle, Int32, Int32, Array2D<TComplex>, Int32)
Computes the Cholesky factorization of a complex Hermitian positive definite band matrix A.
public void BandCholeskyDecompose(
MatrixTriangle uplo,
int n,
int kd,
Array2D<TComplex> ab,
out int info
)
Parameters
- uplo MatrixTriangle
= 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
- n Int32
The order of the matrix A. N >= 0.
- kd Int32
The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
- ab Array2D<TComplex>
AB is TComplex array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian 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,j-kd)<=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 triangular factor U or L from the Cholesky factorization A = UH*U or A = L*LH of the band matrix A, in the same storage format as A.
The leading dimension of the array AB. LDAB >= KD+1.
- info Int32
= 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.
Remarks
The factorization has the form A = UH * U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular.
Further Details:
The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U': On entry: On exit: * * a13 a24 a35 a46 * * u13 u24 u35 u46 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 Similarly, if UPLO = 'L' the format of A is as follows: On entry: On exit: a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * a31 a42 a53 a64 * * l31 l42 l53 l64 * * Array elements marked * are not used by the routine.
Contributors:
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
Authors: Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver, NAG Ltd.
Date: November 2011
BandCholeskyDecompose(MatrixTriangle, Int32, Int32, Span2D<TReal>, Int32)
Computes the Cholesky factorization of a real symmetric positive definite band matrix A.
public void BandCholeskyDecompose(
MatrixTriangle storedTriangle,
int n,
int kd,
Span2D<TReal> ab,
out int info
)
Parameters
- storedTriangle MatrixTriangle
= 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
- n Int32
The order of the matrix A. N >= 0.
- kd Int32
The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
- ab Span2D<TReal>
AB is TReal 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,j-kd)<=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 triangular factor U or L from the Cholesky factorization A = UT*U or A = L*LT of the band matrix A, in the same storage format as A.
The leading dimension of the array AB. LDAB >= KD+1.
- info Int32
= 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.
Remarks
The factorization has the form A = UT * U, if UPLO = 'U', or A = L * LT, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular.
Further Details:
The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U': On entry: On exit: * * a13 a24 a35 a46 * * u13 u24 u35 u46 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 Similarly, if UPLO = 'L' the format of A is as follows: On entry: On exit: a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * a31 a42 a53 a64 * * l31 l42 l53 l64 * * Array elements marked * are not used by the routine.
This method corresponds to the LAPACK routine ?PBTRF.
BandCholeskyDecompose(MatrixTriangle, Int32, Int32, Span2D<TComplex>, Int32)
Computes the Cholesky factorization of a complex Hermitian positive definite band matrix A.
public void BandCholeskyDecompose(
MatrixTriangle uplo,
int n,
int kd,
Span2D<TComplex> ab,
out int info
)
Parameters
- uplo MatrixTriangle
= 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
- n Int32
The order of the matrix A. N >= 0.
- kd Int32
The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
- ab Span2D<TComplex>
AB is TComplex array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian 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,j-kd)<=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 triangular factor U or L from the Cholesky factorization A = UH*U or A = L*LH of the band matrix A, in the same storage format as A.
The leading dimension of the array AB. LDAB >= KD+1.
- info Int32
= 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.
Remarks
The factorization has the form A = UH * U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular.
Further Details:
The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U': On entry: On exit: * * a13 a24 a35 a46 * * u13 u24 u35 u46 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 Similarly, if UPLO = 'L' the format of A is as follows: On entry: On exit: a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * a31 a42 a53 a64 * * l31 l42 l53 l64 * * Array elements marked * are not used by the routine.
Contributors:
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
Authors: Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver, NAG Ltd.
Date: November 2011
BandCholeskyDecompose(MatrixTriangle, Int32, Int32, Span<TReal>, Int32, Int32)
Computes the Cholesky factorization of a real symmetric positive definite band matrix A.
public abstract void BandCholeskyDecompose(
MatrixTriangle storedTriangle,
int n,
int kd,
Span<TReal> ab,
int ldab,
out int info
)
Parameters
- storedTriangle MatrixTriangle
= 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
- n Int32
The order of the matrix A. N >= 0.
- kd Int32
The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
- ab Span<TReal>
AB is TReal 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,j-kd)<=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 triangular factor U or L from the Cholesky factorization A = UT*U or A = L*LT of the band matrix A, in the same storage format as A.
- ldab Int32
The leading dimension of the array AB. LDAB >= KD+1.
- info Int32
= 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.
Remarks
The factorization has the form A = UT * U, if UPLO = 'U', or A = L * LT, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular.
Further Details:
The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U': On entry: On exit: * * a13 a24 a35 a46 * * u13 u24 u35 u46 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 Similarly, if UPLO = 'L' the format of A is as follows: On entry: On exit: a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * a31 a42 a53 a64 * * l31 l42 l53 l64 * * Array elements marked * are not used by the routine.
This method corresponds to the LAPACK routine ?PBTRF.
BandCholeskyDecompose(MatrixTriangle, Int32, Int32, Span<TComplex>, Int32, Int32)
Computes the Cholesky factorization of a complex Hermitian positive definite band matrix A.
public abstract void BandCholeskyDecompose(
MatrixTriangle uplo,
int n,
int kd,
Span<TComplex> ab,
int ldab,
out int info
)
Parameters
- uplo MatrixTriangle
= 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
- n Int32
The order of the matrix A. N >= 0.
- kd Int32
The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
- ab Span<TComplex>
AB is TComplex array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian 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,j-kd)<=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 triangular factor U or L from the Cholesky factorization A = UH*U or A = L*LH of the band matrix A, in the same storage format as A.
- ldab Int32
The leading dimension of the array AB. LDAB >= KD+1.
- info Int32
= 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.
Remarks
The factorization has the form A = UH * U, if UPLO = 'U', or A = L * LH, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular.
Further Details:
The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U': On entry: On exit: * * a13 a24 a35 a46 * * u13 u24 u35 u46 * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 Similarly, if UPLO = 'L' the format of A is as follows: On entry: On exit: a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * a31 a42 a53 a64 * * l31 l42 l53 l64 * * Array elements marked * are not used by the routine.
Contributors:
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
Authors: Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver, NAG Ltd.
Date: November 2011