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// Copyright ©2017 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package gonum
import (
"math"
"gonum.org/v1/gonum/blas"
"gonum.org/v1/gonum/blas/blas64"
)
// Dpbtf2 computes the Cholesky factorization of a symmetric positive banded
// matrix ab. The matrix ab is n×n with kd diagonal bands. The Cholesky
// factorization computed is
//
// A = Uᵀ * U if ul == blas.Upper
// A = L * Lᵀ if ul == blas.Lower
//
// ul also specifies the storage of ab. If ul == blas.Upper, then
// ab is stored as an upper-triangular banded matrix with kd super-diagonals,
// and if ul == blas.Lower, ab is stored as a lower-triangular banded matrix
// with kd sub-diagonals. On exit, the banded matrix U or L is stored in-place
// into ab depending on the value of ul. Dpbtf2 returns whether the factorization
// was successfully completed.
//
// The band storage scheme is illustrated below when n = 6, and kd = 2.
// The resulting Cholesky decomposition is stored in the same elements as the
// input band matrix (a11 becomes u11 or l11, etc.).
//
// ul = blas.Upper
// a11 a12 a13
// a22 a23 a24
// a33 a34 a35
// a44 a45 a46
// a55 a56 *
// a66 * *
//
// ul = blas.Lower
// * * a11
// * a21 a22
// a31 a32 a33
// a42 a43 a44
// a53 a54 a55
// a64 a65 a66
//
// Dpbtf2 is the unblocked version of the algorithm, see Dpbtrf for the blocked
// version.
//
// Dpbtf2 is an internal routine, exported for testing purposes.
func (Implementation) Dpbtf2(uplo blas.Uplo, n, kd int, ab []float64, ldab int) (ok bool) {
switch {
case uplo != blas.Upper && uplo != blas.Lower:
panic(badUplo)
case n < 0:
panic(nLT0)
case kd < 0:
panic(kdLT0)
case ldab < kd+1:
panic(badLdA)
}
// Quick return if possible.
if n == 0 {
return true
}
if len(ab) < (n-1)*ldab+kd+1 {
panic(shortAB)
}
bi := blas64.Implementation()
kld := max(1, ldab-1)
if uplo == blas.Upper {
// Compute the Cholesky factorization A = Uᵀ * U.
for j := 0; j < n; j++ {
// Compute U(j,j) and test for non-positive-definiteness.
ajj := ab[j*ldab]
if ajj <= 0 {
return false
}
ajj = math.Sqrt(ajj)
ab[j*ldab] = ajj
// Compute elements j+1:j+kn of row j and update the trailing submatrix
// within the band.
kn := min(kd, n-j-1)
if kn > 0 {
bi.Dscal(kn, 1/ajj, ab[j*ldab+1:], 1)
bi.Dsyr(blas.Upper, kn, -1, ab[j*ldab+1:], 1, ab[(j+1)*ldab:], kld)
}
}
return true
}
// Compute the Cholesky factorization A = L * Lᵀ.
for j := 0; j < n; j++ {
// Compute L(j,j) and test for non-positive-definiteness.
ajj := ab[j*ldab+kd]
if ajj <= 0 {
return false
}
ajj = math.Sqrt(ajj)
ab[j*ldab+kd] = ajj
// Compute elements j+1:j+kn of column j and update the trailing submatrix
// within the band.
kn := min(kd, n-j-1)
if kn > 0 {
bi.Dscal(kn, 1/ajj, ab[(j+1)*ldab+kd-1:], kld)
bi.Dsyr(blas.Lower, kn, -1, ab[(j+1)*ldab+kd-1:], kld, ab[(j+1)*ldab+kd:], kld)
}
}
return true
}
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