// replaceBland uses the Bland rule to find the indices to swap if the minimum
// move is 0. The indices to be swapped are replace and minIdx (following the
// nomenclature in the main routine).
func replaceBland(A mat64.Matrix, ab *mat64.Dense, xb []float64, basicIdxs, nonBasicIdx []int, r, move []float64) (replace, minIdx int, err error) {
// Use the traditional bland rule, except don't replace a constraint which
// causes the new ab to be singular.
for i, v := range r {
if v > -blandNegTol {
continue
}
minIdx = i
err = computeMove(move, minIdx, A, ab, xb, nonBasicIdx)
if err != nil {
// Either unbounded or something went wrong.
return -1, -1, err
}
replace = floats.MinIdx(move)
if math.Abs(move[replace]) > blandZeroTol {
// Large enough that it shouldn't be a problem
return replace, minIdx, nil
}
// Find a zero index where replacement is non-singular.
biCopy := make([]int, len(basicIdxs))
for replace, v := range move {
if v > blandZeroTol {
continue
}
copy(biCopy, basicIdxs)
biCopy[replace] = nonBasicIdx[minIdx]
abTmp := extractColumns(A, biCopy)
// If the condition number is reasonable, use this index.
if mat64.Cond(abTmp, 1) < 1e16 {
return replace, minIdx, nil
}
}
}
return -1, -1, ErrBland
}
// linearlyDependent returns whether the vector is linearly dependent
// with the columns of A. It assumes that A is a full-rank matrix.
func linearlyDependent(vec *mat64.Vector, A mat64.Matrix) bool {
// Add vec to the columns of A, and see if the condition number is reasonable.
m, n := A.Dims()
aNew := mat64.NewDense(m, n+1, nil)
aNew.Copy(A)
col := mat64.Col(nil, 0, vec)
aNew.SetCol(n, col)
cond := mat64.Cond(aNew, 1)
return cond > 1e12
}