Nullspace

echelonNullspace(S, side, tol)

Returns the Echelon form of the: left nullspace, \(L S = [-L0\ I] P S = 0\) or right nullspace \(S L = S ([-L0\ I] P)^T = 0\).

USAGE:

[L0, I, P, L, p] = echelonNullspace(S, side, tol)

INPUT:

S: m x n stoichiometric matrix

OPTIONAL INPUTS:

side: {‘left’, ‘right’} left or right nullspace, left by default. tol: upper bound on tolerance of linear independence

default no greater than 1e-12

OUTPUTS:

if side == left L0: (m-r) x r matrix which forms the non-trivial part of the

left nullspace in echelon form i.e. \([-L0\ I] P S = 0\).

P: m x m row (permutation matrix I: (m-r) x (m-r) identity matrix p: row permutation which leaves first 1:rankA rows independent and

last rows dependent

if side == right L0: (n-r) x r matrix which forms the non-trivial part of the

right nullspace in echelon form i.e. \(S ([-L0\ I] P)^T = 0\).

P: n x n column permutaion matrix I: (n-r) x (n-r) identity matrix p: column permutation which leaves first 1:rankA columns

independent and last columns dependent

See: Conservation analysis of large biochemical networks Ravishankar Rao Vallabhajosyula , Vijay Chickarmane and Herbert M. Sauro

getNullSpace(S, printLevel)

Calculates the nullspace of S for full(S) or \(rank(S) == m\). If full row rank i.e. rank(S) = m, it’s much faster to work with a sparse LU.

USAGE:

[Z, rankS] = getNullSpace(S, printLevel)

INPUT:

S: m x n stoichiometric matrix

OPTIONAL INPUT:

printLevel: {0, (1)}, 0 means quiet

OUTPUT:

Z: (right) null space of S , S*Z = 0 (when \(m \leq n\), otherwise []) rankS: scalar giving rank of S

nullSpaceOperator(S, scale, printLevel)

Uses LUSOL to compute a nullspace operator nullS We assume S is m x n with \(m < n\) with rank r. First nullS = nullSpaceOperator(S) computes a structure nullS from an m x n sparse matrix S (\(m < n\)). Second, if V is an (n-r) x k sparse matrix \((k \geq 1)\), W = nullSpaceOperatorApply(nullS, V); computes an n x k sparse matrix W from V such that \(S W = 0\).

This is an operator form of finding an n x (n-r) matrix Z such that \(S Z = 0\) and then computing \(W = Z V\). The aim is to obtain W without forming Z explicitly.

nullS.rank returns the rank of S \((r \leq m)\). It doesn’t matter if \(rank < m\).

USAGE:

nullS = nullSpaceOperator(S, scale, printLevel)

INPUTS:

S: m x n matrix scale: {(1), 0} geometric mean scaling of S printLevel: {(1), 0}

OUTPUT:

nullS: nullspace operator to be used with nullSpaceOperatorApply.m

• nullS.rank - rank of S

Note

Requires Nick Henderson’s 64 bit LUSOL interface to be intalled and added to the matlab path. See https://github.com/nwh/lusol see also https://web.stanford.edu/group/SOL/software/lusol/

nullSpaceOperatorApply(nullS, V)

Computes a sparse matrix W from V such that \(S W = 0\). First, nullS = nullSpaceOperator(S) computes a structure nullS from an m x n sparse matrix S \((m < n)\), with rank r.

Second, if V is an (n-r) x k sparse matrix \((k \geq 1)\), W = nullSpaceOperatorApply(nullS, V); computes an n x k sparse matrix W from V such that \(S W = 0\).

This is an operator form of finding an n x (n-r) matrix Z such that \(S Z = 0\) and then computing \(W = Z V\). The aim is to obtain W without forming Z explicitly.

nullspaceLUSOLapply2Modes(mode, m, n, V, nullS)

Computes the matrix vector product with the operator nullspace function handle of the form y = pdMat(mode, m, n, x)

USAGE:

W = nullspaceLUSOLapply2Modes(mode, m, n, V, nullS)

INPUTS:

mode: \(mode = 1\) returns \(W = Z V\), mode = 2 returns \(W = Z^T V\) m: first dimension of the matrix n: second dimension of the matrix V: one of the components of the multiplication nullS: structure nullS from the function nullspaceLUSOLform(S);

where m x n sparse matrix S (\(m < n\)).

OUTPUT:

W: Matrix vector product