Nullspace

• Remote

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