Nullspace

echelonNullspace(S, side, tol)[source]

Returns the Echelon form of the: left nullspace, LS=[L0 I]PS=0 or right nullspace SL=S([L0 I]P)T=0.

USAGE

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

INPUT

Sm 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]PS=0.

  • Pm 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.

  • Pn 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)[source]

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

Sm x n stoichiometric matrix

OPTIONAL INPUT

printLevel – {0, (1)}, 0 means quiet

OUTPUT
  • Z – (right) null space of S , S*Z = 0 (when mn, otherwise [])

  • rankS – scalar giving rank of S

nullSpaceOperator(S, scale, printLevel)[source]

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 (k1), W = nullSpaceOperatorApply(nullS, V); computes an n x k sparse matrix W from V such that SW=0.

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

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

USAGE

nullS = nullSpaceOperator (S, scale, printLevel)

INPUTS
  • Sm 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)[source]

Computes a sparse matrix W from V such that SW=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 (k1), W = nullSpaceOperatorApply(nullS, V); computes an n x k sparse matrix W from V such that SW=0.

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

nullspaceLUSOLapply2Modes(mode, m, n, V, nullS)[source]

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
  • modemode=1 returns W=ZV, mode = 2 returns W=ZTV

  • 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