fMultiplext – IQClab

The function $[P_\mathrm{e},Q_\mathrm{e},S_\mathrm{e},R_\mathrm{e}]=fMultiplext(P_\mathrm{p}, P_\mathrm{d},n_\mathrm{pos} ,n_\mathrm{neg},type)$ constructs the extended full-block multiplier

$P_\mathrm{e}=\left(\begin{array}{cc}Q_\mathrm{e}&S_\mathrm{e}\\ S_\mathrm{e}^T&R_\mathrm{e}\end{array}\right).$

Here the inputs should satisfy the following constraints:

$P_\mathrm{p}=\left(\begin{array}{cc}Q_\mathrm{p}&S_\mathrm{p}\\S_\mathrm{p}^T &R_\mathrm{p}\end{array}\right)$ with $Q_\mathrm{p}\succ0$ and $R_\mathrm{p}\prec0$
$P_\mathrm{d}=\left(\begin{array}{cc}Q_\mathrm{d}&S_\mathrm{d}\\S_\mathrm{d}^T &R_\mathrm{d}\end{array}\right)$ with $Q_\mathrm{d}\succ0$ and $R_\mathrm{d}\prec0$
$n_\mathrm{pos}$ denotes the number of positive eigenvalues of $P_\mathrm{p}$ (this equals the size of $Q_\mathrm{p}$ )
$n_\mathrm{neq}$ denotes the number of negative eigenvalues of $P_\mathrm{p}$ (this equals the size of $R_\mathrm{p}$ )
specifies the type of extention:
- $type\in\{1,2,3\}$ :
  $P_\mathrm{e}=\left(\begin{array}{cc}Q_\mathrm{e}&S_\mathrm{e}\\ S_\mathrm{e}^T&R_\mathrm{e}\end{array}\right)$
  is such that
  $P_\mathrm{e}^{-1}=\left(\begin{array}{cccc}Q_\mathrm{e}&\star&S_\mathrm{e}&\star\\ \star&\star&\star&\star\\ S_\mathrm{e}^T&\star&R_\mathrm{e}&\star\\ \star&\star&\star&\star\end{array}\right)$
- $type\in\{4,5,6\}$ :
  $P_\mathrm{e}=\left(\begin{array}{cc}Q_\mathrm{d}&S_\mathrm{d}\\ S_\mathrm{d}^T&R_\mathrm{d}\end{array}\right)$
  is such that
  $P_\mathrm{e}^{-1}=\left(\begin{array}{cccc}Q_\mathrm{di}&\star&S_\mathrm{di}&\star\\ \star&\star&\star&\star\\ S_\mathrm{di}^T&\star&R_\mathrm{di}&\star\\ \star&\star&\star&\star\end{array}\right)$
  with
  $P_\mathrm{d}^{-1}=\left(\begin{array}{cc}Q_\mathrm{d}&S_\mathrm{d}\\ S_\mathrm{d}^T&R_\mathrm{d}\end{array}\right)^{-1}=\left(\begin{array}{cc}Q_\mathrm{di}&S_\mathrm{di}\\ S_\mathrm{di}^T&R_\mathrm{di}\end{array}\right)$

Let $J=\left(\begin{array}{cc}I_{n_\mathrm{pos}}&0_{n_\mathrm{pos}\times n_\mathrm{neq}}\end{array}\right)^T$ and $\tilde{J}=\left(\begin{array}{cc}0_{n_\mathrm{neg}\times n_\mathrm{pos}}&I_{n_\mathrm{neg}}\end{array}\right)^T$ . Then the different types are defined as follows:

P_\mathrm{i}=(P_\mathrm{p}- P_\mathrm{d}^{-1})^{-1}

T=\left(\begin{array}{cc}T_1&T_2\end{array}\right)

Type	Description
type=1	Define $P_\mathrm{i}=(P_\mathrm{p}- P_\mathrm{d}^{-1})^{-1}$ and let $T=\left(\begin{array}{cc}T_1&T_2\end{array}\right)$ where $T_1$ is the collection of eigenvectors corresponding to the positive eigenvalues of $\mathrm{eig}(P_\mathrm{i}-J(J^TP_\mathrm{p}J)^{-1}J^T)$ and $T_2$ is the collection of eigenvectors corresponding to the negative eigenvalues of $\mathrm{eig}(P_\mathrm{i}-\tilde{J}( \tilde{J}^T P_\mathrm{p} \tilde{J})^{-1}\tilde{J}^T)$ . Then $P_\mathrm{e}=\left(\begin{array}{cc}Q_\mathrm{e}&S_\mathrm{e}\\ S_\mathrm{e}^T&R_\mathrm{e}\end{array}\right)= \left(\begin{array}{cc} P_\mathrm{p}&T\\T^T&T^TP_\mathrm{i}T\end{array}\right)$ .
type=2	Define $P_\mathrm{i}=(P_\mathrm{p}-P_\mathrm{d}^{-1})^{-1}$ and let $T=\left(\begin{array}{cc}T_1&T_2\end{array}\right)$ where $T_1$ is the collection of eigenvectors corresponding to the positive eigenvalues of $\mathrm{eig}(P_\mathrm{i}-P_\mathrm{i}J(J^TP_\mathrm{p}J)^{-1}J^T P_\mathrm{i})$ and $T_2$ is the collection of eigenvectors corresponding to the negative eigenvalues of $\mathrm{eig}(P_\mathrm{i}- P_\mathrm{i}\tilde{J}( \tilde{J}^T P_\mathrm{p} \tilde{J})^{-1}\tilde{J}^TP_\mathrm{i})$ . Then $P_\mathrm{e}=\left(\begin{array}{cc}Q_\mathrm{e}&S_\mathrm{e}\\ S_\mathrm{e}^T&R_\mathrm{e}\end{array}\right)= \left(\begin{array}{cc} P_\mathrm{p}& P_\mathrm{i}T\\T^T P_\mathrm{i} &T^TP_\mathrm{i}T\end{array}\right)$ .
type=3	– Factorize $P_\mathrm{p}=M_\mathrm{p}^TJ_\mathrm{p}M_\mathrm{p}$ and $P_\mathrm{d}=M_\mathrm{d}^TJ_\mathrm{d}M_\mathrm{d}$ with $J_\mathrm{p}=\mathrm{diag}(I,-I)$ and $J_\mathrm{d}=\mathrm{diag}(I,-I)$ respectively and define $M_\mathrm{pi}=M_\mathrm{p}^{-1}$ and $M_\mathrm{di}= M_\mathrm{d}^{-1}$ – Define $U\Sigma V^T=M_\mathrm{di}^TM_\mathrm{pi}-J_\mathrm{d} M_\mathrm{d}M_\mathrm{p}^TJ_\mathrm{p}$ with $U\Sigma V^T$ begin the singular value decomposition. Also denote by $S_\mathrm{d}=U\Sigma^{\frac{1}{2}}$ and $S_\mathrm{p}=C\Sigma^{\frac{1}{2}}$ – Let $T=\left(\begin{array}{cc}T_1&T_2\end{array}\right)$ where $T_1$ and $T_2$ respectively denote the collection of eigenvectors corresponding to the positive and negative eigenvalues of $\mathrm{eig}(-S_\mathrm{d}^{-1}J_\mathrm{d}M_\mathrm{d}M_\mathrm{p} ^T S_\mathrm{p} -S_\mathrm{p}^TM_\mathrm{p}J(J^TP_\mathrm{p}J)^{-1}J^T M_\mathrm{p}^TS_\mathrm{p})$ $\mathrm{eig}(-S_\mathrm{d}^{-1}J_\mathrm{d}M_\mathrm{d}M_\mathrm{p} ^T S_\mathrm{p} -S_\mathrm{p}^TM_\mathrm{p}\tilde{J}( \tilde{J}^T P_\mathrm{p} \tilde{J} )^{-1} \tilde{J}^TM_\mathrm{p}^TS_\mathrm{p})$ Then $P_\mathrm{e}=\left(\begin{array}{cc}Q_\mathrm{e}&S_\mathrm{e}\\ S_\mathrm{e}^T&R_\mathrm{e}\end{array}\right)=$ $=\left(\begin{array}{cc}M_\mathrm{pi}^T&0\\ J_\mathrm{d}M_\mathrm{d} &S_\mathrm{d}T^{-T}\end{array}\right)^{-1}\left(\begin{array}{cc} J_\mathrm{p} M_\mathrm{p}&S_\mathrm{p}T\\ M_\mathrm{di} &0\end{array}\right)$
type=4	Define $P_\mathrm{i}=(P_\mathrm{p}- P_\mathrm{d})^{-1}$ and let $T=\left(\begin{array}{cc}T_1&T_2\end{array}\right)$ where $T_1$ is the collection of eigenvectors corresponding to the positive eigenvalues of $\mathrm{eig}(P_\mathrm{i}-J(J^TP_\mathrm{p}J)^{-1}J^T)$ and $T_2$ is the collection of eigenvectors corresponding to the negative eigenvalues of $\mathrm{eig}(P_\mathrm{i}-\tilde{J}( \tilde{J}^T P_\mathrm{p} \tilde{J})^{-1}\tilde{J}^T)$ . Then $P_\mathrm{e}=\left(\begin{array}{cc}Q_\mathrm{e}&S_\mathrm{e}\\ S_\mathrm{e}^T&R_\mathrm{e}\end{array}\right)= \left(\begin{array}{cc} P_\mathrm{p}&T\\T^T&T^TP_\mathrm{i}T\end{array}\right)$ .
type=5	Define $P_\mathrm{i}=(P_\mathrm{p}-P_\mathrm{d})^{-1}$ and let $T=\left(\begin{array}{cc}T_1&T_2\end{array}\right)$ where $T_1$ is the collection of eigenvectors corresponding to the positive eigenvalues of $\mathrm{eig}(P_\mathrm{i}-P_\mathrm{i}J(J^TP_\mathrm{p}J)^{-1}J^T P_\mathrm{i})$ and $T_2$ is the collection of eigenvectors corresponding to the negative eigenvalues of $\mathrm{eig}(P_\mathrm{i}- P_\mathrm{i}\tilde{J}( \tilde{J}^T P_\mathrm{p} \tilde{J})^{-1}\tilde{J}^TP_\mathrm{i})$ . Then $P_\mathrm{e}=\left(\begin{array}{cc}Q_\mathrm{e}&S_\mathrm{e}\\ S_\mathrm{e}^T&R_\mathrm{e}\end{array}\right)= \left(\begin{array}{cc} P_\mathrm{p}& P_\mathrm{i}T\\T^T P_\mathrm{i} &T^TP_\mathrm{i}T\end{array}\right)$ .
type=6	– Factorize $P_\mathrm{p}=M_\mathrm{p}^TJ_\mathrm{p}M_\mathrm{p}$ and $P_\mathrm{d}=M_\mathrm{d}^TJ_\mathrm{d}M_\mathrm{d}$ with $J_\mathrm{p}=\mathrm{diag}(I,-I)$ and $J_\mathrm{d}=\mathrm{diag}(I,-I)$ respectively and define $M_\mathrm{pi}=M_\mathrm{p}^{-1}$ and $M_\mathrm{di}= M_\mathrm{d}^{-1}$ – Define $U\Sigma V^T=M_\mathrm{d}^TM_\mathrm{pi}-J_\mathrm{d} M_\mathrm{di}M_\mathrm{p}^TJ_\mathrm{p}$ with $U\Sigma V^T$ begin the singular value decomposition. Also denote by $S_\mathrm{d}=U\Sigma^{\frac{1}{2}}$ and $S_\mathrm{p}=C\Sigma^{\frac{1}{2}}$ – Let $T=\left(\begin{array}{cc}T_1&T_2\end{array}\right)$ where $T_1$ and $T_2$ respectively denote the collection of eigenvectors corresponding to the positive and negative eigenvalues of $\mathrm{eig}(-S_\mathrm{d}^{-1}J_\mathrm{d}M_\mathrm{d}M_\mathrm{p}^T S_\mathrm{p} -S_\mathrm{p}^TM_\mathrm{p}J(J^TP_\mathrm{p}J)^{-1}J^T M_\mathrm{p}^TS_\mathrm{p})$ $\mathrm{eig}(-S_\mathrm{d}^{-1}J_\mathrm{d}M_\mathrm{d}M_\mathrm{p} ^T S_\mathrm{p} -S_\mathrm{p}^TM_\mathrm{p}\tilde{J}( \tilde{J}^T P_\mathrm{p} \tilde{J} )^{-1} \tilde{J}^TM_\mathrm{p}^TS_\mathrm{p})$ Then $P_\mathrm{e}=\left(\begin{array}{cc}Q_\mathrm{e}&S_\mathrm{e}\\ S_\mathrm{e}^T&R_\mathrm{e}\end{array}\right)=$ $=\left(\begin{array}{cc}M_\mathrm{pi}^T&0\\ J_\mathrm{d}M_\mathrm{di}^T &S_\mathrm{d}T^{-T}\end{array}\right)^{-1}\left(\begin{array}{cc} J_\mathrm{p} M_\mathrm{p}&S_\mathrm{p}T\\M_\mathrm{d} &0\end{array}\right)$