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Suppose I have a matrix valued function

$$ F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T $$ where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, $\tilde R\in\mathbb{R}^{m\times n}$ upper triangular, $u_1,u_2\in\mathbb{R}^m$ and $v_1,v_2\in\mathbb{R}^n$.

Is there anything that can be said about the QR decomposition of $F(x)=Q(x)R(x)$ depending on $x$?

To give a bit more background: I would like to minimize $$ g(x)=||Q^T_2(x)z||^2_2 $$ for some vector $z$ and $$ F(x)=Q(x)R(x)=\begin{bmatrix}Q_1(x)& Q_2(x)\end{bmatrix}\begin{bmatrix}R_1(x)\\0\end{bmatrix}=Q_1(x)R_1(x). $$

I plotted the function $g$ for a few different cases, and it always looks similar to this:

I have been wrapping my mind around the following two questions:

• Is it just coincidence that I see exactly one local minimum and one local maximum, or might that be proven?
• Might it even be possible to give a direct algorithm that finds the minimum of this function?

It is not to difficult to employ a nonlinear optimizer to find the minimum, however, in that case I would like the guarantee that I in fact only have one local minimum and that my optimizer in case it does not diverge is ensured to find the global optimum.

What I have tried: There are algorithms for updating a QR decomposition with rank 1 matrices, e.g. by Daniel, Gragg, Kaufman and Stewart. I tried to follow those steps symbolically but using a series of Givens Rotations to ensure triangularity of a the matrix $R(x)$ quickly leads to terms that I found not to be good to handle. However, maybe I am just missing a good idea for a clear notation or do not see the system behind.

Any help (even if it is just a pointer to a paper that does something similar) is greatly appreciated.

Suppose I have a matrix valued function

$$ F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T $$ where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, $\tilde R\in\mathbb{R}^{m\times n}$ upper triangular, $u_1,u_2\in\mathbb{R}^m$ and $v_1,v_2\in\mathbb{R}^n$.

Is there anything that can be said about the QR decomposition of $F(x)=Q(x)R(x)$ depending on $x$?

To give a bit more background: I would like to minimize $$ g(x)=||Q^T_2(x)z||^2_2 $$ for some vector $z \in \mathbb{R}^{m} $ and $$ F(x)=Q(x)R(x)=\begin{bmatrix}Q_1(x)& Q_2(x)\end{bmatrix}\begin{bmatrix}R_1(x)\\0\end{bmatrix}=Q_1(x)R_1(x). $$

I plotted the function $g$ for a few different cases, and it always looks similar to this:

I have been wrapping my mind around the following two questions:

• Is it just coincidence that I see exactly one local minimum and one local maximum, or might that be proven?
• Might it even be possible to give a direct algorithm that finds the minimum of this function?

It is not to difficult to employ a nonlinear optimizer to find the minimum, however, in that case I would like the guarantee that I in fact only have one local minimum and that my optimizer in case it does not diverge is ensured to find the global optimum.

What I have tried: There are algorithms for updating a QR decomposition with rank 1 matrices, e.g. by Daniel, Gragg, Kaufman and Stewart. I tried to follow those steps symbolically but using a series of Givens Rotations to ensure triangularity of a the matrix $R(x)$ quickly leads to terms that I found not to be good to handle. However, maybe I am just missing a good idea for a clear notation or do not see the system behind.

Any help (even if it is just a pointer to a paper that does something similar) is greatly appreciated.

Suppose I have a matrix valued function

$$ F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T $$ where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, $\tilde R\in\mathbb{R}^{m\times n}$ upper triangular, $u_1,u_2\in\mathbb{R}^m$ and $v_1,v_2\in\mathbb{R}^n$.

Is there anything that can be said about the QR decomposition of $F(x)=Q(x)R(x)$ depending on $x$?

To give a bit more background: I would like to minimize $$ g(x)=||Q_2(x)z||^2_2 $$ for some vector $z$ and $$ F(x)=Q(x)R(x)=\begin{bmatrix}Q_1(x)& Q_2(x)\end{bmatrix}\begin{bmatrix}R_1(x)\\0\end{bmatrix}=Q_1(x)R_1(x). $$

I plotted the function $g$ for a few different cases, and it always looks similar to this:

I have been wrapping my mind around the following two questions:

• Is it just coincidence that I see exactly one local minimum and one local maximum, or might that be proven?
• Might it even be possible to give a direct algorithm that finds the minimum of this function?

It is not to difficult to employ a nonlinear optimizer to find the minimum, however, in that case I would like the guarantee that I in fact only have one local minimum and that my optimizer in case it does not diverge is ensured to find the global optimum.

What I have tried: There are algorithms for updating a QR decomposition with rank 1 matrices, e.g. by Daniel, Gragg, Kaufman and Stewart. I tried to follow those steps symbolically but using a series of Givens Rotations to ensure triangularity of a the matrix $R(x)$ quickly leads to terms that I found not to be good to handle. However, maybe I am just missing a good idea for a clear notation or do not see the system behind.

Any help (even if it is just a pointer to a paper that does something similar) is greatly appreciated.

Suppose I have a matrix valued function

$$ F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T $$ where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, $\tilde R\in\mathbb{R}^{m\times n}$ upper triangular, $u_1,u_2\in\mathbb{R}^m$ and $v_1,v_2\in\mathbb{R}^n$.

Is there anything that can be said about the QR decomposition of $F(x)=Q(x)R(x)$ depending on $x$?

To give a bit more background: I would like to minimize $$ g(x)=||Q^T_2(x)z||^2_2 $$ for some vector $z$ and $$ F(x)=Q(x)R(x)=\begin{bmatrix}Q_1(x)& Q_2(x)\end{bmatrix}\begin{bmatrix}R_1(x)\\0\end{bmatrix}=Q_1(x)R_1(x). $$

I plotted the function $g$ for a few different cases, and it always looks similar to this:

I have been wrapping my mind around the following two questions:

• Is it just coincidence that I see exactly one local minimum and one local maximum, or might that be proven?
• Might it even be possible to give a direct algorithm that finds the minimum of this function?

It is not to difficult to employ a nonlinear optimizer to find the minimum, however, in that case I would like the guarantee that I in fact only have one local minimum and that my optimizer in case it does not diverge is ensured to find the global optimum.

What I have tried: There are algorithms for updating a QR decomposition with rank 1 matrices, e.g. by Daniel, Gragg, Kaufman and Stewart. I tried to follow those steps symbolically but using a series of Givens Rotations to ensure triangularity of a the matrix $R(x)$ quickly leads to terms that I found not to be good to handle. However, maybe I am just missing a good idea for a clear notation or do not see the system behind.

Any help (even if it is just a pointer to a paper that does something similar) is greatly appreciated.

Suppose I have a matrix valued function

$$ F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T $$ where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, $\tilde R\in\mathbb{R}^{m\times n}$ upper triangular, $u_1,u_2\in\mathbb{R}^m$ and $v_1,v_2\in\mathbb{R}^n$.

Is there anything that can be said about the QR decomposition of $F(x)=Q(x)R(x)$ depending on $x$?

To give a bit more background: I would like to minimize $$ g(x)=||Q_1(x)z||^2_2 $$ for some vector $z$ and $$ F(x)=Q(x)R(x)=\begin{bmatrix}Q_1(x)& Q_2(x)\end{bmatrix}\begin{bmatrix}R_1(x)\\0\end{bmatrix}=Q_1(x)R_1(x). $$

I plotted the function $g$ for a few different cases, and it always looks similar to this:

I have been wrapping my mind around the following two questions:

• Is it just coincidence that I see exactly one local minimum and one local maximum, or might that be proven?
• Might it even be possible to give a direct algorithm that finds the minimum of this function?

It is not to difficult to employ a nonlinear optimizer to find the minimum, however, in that case I would like the guarantee that I in fact only have one local minimum and that my optimizer in case it does not diverge is ensured to find the global optimum.

What I have tried: There are algorithms for updating a QR decomposition with rank 1 matrices, e.g. by Daniel, Gragg, Kaufman and Stewart. I tried to follow those steps symbolically but using a series of Givens Rotations to ensure triangularity of a the matrix $R(x)$ quickly leads to terms that I found not to be good to handle. However, maybe I am just missing a good idea for a clear notation or do not see the system behind.

Any help (even if it is just a pointer to a paper that does something similar) is greatly appreciated.

Suppose I have a matrix valued function

$$ F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T $$ where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, $\tilde R\in\mathbb{R}^{m\times n}$ upper triangular, $u_1,u_2\in\mathbb{R}^m$ and $v_1,v_2\in\mathbb{R}^n$.

Is there anything that can be said about the QR decomposition of $F(x)=Q(x)R(x)$ depending on $x$?

To give a bit more background: I would like to minimize $$ g(x)=||Q_2(x)z||^2_2 $$ for some vector $z$ and $$ F(x)=Q(x)R(x)=\begin{bmatrix}Q_1(x)& Q_2(x)\end{bmatrix}\begin{bmatrix}R_1(x)\\0\end{bmatrix}=Q_1(x)R_1(x). $$

I plotted the function $g$ for a few different cases, and it always looks similar to this:

I have been wrapping my mind around the following two questions:

• Is it just coincidence that I see exactly one local minimum and one local maximum, or might that be proven?
• Might it even be possible to give a direct algorithm that finds the minimum of this function?

It is not to difficult to employ a nonlinear optimizer to find the minimum, however, in that case I would like the guarantee that I in fact only have one local minimum and that my optimizer in case it does not diverge is ensured to find the global optimum.

What I have tried: There are algorithms for updating a QR decomposition with rank 1 matrices, e.g. by Daniel, Gragg, Kaufman and Stewart. I tried to follow those steps symbolically but using a series of Givens Rotations to ensure triangularity of a the matrix $R(x)$ quickly leads to terms that I found not to be good to handle. However, maybe I am just missing a good idea for a clear notation or do not see the system behind.

Any help (even if it is just a pointer to a paper that does something similar) is greatly appreciated.