That concludes general Questions. Before we move to First Minister's Questions, I invite members to join me in welcoming to the gallery Abbie Taylor, Presiding Officer of the Tobago House of Assembly. I would also invite Members to join me in welcoming his Excellency Dr Robert Andrejac, Ambassador of Slovakia to the United Kingdom. Now predict that bodyywong cameras will be rolled out next year. Humza Yousaf didn'ti sail them when he was just as Secretary. Will he let police officers down again? First and foremost of course, we have increased the budget for Police Scotland. Please, that道glas Ross seems to have accepted the point that the Scottish Government has increased the budget for Police Scotland by 6.3 per cent and an additional £80 million. cólad ei fod yn sicr mae'n gwyld hwn o'r bau cyffinffiol nawr, Llywydd, asîr, pan fe fydd ni bawb eich fadith gwrs a ph checked ei fod yn ddiwedd iawn, fyrdawch yn y bapddol ni yw talked wid coupledara llw Гwanfaith roedd Feordeд Ffwrdd ynРИwc Romun Gar赤llad iawn, aconst homo'r fadlaid iawn trumpet yn flu gyrfaedd iawn o'r bau heb. Fod oherwydd i ddwygo'r bod y hefyd yn rhan oedd onboard dieseb ddiwedd holech a'r ffacol ar gyfer mae'n cyffinogi wedi'n de 좋아 the budget for Police Scotland, but much like Chief Officer David Page said yesterday, the plans are to roll out our national body worn video cameras for our police officers, who of course are the ones who should be credited for the fact that we have seen the lowest crime rates in Scotland, one of the lowest crime rates in Scotland for almost 50 years. Douglas Ross. The evidence was very clear that next year they will not be fully rolled out. Humza use have said in his programme for government last week that they were a priority, but Police Scotland now say they can't guarantee it because there is a £300 million black hole in the capital budget. The First Minister likes to quote from Police Scotland, so let me quote a submission that they put to this Parliament's Justice Committee for yesterday's session. Our current capital allocation is among the lowest in UK policing. This is significantly short of the funding that is required to improve conditions and equipment for the wellbeing of officers and staff. If this really is a priority for Humza use, will he now treat it like one? First Minister. Let's deal with the facts. The fact is that the UK Government has cut our capital budget by over 6.5 per cent. Here are some more facts for Douglas Ross. The Scottish Government has increased Police Scotland's budget by £80 million this year. Fact of the matter is that we have more police officers per head in Scotland than England. Fact. Of course, when we look at recorded crime levels, they are at one of their lowest in Scotland in almost 50 years. Fact. In fact, recorded crime has fallen by 42 per cent since the SNP came into power. That is a fact. So when it comes to looking at whether or not Scotland is safer under the SNP, there is simply no denying that Scotland is undoubtedly safer under this SNP government. That is a fact. Douglas Ross. Budget are so stretched that the police say that officer redundancies are now on the table. Let's look at facts that the First Minister wants to reiterate. £300 million black hole in Police Scotland's capital budget, fact. Officer numbers in Scotland at their lowest levels in 14 years, fact, First Minister. They started to slip when he was justice secretary and now they are in free fall as he is First Minister. Without further funding, Police Scotland warned yesterday that 2,000 police officers could be let go in the next five years. 600 could be off our streets by April next year. So why is the Government even considering these cuts, which will stretch policing in Scotland to breaking point? First of all, can I just remind Douglas Ross once again that this year Police Scotland has seen a significant increase to their budget?  Let's look at how we are spending that money, not only are we ensuring that we have more police officers on the beat per head than England. If we look at fair pay, we have just concluded a deal with our police officers for a generous and a fair pay increase to them. That means in Scotland that the minimum and the maximum salary is now higher for police officers in all ranks in Scotland compared to their counterparts in England and Wales. When it comes to the budget, we will negotiate and engage with all political parties right across the chamber in relation to the budget. What Douglas Ross cannot do is demand more money that is spent and at the same time demand that we cut taxes. That is simply not a credible position. The First Minister is getting angry again. Either a civil servant has not put this stat in his briefing pack or he is not reading it. Police officer numbers in Scotland under the SNP are now at the lowest level in 14 years. That is a fact, and that is what we are facing on the streets of Scotland right now. Police Scotland will be the last force in the UK to roll out body warm cameras. Officers no longer have the resources to investigate every crime. Their workplaces are not fit for purpose, and now the number of officers could be cut by 2000. The thin blue line is barely visible at this rate, so does the First Minister accept that, with falling officer numbers, communities in Scotland will feel less safe? I do not accept that in the slightest. Around 1,280 new recruits have joined Police Scotland in the last 18 months. Over 350 more police officers than we took office in 2007. When it comes to real-terms increases in Police Scotland's budget, that is a fair enough call for any political party to make. Of course, we will give that consideration when it comes to the budget. Has Douglas Ross ever asked himself why a real-terms increase was difficult this year? Of course, it was difficult because inflation is sky-high because of his party's complete economic mismanagement of the public finances. When it comes to real-terms increases, of course, we will give that consideration. However, it was Douglas Ross a year ago that stood here that demanded the Scottish Government to copy Liz Truss's disastrous many budget. If we had done that, we would not have money to spend on our police service or on any of our public services. Thank goodness that Douglas Ross is nowhere near the public finances of Scotland. 2. Anna Sarwar Last year, the then Health Secretary promised no-one would be waiting more than 18 months for an NHS procedure by September 2023. However, the most recent figures show that there are still 17,200 people on an NHS waiting list who have already waited over 18 months for hospital procedures, yet another broken promise from a health secretary that has failed upwards. The result is people turning to private care. Figures out this week show that 40 per cent of all knee and hip replacements being carried out in Scotland are being self-funded by patients. Why does Humza Yousaf think that it is acceptable that almost 5,000 patients last year were forced to cover the costs of their hip or knee replacements because of his failures? First Minister, first of all, Anna Sarwar of course has every right to ask around the waiting times that our NHS are experiencing, but what I would say to Anna Sarwar is that every time he asks the question, he fails to mention of course the biggest shock our NHS has ever faced in its 75-year existence. Health services right across the UK, including in Scotland, are of course ensuring that we recover. When it comes to the longest waits, again, just to inject some facts into this exchange. When we look at the two-year waits, the number waiting over two years for a new outpatient appointment is down 59 per cent in Scotland. When we look at the 18-month wait that is reduced by 40.6 per cent, we look at those waiting over a year, 34 per cent of specialities had fewer than 10 patients waiting over 52 weeks. So there is progress, there is recovery and of course that recovery is going to take time. What of course helps with that recovery is making sure that we have NHS staffing at near record levels as they are under this Government, but also making sure that we don't lose a single day of the NHS to industrial action. I'm pleased that this Scottish Government has ensured that our NHS staff remain the best paid in the entire UK. Anna Sarwar, Presiding Officer, surely the First Minister knew about Covid when he made the promise last year that we are eradicated by this time this year? The First Minister may not like it, but this is happening on his watch. In the last financial year alone, there were 43,000 patients treated privately in Scotland, an 8 per cent increase in one year. 4,739 hip and knee replacements paid for privately. 7,805 cataract surgeries paid for privately. 1,980 colonoscopies, 2,055 endoscopies, 995 hernia repairs and, most shockingly of all, 1,745 rounds of chemotherapy paid for privately by patients. Healthcare free at the point of need, a founding principle of our NHS. How can the SNP have let it get so bad that patients are having to find the cash to pay for life-saving cancer treatment? Of course, we don't want people to have to fund care out of their own pockets. We know, of course, the impact that the pandemic has had on our health service and, indeed, on those waiting lists. I go back to the point, Anna Sarwar, that this is something that has been seen right across health systems, across the globe, let alone across the UK. When we look at those happening here in Scotland as well as across the UK, when I look at the figures from the private health information network over the first quarter of 2023, of course, it shows worrying trends in relation to those who are accessing private healthcare, but it does also show that Scotland has a lower rate of take-up of private healthcare compared to England and Wales, the rate of people who are self-funding for private inpatient daycare is 16 per cent higher in England and, in Wales, it is 51 per cent higher where the Labour Party is in charge. Of course, that will be cold comfort for people who have to dig deep into their own pockets to pay for healthcare, but what I would say is that it is not unique here to Scotland. What we will do is continue to see that recovery, continue to reduce waiting lists and what we will continue to do is make sure that we, our NHS staff, are the best paid in the UK so that they continue to provide the excellent service that they do to patients up and down Scotland. Anna Sarwar. The number of people going private each year is going up and those answers from the First Minister are going to mean no comfort to the 43,000 patients who are being forced to pay privately. One person I spoke to in just a few weeks ago was £15,000 for a hip replacement because he had to wait three years, completely and utterly unacceptable. In the middle of a cost-of-living crisis, when people are struggling to pay the bills, they shouldn't also have to worry about the cost of getting sick. On the SNP's watch, 43,000 Scots are being forced to find £15,000 for a hip or knee replacement, £3,000 for cataract surgery, £3,000 for a colonoscopy, £4,000 for a hernia repair and even cancer patients forced to find thousands of pounds for their chemotherapy. Patients in pain and heartbroken families trying to scrape together the cash, some even being forced to remortgage their homes to pay for the care that they should be getting on the NHS. First Minister, why are more and more Scots being forced to pay the price for SNP incompetence and failure? First Minister, what we're saying, of course, is the impacts of that global pandemic, which has impacted every single health service, including, of course, the health service here in Scotland. If Anna Sarwar wants evidence of the recovery, we are seeing more and more activity within the NHS. When I look at the activity in relation to inpatient daycare, activity for quarter to two was at its highest since the start of the pandemic. That's not the first, not the second, but actually the six quarterly increase in a row with 58,813 patients seen in Q2. We're seeing more and more patients in patients in this case and daycare patients being seen. I'm pleased that Anna Sarwar mentioned the cost of living crisis because we are taking action to tackle that cost of living crisis. Because of the action that we are taking, of course, it's estimated that 90,000 fewer children will be in absolute or relative poverty. In stark contrast to the position of Anna Sarwar of a summer of U-turns, where he has aligned himself with cruel Tory policy after cruel Tory policy, now we are unashamedly anti-poverty, unashamedly pro-growth. The only thing Anna Sarwar is unashamedly pro-starmer, Presiding Officer. Question number three, Ross Greer. Thank you. To ask the First Minister what action the Scottish Government is taking in response to the first UN global stock-take report on the Paris climate agreement. It's simply no denying that the conclusions of the global stock-take report are deeply concerning and that all countries, including Scotland, must seek to do more. The report found that nothing less than radical decarbonisation and systemic transformation will do. Scotland will, of course, play her part. Internationally, we will continue to urge greater ambition, greater action towards a just transition to net zero through our core chairmanship of the under two coalition. I will highlight this as a matter of urgency while at the New York climate week, and we look forward to COP 28 to make clear decisions on fossil fuels, renewable energy and climate finance must be progressed with urgency. At home, Scotland has made progress on our net zero journey, though we know that more has to be done. We will continue to work with communities, businesses and international partners towards a net zero future. As First Minister said, Scotland has a deserved international reputation as a leader on climate action, but our domestic record needs to match up to that reputation. With Scottish green policies like removing peak time rail fares from a few weeks from now, we are taking action to reduce emissions. The First Minister said that he is about to join other world leaders at climate week and then at COP. What new actions is he bringing to the table to demonstrate Scotland's commitment to tackle this global emergency? First Minister. Ross Greer is absolutely right. It is incumbent upon Governments right across the world to ensure that they are bringing urgency, pace and indeed initiative to tackling the climate. Emergency Scotland has already had a big impact on the global stage, particularly through our pioneering commitment to loss and damages, and I am determined to keep building on that. In terms of new initiatives, that is my programme for government committed to serious climate action. Cheaper public transport through free bus travel, trialling the scrapping of peak rail fares on ScotRail, as Ross Greer has mentioned, investing almost £5 billion in the net zero energy economy, a new climate change adaptation programme, a heating buildings bill and the natural environment bill. Much more on top of that that we will continue to build upon as we develop our new climate plan. That is why I am looking forward to hosting all party leaders so that we can get some sort of consensus, I hope, on the urgency that is required to tackle the climate emergency. Given that the Scottish Government has consistently failed to live up to our climate target, which is bad news for the 38 per cent of households living in fuel poverty whose energy bills have increased, what does he say to the millions of people across the world and here in Edinburgh who will be campaigning on the streets this weekend to demand a just transition, given that his Government has flip-flopped an oil and gas and failed to deliver retrofitting of people's homes who are living in fuel poverty? I say to Sarah Boyack that she would have more credibility on this issue if her party did not U-turn on its support on low emission zones or indeed it did not dump its £28 billion green prosperity fund. That is the very, very point here. Time and time again, the Scottish Government does not just talk the talk, we are prepared to walk the walk. We will bring forward serious action to tackle the climate emergency, but what would really help, Presiding Officer, is that those parties that demand we meet our targets do not then oppose every single measure that we bring to this chamber. So if Sarah Boyack is serious and the Labour Party is serious about tackling the climate emergency, join with us, support us, bring forward ideas that will help us collectively to tackle the climate emergency. Do not simply just dump the policies that you had and do not simply just oppose those policies that we bring forward and support us, because collective action is needed to tackle the climate emergency. Finlay Carson. Thank you, Presiding Officer. Could the First Minister update the chamber on when we should expect to see the published update to the climate change plan? First Minister. At the end of this year, Presiding Officer. Thank you, and we move to question 4, and I call Jackie Dunbar. Thank you, Presiding Officer. To ask the First Minister, in light of reports that no bids were received for offshore wind projects in the UK Government's latest contract for difference round, what action the Scottish Government has taken to encourage investment in offshore wind and hydrogen power in Scotland? First Minister. Jackie Dunbar absolutely exposes the UK Government's complete failure to capture one of the greatest opportunities for Scotland's net zero economy. Their inexplicable stance is both anti-climate, but it's also anti-growth. Industry has sent the UK Government a clear signal that the strike price must increase and we back those calls 100%. To ensure that we seize this opportunity, the Scottish Government supports the strategic investment model with public and private sector partners, specifically aimed at enabling a move from project-led to sector-level investment. Launching later this year will be the next tranche of hydrogen funding. The green hydrogen fund, this flagship fund, will focus on supporting renewable hydrogen production from Scotland's abundant renewable energy resources, seeking to give us first mover advantage. However, I hope that on the back of industry calls that the UK Government will listen to the industry and, of course, hopefully will have a successful future round. A new report from Aberdeens Robert Gordon University published just this week further reinforced that we cannot afford to get this wrong and we must see investment in renewable technologies gather pace. The SNP Scottish Government has shown its ambition by committing £500 million to a just transition fund, but this has to be matched by Westminster. Can the First Minister provide his response to this new report and any update on discussions with the UK Government to match the Scottish Government's just transition fund? I do welcome the report from Robert Gordon University and Jackie Dunbarh, who is absolutely right. Successive UK Governments have taken billions and billions and billions from our north-east and are not prepared to put a fraction back in. The very least the UK Government should do is match our £500 million just transition fund. What I would also say to the UK Government is that the latest round is an abject failure to capitalise on the huge and vast renewable potential that Scotland has. What I would say to the UK Government is that it is not just that it is harming our economy, but that it is seriously harming our planet. Over a summer where we have seen the disastrous visual impacts of the climate emergency, this is a time for climate leadership. What we are getting from the UK Government is, of course, complete political abdication. The fact that there are no bids received should be a wake-up call to this Government. Offshore wind should be part of an energy mix. Instead, it is putting all its eggs into one basket, a basket that is considerably more expensive than the Scottish Government has done to it. Will the First Minister look again at having a more balanced energy mix that includes nuclear energy? The member rightly looks embarrassed, as he should be. The Scottish Government is somehow responsible for the failure of the contract for different auction that the UK Government controls. It is seriously desperate from the member. If he does not want to listen to me, he should listen to the likes of Keith Anderson. I will quote directly, that this is a multi-billion-pound lost opportunity to deliver low-cost energy for consumers and a wake-up call for the UK Government. Let us listen to Dan McRail, the CEO of Renewable UK, who says that the results of the contract for difference auction round should set alarm bells ringing in the Government, as the UK's energy security and net zero goals can only be met if we have offshore wind as the backbone of our future energy system. I agree with Dan McRail. I agree with Keith Anderson. The Conservatives had any influence in that regard. They would make sure that the UK Government listens to the industry as opposed to those who back fossil fuels alone. Beatrice Wishart Offshore wind is a sustainable source of energy and a key part of Scottish Government plans to reach net zero targets, but the Scottish Government's first round of leasing sold our natural environment short with a cap price per square kilometre. Will the Scottish Government ensure that any future leasing grounds are not undersold in order to get the maximum amount possible to reinvest in public services? The first involved leasing round for offshore wind is, of course, development of Scottish waters. The first leasing round in Scotland in a decade will make a significant contribution in terms of our public finances, not only the £750 million in revenues to the public purse, but the billions of pounds that it will create in terms of opportunity to the supply chain. The cabinet secretary will be more than happy to speak to the member in relation to what more we can do to improve future leasing rounds. The point is that the Scottish Government believes in the vast potential of our renewables industry and our renewables potential in the face of a UK Government that has turned its back of Scotland's renewable potential. I hope that members, such as Beatrice Wishart and other members across political parties, will join with the SNP-led Scottish Government to put pressure on the UK Government to do more to unlock the huge potential that Scotland has for renewables. To ask the First Minister what the Scottish Government's responses to statistics reportedly showing that the number of complaints against Social Security Scotland has increased by 350 per cent in one year. Social Security Scotland has introduced 13 Scottish Government benefits, seven of which are available only in Scotland, thanks to a total investment of £12 billion, supporting people who need assistance. In 2022-23, it received almost 370,000 applications, hundreds of thousands of phone calls and webchats too. Social Security Scotland is a fundamental human right, which is why we have established a very radically different benefit system built on dignity, built on fairness, built on respect. Our system vigorously promotes benefit take-up to make sure that as many people as possible get what they are entitled to, while mitigating, of course, the austerity impact of UK Government policies within our fixed budgets. In 2022-23, there were a total of 1,790 complaints, so complaint numbers do remain low. In fact, in relation to total applications, it is less than a half per cent. I am sure that the First Minister must agree that the situation is not acceptable. MSPs from across Parliament will be aware of increasing costs and complaints experienced by our constituents. SNP ministers have said all is well, but we are seeing increased processing times, delayed delays in benefit roll-out and the possibility of the full devolution of all benefits not happening until 2026. Therefore, when does the First Minister expect all benefits to be devolved to Social Security Scotland? The more than happy for the cabinet secretary to write in detail to Miles Briggs, not just around the question that he asked specifically, but also on the significant improvements that we are making in relation to application processing times. Social Security Scotland is a success story. It is because of Social Security Scotland, because of the actions that we have taken in relation to Social Security and indeed progressive taxation that it is estimated that 90,000 children will be lifted out of absolute and relative poverty this year. It is the reasons why, of course, poverty levels in Scotland are still too high, but we have seen rises in poverty across the UK with the situation in Scotland has stabilised. We want to make sure that poverty reduces. The final thing that I would say to Miles Briggs is that we will furnish him with further detail. When it comes to the satisfaction levels of those who interact with Social Security Scotland, 81 per cent, over 8 out of 10 respondents, agreed that it was easy to contact Social Security Scotland and 94 per cent. Over 9 out of 10 respondents agreed that Social Security staff treated them with kindness. That is expected. There is a stark difference to the DWP. Can the First Minister confirm that user survey results demonstrate Scotland's Social Security system as adhering to the founding principles of dignity, fairness and respect so that people are benefiting from the powers over Social Security being in the hands of this Parliament? The First Minister is absolutely right that the survey shows just that. When I look at satisfaction levels, almost 9 out of 10, 89 per cent of respondents said that Social Security staff were able to help them. 89 per cent of respondents said that Social Security staff are knowledgeable about benefits. As I have already said, through the actions that this Government has taken through the hard work of the excellent team at Social Security Scotland, we have seen significant results when it comes to tackling poverty and child poverty in particular. That would not have been possible. It is not for the hard work of the excellent staff at Social Security Scotland and, of course, the ethos that we have had from the very beginning around dignity, fairness and respect. To ask the First Minister what the Scottish Government is doing to tackle repeat offending in the light of a recent report that found that people who experience frequent and serious repeat violence often do not bring that to the attention of the authorities. The Scottish Government commissioned the report that the member refers to precisely because we knew that victims of repeat violence do not report their experiences, and it is, of course, important for us to understand the reasons and rationale behind why and how we can help them to access that support. While there are fewer crimes and fewer victims than over a decade ago, the impact on victims, particularly of sexual and violent crimes, can be devastating. That is why we are investing in law enforcement, crime prevention and reducing offending and reoffending through a range of activity, including our violence prevention framework. We are also putting victims of crime at the heart of our justice system through our victims' witnesses and justice reform bill, and we have invested 93 million over the past five years to support victims. I welcome the report by the Scottish Government, which seeks to better understand repeat violent victimisation. It found that, of those men who experience and repeat visible violence in Scotland, they live in deprived areas, have experienced homelessness or are living in supported accommodation and are in recovery from addiction. Whereas intimate partner abuse was the highest among women aged 16 to 24, also living in deprived areas. The report talks about local communities led by mentors, peer mentors and those with lived experience, where they are uniquely positioned to support people experiencing repeat violence. It is, First Minister, a slightly different perspective on crime, so I ask the First Minister with that in mind what can be done to support the grassroots organisations that the research found is quite limited by insecure funding. I think that Pauline McNeill makes some very important points indeed. The point that she makes is exactly why we will continue to support programmes such as Medics Against Violence and, of course, the delivery of the excellent navigator programme. It is why we have invested and continue to invest in the violence reduction unit, which has been hailed as a great exemplar globally. In terms of supporting grassroots organisations, I cannot agree more with Pauline McNeill. I think that the investment in those third sector partners is absolutely crucial. That is why more than 20 organisations have received awards from our victim-centred approach funding, including Victim Support Scotland, who were awarded over £18 million to support families bereaved by crime service. We will continue to invest in those grassroots interventions. What I would say to Pauline McNeill is that Justice Secretary will be more than happy to continue the conversation, Pauline McNeill. We have received that report. It is really important that we act upon what that excellent report is telling us. We move to general and constituency supplementaries, and I call Clare Adamson. Thank you very much, Presiding Officer. Does the First Minister share my dismay that Motherwell Concert Hall and Theatre was closed with immediate effect until further notice due to the discovery of rack concrete in the roof of the building? Of course, safety is absolutely paramount. Motherwell Theatre and Concert Hall is a beloved venue both by professional and amateur performers, not least the cast of the acclaimed Christmas Pantel. Can I ask the First Minister if the Scottish Government will engage with the local authority on the challenges facing the future of this important local cultural venue? Yes, we will continue to engage with local authorities, as we have done throughout the course of the last not just few weeks, but few months. I am more than happy for the cabinet secretary to speak directly to Clare Adamson. Clare Adamson is right. Those are difficult issues. It is really important that local authorities and others continue to align themselves with the Institute of Structural Engineers guidance. That is what partners are seeking to do. Of course, that empowers them to take the appropriate mitigations. That can include, at times, unfortunately having to close down buildings where they feel that that is necessary, but I am more than happy to ensure that the cabinet secretary writes to Clare Adamson. I am sure that I should be happy to engage with Clare Adamson, too, on this very important issue. Minister, childcare formed the cornerstone of your programme for government. It is also central for child development and allowing parents and carers to get back to work. Yesterday, out of the blue, curry primary school families lost their before-and-after school clubs. Parents have contacted me desperate for help, with one unsure how they can balance childcare and their job as a teacher in Dalkeith. First Minister, will you meet me to discuss solutions to this very pressing local issue? I am more than happy for Sue Weber to furnish the Government with the full details. She is given a synopsis of the situation, and the cabinet secretary will be more than happy to meet Sue Weber. Sue Weber is right. Child care was a cornerstone, not just of my programme for government, but it has been a cornerstone of this Government since 2007. We now have a very generous childcare offer, one that I am keen to build upon. That is why the staff who do a phenomenal job in providing childcare will ensure that the funds are available to pay them £12 an hour in the PVI sector. However, if Sue Weber can furnish the Government with full details, I will ensure that the cabinet secretary engages with her on this issue. This week, disabled people in Glasgow have written to all MSPs in the region, including the First Minister, to say that, and I quote, "...despite our large numbers, disabled people have never been a priority for the Scottish Government. The Glasgow Centre for Population Health has described the situation as a human catastrophe and the CEO of Glasgow Disability Alliance has said that the result is that disabled people are dying of poverty, nobody knows and nobody cares." Can I ask the First Minister, does he know that it is this bad for disabled people in the city that he represents, and does he care enough to commit to taking action to address it, including to end non-residential care charges? I do understand, of course, because I met with the Glasgow Disability Alliance just a matter of weeks ago. I heard from the members of the board of the Glasgow Disability Alliance and Tresor Burke, who very powerfully, over the years, has articulated the concerns for disabled people living in Scotland. That is why my programme for government, of course, committed to reopen the independent living fund, which was an ask from disabled people's organisations. Of course, that is why I will continue to work with them to see what more we can do, including, of course, the ending of non-residential care charges, which we are still committed to do by the end of this parliamentary term. However, I would say to Pam Duncan Glancy that that will come at a significant cost, so it is really important that, when Labour does engage with us, it does not turn their back on progressive taxation, as Anna Sauer seems to have done, and to turn their backs on any ability for us to raise any revenue, because all of these initiatives, well-intentioned, can make a difference, will cost money. It is so important that we engage in a credible discussion on how we raise that revenue. The Linda Norgrove Foundation in my constituency has been desperately trying to get 20 female medical students out of Taliban-ruled Afghanistan and into medical schools in Scotland. However, they are being stymied by UK visa issues. I know that the foundation was extremely appreciative of the cabinet secretary's help when they met with her last month. Is there anything further that the Scottish Government can do to help these inspirational women? I hope that all of us as a Parliament can unite to back this particular cause. I want to pay tribute to the excellent work that Linda Norgrove Foundation does. It is incumbent on all of us to put pressure on the UK Government on the Home Office to help, given that we all know how difficult the plight is for women in Afghanistan. Jenny Goldruth, I know that the cabinet secretary has engaged with the foundation. We have also understood that he has engaged and written to the UK Government and the Home Office on this particular matter. I will see if there is anything more that we can possibly do on this issue. However, I would hope that we can collectively join as a Parliament to say to the UK Government that the response, the foot dragging on this particular issue, I am afraid, is simply not good enough and is letting down women and girls in Afghanistan. Graham Simpson Thank you. The cost of building the Glen Sannocks and Glen Rosa is an ongoing scandal. We don't know when they'll be finished, we don't know what the final bill will be, and we don't know what's happened to all the money. The Auditor General says he needs extra powers in order to find out what's happened to £128 million of money that was spent by FML. Will the First Minister grant those powers? The First Minister There has been a number of inquiries to which, of course, the Scottish Government and those at Ferguson's have co-operated fully. Hundreds of documents have been published and put into the public domain around what is happening at Ferguson's. I will not hesitate and apologise to our island communities for the fact that they are still waiting for the completion of both 801 and 802. What I won't do, Presiding Officer, is apologise for the fact that the Scottish Government did step in and made sure that we secured and saved hundreds of jobs in Inverclyde. Faisal Choudhury Thank you, Presiding Officer. Police Scotland recently released a report highlighting that this year's funding allocation represents a real redaction. As a result, the number of police officers has been reduced to 16,600 in Scotland. At the same time, the number of calls being made to Police Scotland are reportedly only increasing. Will the First Minister advise what action that the Scottish Government is taking to ensure that Police Scotland is suitably funded to deal with the demand? It is the member that we are funding Police Scotland suitably because, of course, they have received an increase of 6.3 per cent, an £80 million increase to their budget. Not only have they had an increase to their budget, of course, recorded crime is at one of its lowest levels in almost 50 years. When we look at police officer pay, they are, of course, the best-paid police officers in the UK. We are making sure that Police Scotland is appropriately funded. If Faisal Choudhury thinks that there should be an increase to their budget, what he cannot do, what his party cannot do, is turn their back on progressive taxation or, indeed, any way in which we can raise revenue to ensure that our public services are appropriately funded. Kenneth Gibson Thank you, Presiding Officer. First Minister, the Aquafill factory in Kilburney, which carries out interlacing and twisting for the carpet yarn industry, has been unexpectedly earmarked for closure as early as next month, with a loss of up to 40 jobs. This will be a devastating blow to the workers and wider community. Aquafill blamed, and I quote, Covid and Brexit for the challenging conditions in the UK and a doubling of energy costs. I'll meet with the company on-site tomorrow. What steps can the Scottish Government take to assist the company, the workforce and lessen the impact of any proposed redundancies? Neil Gray Will Neil Gray be happy to meet the member around Aquafilm? We'll see what can be done where necessary to hopefully prevent job losses or where that unfortunately becomes a material reality where we can help those into alternative employment. It's interesting that Kenny Gibson says that there are three factors that have contributed to Covid, Brexit and, indeed, high energy costs. Of course, Covid is affecting business right across the globe, but the other two factors are very unique to the UK Government. They are very unique to the UK and self-inflicted damage that Brexit has done. The utter mismanagement of the UK public finances is why, of course, we are seeing high energy and high inflation costs, which are damaging businesses right across Scotland. Liam Kerr Thank you, Presiding Officer. During FMQs on 22 June, I raised a point of order that the First Minister had misled the chamber. On 29 August, the First Minister wrote telling me what he had intended to say, but what he has stated to me in writing is not true. A freedom of information request reveals that weeks of civil service time and effort was spent trying to engineer a face-saving response rather than the First Minister simply admitting that he had misled us all. Does the First Minister share people's concerns that the Scottish civil service appears to be increasingly politicised, which could destroy public trust? What will he do to prevent such partisan protection for ministers who intentionally mislead in the future? First Minister It is incredible, Presiding Officer, that I have a member who is telling me what I was actually meaning to say and meaning to think. It is incredible that we have Liam Kerr talking about disrespecting Parliament. I am meant to take a lecture from the Conservatives about disrespecting Parliament. Sorry, let's treat one another with courtesy and respect. Let's hear the First Minister. I don't want to get to them about courtesy and respect, Presiding Officer, because I am meant to take a lecture about disrespecting Parliament from the Conservatives who illegally shut down the Parliament. I am meant to take lectures about truth and honesty from the party that gave us Boris Johnson, Presiding Officer. The very difference between myself and Liam Kerr, between the SNP and the Conservatives, is that the SNP wants to empower our just transition, wants to empower our offshore renewable sector. The Tories, I am afraid, just want to hold us back, Presiding Officer. This morning, new data shows the vast majority of Scots view immigration as a positive benefit that enriches Scotland culturally and economically. Can the First Minister update the chamber on his interactions with the UK Government on labour shortages in key sectors as a result of the UK's archaic immigration policies? Articulates very well the positive benefits of migration. Virtually every single study under the sun shows that migrants who have come to the UK have contributed more than they have taken it. I have to say that the absolute hostile environment that has been created by successive UK Governments is woeful. The UK has become a pathetically insular country, not only is it morally wrong, but it undoubtedly cuts our nose despite our face in terms of the economic potential that migrants could bring to Scotland and to the rest of the UK. I urge political parties to frankly have lost hope with the UK Government, the Conservative UK Government when it comes to migration. I would plead with Anna Sauer to use any influence he has when it comes to Keir Starmer and the UK Labour Party who seem to be engaged in a race to the bottom when it comes to the issue of migration. That is not going to help the UK, certainly it is not going to help Scotland at all. Murdo Fraser Thank you. It has been a persistent issue over many years with the short running of trains, serving the Fife to Edinburgh service well used by my constituents. I raised this previously with ScotRail and with the previous transport minister. I was assured that this would be resolved, but it continues. Today, for example, the 0811 service from Inver Keithing, a very busy service, had just two carriages, leaving passengers stranded on the platform having to wait 20 minutes for the next service, meaning that they would be late arriving at their place of work. It is very welcome that we are seeing a reduction in peak-time fares, but that is no use to people if they cannot get on the train in the first place, so when is this finally going to be brought to an end? We have invested significantly in the rolling stock in Scotland in the time that we have been in office. I know that the previous transport minister held a meeting with Fife MSPs in relation to the issues that Murdo Fraser mentioned earlier in the year in spring time, but I am more than happy for the cabinet secretary to write to Murdo Fraser on where appropriate meet with Fife MSPs so that we can ensure that the people of Fife get the service that they absolutely need and require. That concludes First Minister's questions. The next item of business is a member's business debate in the name of Julian Mackay. There will be a short suspension to allow those leaving the chamber and public gallery to do so.


Hi, we are studying numerical methods for computing eigenvalues and eigenvectors of a given matrix. In this we have introduced power method in the last class and also we learnt convergence theorem for power method. In this class we will study a small variant of power method called inverse power method. Let us first recall a result from linear algebra which states that if lambda is an eigenvalue of an invertible matrix A, then 1 by lambda is an eigenvalue of A inverse. This is not very difficult for us to see you just take lambda as the eigenvalue of a matrix A, then you can write A into V is equal to lambda V, where V is an eigenvector corresponding to the eigenvalue lambda. Since A is invertible, you can write V is equal to lambda times A inverse into V. That implies A inverse into V is equal to 1 by lambda times V. Here you can note that there is no change in the eigenvector, only the eigenvalue becomes the reciprocal of the eigenvalue of the matrix A. So, that is very important for us to note. Using this theorem, we can in fact compute the smallest eigenvalue of a matrix A. When I say smallest eigenvalue, it should be smallest in the absolute value. Say for instance suppose we have a matrix A whose eigenvalues are rearranged such that modulus of lambda 1 is greater than or equal to modulus of lambda 2 and so on up to lambda n where mod lambda n is strictly less than mod lambda n minus 1. Now you can see that if A has such a eigenvalues, then lambda n which is the smallest eigenvalue is unique. That is the only eigenvalue which is smallest in this sense. Now let us see how to compute this eigenvalue lambda n using power method for that what we will do is instead of applying power method to A which will capture the dominant eigenvalue. Now we will apply the power method to A inverse. Why we do that? Well we have seen from the above theorem that if lambda n is the smallest eigenvalue of A, then 1 by lambda n will be the dominant eigenvalue of A inverse. That makes us to apply power method on A inverse in order to compute 1 by lambda n and therefore, we can obtain this smallest eigenvalue of A. That is the idea. Remember once we compute 1 by lambda n, the power method applied on A inverse will also capture an eigenvector corresponding to 1 by lambda n. That is the same as the eigenvector corresponding to lambda n also. So, in that way we can obtain the smallest eigenvalue of A and a corresponding eigenvector of the eigenvalue lambda n. So, that is the interesting part of power method and this way of computing the smallest eigenvalue of a matrix is what we call as the inverse power method. Now let us give the iterative procedure for inverse power method. You can see that there is nothing much difference in inverse power method when compared to power method. The only difference is instead of applying the power method to the matrix A, now we will be applying it to A inverse. That is for every iteration we will compute y k plus 1 as A inverse x k. Remember in power method we will define y k plus 1 is equal to A into x k right, but here we will apply A inverse to get y k plus 1. Well this is not something which is straight forward for us because we are given the matrix A right. Now in order to compute y k plus 1 we need to invert the matrix A which may be computationally costly. So, what we can do is you can write this equation as A into y k plus 1 equal to x k. Here you can observe that once you choose x naught you will plug in x naught on the right hand side which is a known vector and you will then solve this linear system to obtain the unknown y k plus 1. That is a major difference in inverse power method when compared to the power method. This complicated step is not there in power method because y k plus 1 is just defined as A into x k. x k is known to us and A is also given to us therefore, y k plus 1 can be explicitly computed from there whereas, in inverse power method y k plus 1 has to be computed by solving this linear system. That is we have to solve this linear system y 1. Once you have this then you will have x 1 and that has to be then taken as the right hand side to get y 2 and so on right. So, if you see here the right hand side vectors will only change whereas, the coefficient matrix A will remain unchanged. Therefore, what you can do is you go for a LU factorization because once you do the LU factorization A is equal to L into U then the system A y is equal to x can be solved by first doing a forward substitution L z is equal to x and that gives you z then we can plug in that into U y is equal to z. So, you bring that to the right hand side and that gives us the required vector y. We have learnt it in one of our previous classes where we have discussed LU factorization you just recall that. So, the advantage of going for LU factorization is that once you make the factorization L into U for the matrix A then at every iteration you just have to plug in the right hand side vector and do one forward substitution and one backward substitution you will get your y that will be computationally relatively cheaper. So, once you get this y k plus 1 rest of the algorithm reminds the same as we did in the power method. What you have to do you take this vector y and find the maximum norm of that and in whichever index the maximum norm is achieved you take that value of the y coordinate and define that as mu k plus 1. For instance if y k plus 1 is equal to say minus 2, 3 and minus 5 then your capital J which is nothing, but the minimum of all the j's at which the maximum norm is achieved. So, here it is achieved only at the third coordinate therefore, mu k plus 1 will be y 3 k plus 1 in this particular example it is minus 5. So, this is just an example. So, in that way only you have to choose mu k plus 1 once you have mu k plus 1 well you can find the x k plus 1 very easily by taking y k plus 1 and divided by mu k plus 1 which will be a unit vector. So, we have seen all this when we were discussing power method and this is the iterative procedure for the inverse power method. The only difference and the complication is in computing y k plus 1 otherwise all these steps reminds the same as in the power method. Well the next question is if these sequences that is sequence mu k and sequence x k if both these sequences converge then where do they converge? That is the question. In a sense we have already answered this question, but we will make it more precise. Of course, we have to keep in mind that the initial guess x naught should be chosen in such a way that C 1 is not equal to 0. What is C 1? C 1 is nothing but the coefficient in the representation of x naught in terms of the eigenvectors C 1 v 1 plus dash dash dash up to C n v n in that the coefficient C 1 should be non-zero that is for the theoretical reason, but assume that everything went well with our choice of x naught. Then our question is where these two sequences will converge? Well the sequence mu k will converge to the dominant eigenvalue of A inverse which is nothing but the reciprocal of the smallest eigenvalue of A. In our rearranged form it is lambda n therefore, mu k will converge to 1 by lambda n. Then where does the sequence x k converge? x k will converge to an eigenvector of lambda n. Remember it will converge to an eigenvector of 1 by lambda n which is of course, the same as the eigenvector of lambda n also. Let us see an example. Let us take the matrix A given like this. I am just giving you only 4 digits after the decimal place. Well as I told you we prefer to first decompose the matrix A into L into U. I have just taken the do little factorization. You can just observe it here all the diagonal elements are 1. So, I have just taken the do little factorization for A. So, once you have L and U then A into y equal to x can be done with a forward substitution and one backward substitution and at each iteration you can get the vector y. Once you get the vector y then you will look for that coordinate of y at which the maximum norm is achieved. If it is achieved at multiple coordinates then you will take the one which is with the minimum coordinate. So, in that sense the maximum norm is achieved at the first coordinate of y 1 and therefore, mu 1 is given by minus 2.14. Remember it should be minus naught plus and once you get that then simply take x 1 as y 1 divided by mu 1. So, that is your x 1 and that is given by this and once you have x 1 again you plug in x 1 into L into U into y 1 is equal to x 1 right. A forward substitution and the backward substitution will give you y 2 sorry this is y 2 right. So, you will get y 2 from y 2 again you will find the coordinate at which the maximum norm is achieved that is mu 2 and then you divide mu 2 y 2 by mu 2 you will get x 2. Now, you repeat this process I am just going head with the iteration you can observe that mu is converging slowly to the number 1.63. Remember if mu is converging to 1.63 it means what it is actually converging to 1 by lambda 3 right. What is lambda 3 well we have not given what are all the Eigen values of A therefore, we do not know that is the smallest Eigen value of A right. When I say smallest it is the smallest value of all the Eigen values in the absolute sense right. Therefore, this is for us given as 1.63 when you take the reciprocal of that that gives you 0.61 and what is the corresponding Eigen vector well for Eigen vector you do not need to do anything just comes as it is the Eigen vector is taken as minus 0.81 and 0.03. So, this is the smallest Eigen value of the matrix A and a corresponding Eigen vector of this Eigen value is given like this and this is how the inverse power method will go on well for this we have chosen the initial guess as x naught is equal to 1 1 1 I am sorry I forgot to mention this. The next question is ok we learnt how to capture the dominant Eigen value of a matrix and the smallest Eigen value of a matrix. Now is there a way to find some of the other Eigen values of A also that is the question well this can be done by shifting the matrix A appropriately and that results into shifted inverse power method. Let us try to understand what is shifted inverse power method assume that we are interested in capturing an Eigen value lambda j and one of its corresponding Eigen vector say v j. Now what you do is you choose a nu some real number nu such that mod lambda j minus nu is strictly less than mod lambda i minus nu where i ranges from 1 to n with i naught equal to j. It means you try to find a nu such that when you take this quantity that will be the smallest value of all such quantity when applied to all other Eigen values of the matrix A well practically this is not possible to get because simply we do not know what is lambda j. However in certain situations we may use Grashgorian disc to get an estimate of nu. For instance suppose your matrix A is such that the Grashgorian disc or something like this disc 1, disc 2 and so on, disc 3 and then disc 4 like that, disc 5 like that and then in between you have a small disc which is disjoint from here and you want to find this Eigen value let us call this as our x lambda j. In such cases you know how to choose your nu say you take this interval and suppose this interval is very small then you have an idea of how to choose this nu. So, like this in certain restricted situations you may be able to choose such nu using Grashgorian theorem if such a favorable situation is not happening with the matrix A then you can also try with A transpose sometimes that may give you a better information about how the Eigen values are distributed. So, if such a favorable situation happens then one may go to use shifted inverse power method otherwise it is not that easy for us to use this method in any practical situation ok. Assume that we got this nu somehow then how to compute lambda j is our question now. Once you have such a nu then probably by looking at this you can get an idea of how we are going to capture lambda well what you can do is you apply the inverse power method to A minus nu i. So, this is called the shifted matrix and you apply inverse power method to this matrix inverse power method is applied to A minus nu i that is equivalent to saying that apply power method to A minus nu i inverse right. That is the idea we do not need to give the procedure for this because it is just the same as what we have given in the inverse power method only thing is instead of A inverse now we have to put A minus nu i inverse that is the only difference. Well let us consider this example now once you have this you will have to use a appropriate nu as I told you this is generally not possible perhaps if you are very lucky you may get an idea of how to choose this nu from Grashgorand deals of either A or A transpose. Well I will leave it to you to think how to do that for some appropriate matrices we will give you some such matrices in exercise problems, but here to keep our discussion very simple let us know what are all the Eigen values and the corresponding Eigen vectors of this matrix A these are given by lambda 1 lambda 2 and lambda 3 and the corresponding Eigen vectors are given here. Now let us try to capture lambda 2 and Eigen vectors of lambda 2 let us see how this is going to happen for that we have to choose a nu right let me choose nu as minus 2.4 you can choose any number in such a way that lambda 2 minus nu is strictly less than lambda 1 minus nu and this is strictly less than lambda 3 minus nu also. So, this choice of nu is going to work therefore, I chose like this you can also choose any value and now we have to apply the inverse power method for A minus nu i that is A plus 2.4 i. You can clearly see that the Eigen values of A minus nu i are given like this. Now we have to find the LU factorization of A minus nu i I have gone for the du-little factorization you can go for either Gaussian elimination or Crude factorization or even if the matrix is symmetric and positive definite you can go for Cholsky's factorization also right, but I have gone here for du-little factorization as per this the lower triangular and the upper triangular matrices are given like this and the iterations are given like this. I have stopped this iteration procedure at the fifth iteration because you can see that from the fourth iteration to fifth iteration there was no much improvement at least to the number of digits that we have shown here that is why I have stopped up to here. If you want more accurate then you have to go for more iterations that is the only need here, but I am happy with this because it is just an illustration. So, I stop here therefore, your iteration mu k sequence is converging to remember it is going to converge to 1 by lambda 2 minus nu right. Remember we are now working with shifted matrix therefore, your mu k is going to converge to 1 by lambda 2 nu and since mu phi is given by minus 7.61 you can see that 1 by mu phi is something like of course, 1 minus 1 by 7.61 that is in my calculation approximately equal to minus 0.1314 right. So, that is going to be approximately equal to lambda 2 minus nu right because mu is going to converge to lambda 2 minus nu therefore, 1 by mu k will converge to lambda 2 minus nu I have just stopped at the fifth iteration therefore, I am going to consider this value as a approximate value to lambda 2 minus nu, but remember nu is taken as minus 2.4 right therefore, lambda 2 is equal to minus 0.1314 right plus nu that is nothing but minus 2.4 and that is equal to minus 2.5314. So, that is the lambda that we are trying to obtain from the shifted inverse power method. Let us see what is the value of lambda that we have taken previously that is also approximately the same it is minus 2.5313 what we got is minus 2.5314 right. So, we are pretty good in our approximation and the corresponding Eigen vector captured by our shifted inverse power method is 1 comma minus 0.06 comma 0.04 you can also see that it will be a scalar multiple of the corresponding Eigen vector that we have chosen in our first slide right. So, this gives us an idea of how we may capture other Eigen values of a given matrix, but there is a serious disadvantage in this approach that we do not know how to choose this scaling parameter nu. Apart from that the method is very interesting with this we will finish our discussion or power methods. I thank you for your attention.